- ELL Support Solving Sstems Using Tables and Graphs Use the chart below to review vocabular. These vocabular words will help ou complete this page. Word Classif (verb) KLAS uh fah Classified (verb) KLAS uh fahd Solution (noun) suh Loo shun Solve (verb) sahlv Eplanations To arrange into groups that have similar characteristics Arranged into groups The ordered pair that makes both equations true To work out a correct answer to the problem Eamples We classif linear sstems based upon how man solutions the sstem has. Linear sstems are classified as consistent or inconsistent. (, ) is the solution to the sstem of equations and. Solve the sstem of equations using a table. Use the vocabular above to fill in the blanks.. On the test tomorrow, ou will classif linear sstems according to how man solutions each sstem has.. Linear sstems are classified in order to give us information about them.. The solution to a sstem of equations is a point of intersection on their graphs.. You can solve a sstem of equations using a table or a graph. Eample Linear sstems are classified as consistent or inconsistent. solution consistent no solution inconsistent Solve each sstem of equations and classif the sstem as consistent or inconsistent.. e 9 no solution; inconsistent. e (, ); consistent Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems Using Tables and Graphs Sports You can choose between two tennis courts at two universit campuses to learn how to pla tennis. One campus charges $ per hour. The other campus charges $ per hour plus a one-time registration fee of $. a. Write a sstem of equations to represent the cost c for h hours of court use at each campus. b. Graphing Calculator Find the number of hours for which the costs are the same. c. Reasoning If ou want to practice for a total of hours, which universit campus should ou choose? Eplain.. What is an equation that represents the cost c for h hours of court use for the first campus? c h. What is an equation that represents the cost c for h hours of court use for the second campus? c h. What is one method ou can use to find the number of hours for which the costs are the same? Answers ma var. Sample: Graph both equations and find the point of intersection.. What is another method ou can use to find the number of hours for which the costs are the same? Answers ma var. Sample: Use a table to list -values until the corresponding -values match each other.. Use one of our methods to find the number of hours for which the costs are the same. h. What happens to the cost at the two campuses after ou have practiced for the number of hours ou found in Eercise? The cost at the first campus increases b $ per hour, while the cost at the second campus increases b $ per hour.. If ou want to practice for a total of hours, which universit campus should ou choose? Eplain. The second campus; after hours, the cost of the first campus will alwas be greater than the cost of the second campus, because the slope of the equation for the first campus is greater than the slope of the equation for the second campus. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems Using Tables and Graphs Solve each sstem b graphing or using a table. Check our answers.. e (, ). e (, ). e (, ). e (, ). e (, ). e (, ). e (, ). e (, ) 9. e Q, R. e (, ). e (, ). e (, ) Write and solve a sstem of equations for each situation. Check our answers.. Your school sells tickets for its winter concert. Student tickets are $ and adult tickets are $. If our school sells tickets and makes $, how man of each ticket did the sell?, ; student tickets, adult tickets. A grocer store has small bags of apples for $ and large bags of apples for $. If ou bu bags and spend $, how man of each size bag did ou bu?, ; small bag, large bags. The spreadsheet below shows the monthl income and epenses for a new business. a. Use our graphing calculator to find linear models for income and epenses as functions of the number of the month. b. In what month will income equal epenses? income: ; epenses:, A B C September Month Income Epenses Ma $ $, June $ $, Jul $ $, August $ $, Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems Using Tables and Graphs Without graphing, classif each sstem as independent, dependent, or inconsistent.. e independent. e inconsistent. e 9 dependent 9. e independent. e 9 dependent 9. e 9 independent. e independent. e inconsistent. e independent. e inconsistent. e. You and our business partner are mailing advertising flers to our customers. You address flers each minute and have alread done. Your partner addresses flers each minute and has alread done. Graph and solve a sstem of equations to find when the two of ou will have addressed equal numbers of flers., ; minutes independent. e dependent You 9. You are going on vacation and leaving our dog in a kennel. Kennel A charges $ per da which includes a one-time grooming treatment. Kennel B charges $ per da and a one-time fee of $ for grooming. a. Write a sstem of equations to represent the cost c for d das that our dog will sta at the kennel. c d, c d b. If our vacation is das long, which kennel should ou choose? Eplain. Kennel B; it costs $, while Kennel A costs $. Open-Ended Write a second equation for each sstem so that the sstem will have the indicated number of solutions.. one an equation. none an equation. an infinite number e with a e with the same e an equivalent? different? slope but? equation slope a different -intercept. Multiple Choice Which ordered pair of numbers is the solution of the sstem? e B (, ) (, ) (, ) (, ) O Your Partner Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems Using Tables and Graphs Solve each sstem b graphing or using a table. Check our answers.. e. e. e (, ) (, ) (, ) Write and solve a sstem of equations for each situation. Check our answers.. Each morning ou do a combination of aerobics, which burns about calories per minute, and stretching, which burns about calories per minute. Your goal is to burn calories during a -minute workout. How long should ou spend on each tpe of eercise to burn the calories? ; ; minutes aerobics; minutes stretching. Suppose members of our class went on a rafting trip. Class members could either rent canoes for $ each or rent kaaks for $9 each. The class spent a total $9. How man people rented canoes and how man people rented kaaks?, 9 9; rented canoes, rented kaaks For Eercise, use our graphing calculator to find a linear model for the set of data. In what ear will the two quantities be equal?. Winning Times for the Olmpic -Meter Butterfl Year 9 9 9 9 9 99 99 Male Times (s)...9....... Female Times (s)... 9. 9.. 9.... SOURCE: www.infoplease.com Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Solving Sstems Using Tables and Graphs Without graphing, does each sstem have zero, one, or infinitel man solutions? To start, rewrite each equation in slope-intercept form.. e e. e 9. e zero solutions infinitel man solutions one solution Graph and solve each sstem. 9. e (, ). e (, ). e no solution O O O 9. Your business needs to ship a package to another store. Compan A charges $. per pound plus a $ service charge. Compan B charges $. per pound without an service charge. a. At what weight does it cost the same for both companies? pounds b. If our package weighs pounds, which shipping compan should our business use? Compan A. Reasoning Is it possible for a linear sstem with infinitel man solutions to contain two lines with different -intercepts? No; a sstem with infinitel man solutions contains the same line. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Solving Sstems Using Tables and Graphs Multiple Choice For Eercises, choose the correct letter.. Which sstem of equations is inconsistent? e e C e e. Which ordered pair of numbers is the solution of the sstem? e G (, ) (, ) (, ) (, ). Which of the following graphs shows the solution of the sstem? e D O O O O. You and our friend are both knitting scarves for charit. You knit rows each minute and alread have knitted rows. Your friend knits rows each minute and has alread knitted 9 rows. When will ou both have knitted the same number of rows? G. minutes minutes 9. minutes minutes Short Response. The sides of an angle are two lines whose equations are and. An angle has its verte at the point where the lines meet. Use a graph to determine the coordinates of the verte. What are the coordinates of the verte? [] graph at right; (, ) [] correct coordinates, without graph [] no answer given O Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Enrichment Solving Sstems Using Tables and Graphs When navigating their crafts, ship captains might be seen drawing lines on a large map using a straightedge. This procedure is used to plot intercept courses, which help to predict where the craft might meet an oncoming plane or ship. Air traffic controllers must also plot these intercept courses so that an oncoming plane does not meet another plane. To determine whether two ships or planes are on a collision course, it is important to know along which lines the are moving, where the are at a given moment, and how fast the are traveling. Solve. You might wish to use a drawing to help ou solve the problems.. Plane A is mi south and mi east of Plane B. Plane A is fling mi west for ever mile it flies north, while Plane B is fling mi east for ever mile it flies south. a. Where do their paths cross? mi south and mi east of Plane B s current location b. Which plane must fl farther? Plane B c. What ratio of the speed of Plane B to the speed of Plane A would produce a midair collision? " : or ". A commercial freight carrier is fling at a constant speed of mi/h and is traveling mi east for ever mi north. A private plane is observed to be mi due east of the commercial carrier, traveling mi north for ever mi west. a. If it is known that the two planes are on a collision course, how fast is the private plane fling? mi/h b. When will the collision take place if it is not averted? in hour. A cruise ship is traveling in the Atlantic Ocean at a constant rate of mi/h cruise ship: and is traveling mi east for ever mi north. An oil tanker is mi due about 9. north of the cruise ship and is traveling mi east for ever mi south. mi; oil tanker: a. How far is each ship from the point at which their paths cross? about. mi b. What rate of speed for the oil tanker would put it on a collision course with the cruise ship? about mi/h Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems Using Tables and Graphs As ou solve a sstem of equations, remember the following ideas. Lines that have the same slopes but different -intercepts are parallel and will never intersect. These sstems are inconsistent. Lines that have both the same slope and the same -intercept are the same line and will intersect at ever point. These sstems are dependent. Lines that have different slopes will intersect, and the sstem will have one solution. These sstems are independent. Problem Using a graph or a table, what is the solution of the sstem of equations? e Write both equations in m b form. O Graph the line. Graph the line. Circle the point of intersection., Determine the - and -coordinates of the point of intersection. The solution is the ordered pair (, ). Check () Check b substituting the solution into both equations. Eercises Solve each sstem b graphing or using a table. Check our answers.. e. e (, ) (, ). e. e. Which point lies on both Line and Line? (, ) (.,.) (.,.) (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 C (, ) (, ). e. e Line O Line (, ) (, )
- Reteaching (continued) Solving Sstems Using Tables and Graphs Problem The table shows the winning times for the Olmpic -M dash. Use our graphing calculator to find linear models for women s and men s winning times. Assuming the trends in the table continue, when will the women s winning time and the men s winning time be equal? What will that winning time be? Winning Times for the Olmpic -M Dash (seconds) Year 9 9 9 9 9 9 99 99 Men s Times Women s Times..... 9.9.........9.. 9. Step Enter the data into lists on our calculator. L: number of ears since 9 (value for ) L: men s winning times in seconds (value for ) L: women s winning times in seconds (value for ) Step Use LinReg(a b) to find linear models. This determines the equation of the lines of best fit for the selected data. Use L and L for the men s winning times. Use L and L for the women s winning times. Step Graph each model. Use the Intersect feature on the graphing calculator to find the solution of the sstem. The solution is 99.9 and.. The linear model shows that if the table s trends continue, the times for men and women will be equal about ears after 9, in. The winning time will be about seconds. Eercise SOURCE: International Olmpic Committee Intersection 99.9.. The table shows the winning times for Olmpic -M speed skating. Assuming these trends continue, when will the women s winning time equal the men s winning time? What will that winning time be? ;. seconds Winning Times for the Olmpic -M Speed Skating (seconds) Year 9 9 9 9 9 9 99 99 Men s Times Women s Times.. 9.. 9.....9.. 9.... 9. 99.9. SOURCE: International Olmpic Committee Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- ELL Support Solving Sstems Algebraicall The column on the left shows the steps used to solve a problem using a sstem of equations. Use the column on the left to answer each question in the column on the right. Problem Solve b Setting up and Solving a Sstem of Equations. A girl received $ for her birthda. At the store she can bu video game and DVDs for $. She could also bu video games and DVD for $. Write and solve a sstem of equations to determine the cost of a video game and the cost of a DVD.. Read the title of the Problem. What process are ou going use to solve the problem? Relate. Wh is the cost of one DVD cost of one a cost of one b multiplied b? video game DVD Answers ma var. Sample: avideo cost of game one b a cost DVD of one b Answers ma var. Sample: Write and solve a sstem of equations. There are three DVDs. Define Let v cost of one video game and d cost of one DVD.. Eplain wh the cost of a video game and the cost of a DVD are represented b variables. Answers ma var. Sample: These are unknowns. Write v d Choose one equation and. Wh was the equation solved for v rather than d? v d solve for v. Answers ma var. Sample: v d Write the other equation. Solving for d would involve (d) d Substitute the epression for v into the other fractions. d equation. Solve. v () Substitute the value for d into one of the equations. v Solve for v. Check Substitute d and v into the original equations. () (). Eplain wh the solution d and v should be checked in both equations. Answers ma var. Sample: It must be a solution to both equations. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems Algebraicall Chemistr A scientist wants to make milliliters of a % sulfuric acid solution. The solution is to be made from a combination of a % sulfuric acid solution and a % sulfuric acid solution. How man milliliters of each solution must be combined to make the % solution? Know. The scientist will begin with z z% and z z% solutions.. The scientist wants to make z zml of % solution. Need. To solve the problem ou need to define: ml of % solution ml of % solution Plan. What are two equations ou can write to model the situation?... and. Which method should ou use to solve the sstem of equations? Eplain. Answers ma var. Sample: Substitution; it is eas to solve one of the equations ( ) and get one of the variables in terms of the other variable.. Solve the sstem of equations. ;. How can ou interpret the solutions in the contet of the problem? ml of the % solution and ml of the % solution should be combined.. Do our solutions check? Eplain. es; ml ml ml, and ml?. ml?.. ml ml. ml ml?. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems Algebraicall Solve each sstem b substitution. Check our answers.. e (, ). e (, ). e. e (, ) (, ). e. e (, ) (, ). e 9 (, ). e (, ) 9. e (, ). Suppose ou bought eight oranges and one grapefruit for a total of $.. Later that da, ou bought si oranges and three grapefruits for a total of $.. What is the price of each tpe of fruit? orange costs $.; grapefruit costs $. Solve each sstem b elimination.. e. e (, ) (, ). e. e 9 (, ). e (, ) (, ). e Q, R. e (, ). e (, ) 9. e (, ). There are a total of apartments in two buildings. The difference of two times the number of apartments in the first building and three times the number of apartments in the second building is. a. Write a sstem of equations to model the relationship between the number of apartments in the first building f and the number of apartments in the second building s. f s, f s b. How man apartments are in each building? in the first; in the second Solve each sstem b elimination.. e 9 (, ). e (, ). e (, ). e (, ). e. e e (, ): f. e (, ) No solution.... e...... 9. e...9 (., ) (, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems Algebraicall. Writing Eplain what it means when elimination results in an equation that is alwas true. When elimination results in an equation that is alwas true, it means the equations in the sstem represent the same line. For each sstem, choose the solution method that seems easier to use. Eplain wh ou made each choice. Solve each sstem.. e b a b a. e p q. e p q. e j k j k. Error Analsis You and our friend are solving the sstem and. Your friend sas there is no solution, and ou sa the solution is (, ). Who is correct? Eplain. You are correct; our friend mistakenl thought both and had opposite coefficients, causing both variables to drop out, giving a sstem with no solution.. You can bu DVDs at a local store for $.9 each. You can bu them at an online store for $.99 each plus $ for shipping. How man DVDs can ou bu for the same amount at the two stores?. Last ear, a baseball team paid $ per bat and $ per glove, spending a total of $. This ear, the prices went up to $ per bat and $ per glove. The team spent $ to purchase the same amount of equipment as last ear. How man bats and gloves did the team purchase each ear? bats, gloves. If the perimeter of the square at the right is units, what are the values of and? 9 units, units. Substitution; one variable is alread defined in terms of the other, so it is eas to substitute one equation into the second; a, b.. Elimination; because both equations have the same -term coefficient, ou can multipl one equation b and add the equations to eliminate the -term; 9,.. Elimination; ou can multipl the second equation b and add the equations; OR substitution; ou can rewrite the second equation in terms of q and substitute into the first equation; p, q.. Substitution; one variable is alread defined in terms of the other, so it is eas to substitute one equation into the second; j, k. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems Algebraicall Solve each sstem b substitution. Check our answers. To start, solve one equation for and substitute into the other equation. 9. e ( ) 9,. e. e, 9,. Your internet provider offers two different plans. One plan costs $. per email plus a $9 monthl service charge. The other plan costs $. per email with no service charge. Variable ma var. Sample:. 9,. a. Write a sstem of equations to model the cost of the two internet plans. b. For how man email messages will both plans cost the same? messages c. If ou send and receive about email messages per month, which plan should ou use? the plan that costs $. per email plus the $9 monthl service charge. A boat can travel mi in h when traveling with the current. When traveling against the same current, the boat can travel onl mi in h. Find the rate of the current and the rate of the boat in still water. mi/h, mi/h. Writing Eplain how ou would solve the sstem e using substitution. Answers ma var. Sample: Substitute for in the first equation and solve for. Then substitute into either of the original equations and solve for. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Solving Sstems Algebraicall. Error Analsis You and our friend are solving the sstem e. You sa there are infinitel man solutions and our friend sas the solution is Q, R. Which of ou is correct? What mistake was made? You are correct; our friend forgot to multipl b when substituting the value for into the second equation. Solve each sstem b elimination. Check our answers.. e, 9. e,. e,. e. e. e,,,. Writing Eplain how ou would solve the sstem e using elimination. Answers ma var. Sample: First, multipl b. Then add the equations to eliminate the variable. Net, solve for. Finall, substitute for into one of the equations and solve for.. Open-Ended Write a sstem of equations in which one equation could be multiplied b and the other equation could be multiplied b in order to solve the sstem using elimination. Answers ma var. Sample: e Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Solving Sstems Algebraicall Multiple Choice For Eercises, choose the correct letter. Use the sstem of equations for Eercises and. e. What is the value of in the solution? 9 D. What is the value of in the solution? G. Which of the following sstems of equations has the solution (, )? e e e 9 e A. At a bookstore, used hardcover books sell for $ each and used softcover books sell for $ each. You purchase used books and spend $. How man softcover books do ou bu? I 9 Etended Response. A local cell phone compan offers two different calling plans. In the first plan, ou pa a monthl fee of $ and $. per minute. In the second plan ou pa a monthl fee of $99 and $. per minute. a. Write a sstem of equations showing the two calling plans. b. When is it better to use the first calling plan? c. When is it better to use the second calling plan? d. How much does it cost when the calling plans are equal? [].,. 99 where number minutes per month and total plan cost; the first plan is better when ou use less than minutes; the second plan is better when ou use more than minutes; $. when plans are equal. [] correct equations, but with one computational error [] incorrect equation OR multiple computational errors [] correct answers, without work shown [] no answers given Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Enrichment Solving Sstems Algebraicall Often, two separate items in a restaurant or a store will be made from essentiall the same ingredients. The difference between two items ma involve different proportions of the same ingredients. Solve each problem b writing two equations using two variables.. Each order of lasagna or ravioli at Casa Italia weighs lb and consists of meat filling wrapped in pasta. On Tuesda evening, the restaurant served orders of ravioli and orders of lasagna and used lb of meat filling. On Wednesda evening, it sold orders of ravioli and orders of lasagna and used lb of meat filling. a. How much meat filling is used in an order of ravioli? b. How much meat filling is used in an order of lasagna? lb lb. Foot Friends uses the same tpe of leather in their men s and women s hiking boots. During one week the compan used 9 ft of leather to make pairs of size men s hiking boots and pairs of size women s hiking boots. Another week the compan used ft of leather to make pairs of size men s hiking boots and pairs of size women s hiking boots. a. How much leather is used to make pair of size men s hiking boots?. ft b. How much leather is used to make pair of size women s hiking boots?. ft. The Fruit Emporium sells a dish of two flavors of ogurt with one serving of fruit toppings for $.. It also sells a dish of three flavors of ogurt with one serving of fruit toppings for $.9. a. How much is one flavor of ogurt with one serving of fruit toppings? $. b. How much does a customer pa for the one serving of fruit toppings? $.. A health-food store sells trail mi made with granola and dried fruit. The store bus granola at $./lb and dried fruit at $./lb, and it sells these items at a % markup. If the trail mi sells for $./lb, what is the recipe for trail mi? lb granola and lb dried fruit make lb of trail mi.. The same health-food store sells two special mitures of granola and dried fruit. One customer bus lb of Miture A and lb of Miture B and pas $.. Another customer reverses the proportions and pas a dollar more. a. What is the cost per lb of Miture A? $. b. What is the recipe for Miture A? lb granola and lb dried fruit make lb of Miture A. c. What is the cost per lb of Miture B? $. d. What is the recipe for Miture B? lb granola and lb dried fruit make lb of Miture B. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems Algebraicall Follow these steps when solving b substitution. Step Solve one equation for one of the variables. Step Substitute the epression for this first variable into the other equation. Solve for the second variable. Step Substitute the second variable s value into either equation. Solve for the first variable. Step Check the solution in the other original equation. Problem What is the solution of the sstem of equations? e Step Solve one equation for. Step ( ) Substitute the epression for into the other equation. Distribute. Combine like terms. Solve for. Step () Substitute the value into either equation. Simplif. Solve for. Step () () Check the solution in the other equation. Simplif. The solution is (, ). Eercises Solve each sstem b substitution.. e. e a b a. e m n m n. e 9, a, b m, n, Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Solving Sstems Algebraicall Follow these steps when solving b elimination. Step Step Step Step Step Step Arrange the equations with like terms in columns. Circle the like terms for which ou want to obtain coefficients that are opposites. Multipl each term of one or both equations b an appropriate number. Add the equations. Solve for the remaining variable. Substitute the value obtained in step into either of the original equations, and solve for the other variable. Check the solution in the other original equation. Problem What is the solution of the sstem of equations? e Step Circle the terms that ou want to make opposite. Step Multipl each term of the first equation b. Multipl each term of the second equation b. Step 9 Add the equations. Step Solve for the remaining variable. Step () Substitute for to solve for. Step () () Check using the other equation. The solution is (, ). You can also check the solution b using a graphing calculator. Eercises Solve each sstem b elimination. f m. e. e. e. e 9 f m, f, m no solution, where is an real number 9. Reasoning Wh does a sstem with no solution represent parallel lines? If there is no solution, then there are no values of the variables that will make both equations true. This means there is no point that lies on both lines, so the lines never meet and are therefore parallel. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- ELL Support Sstems of Inequalities There are two sets of note cards that show how to solve the sstem of inequalities below. The set on the left eplains the thinking. The set on the right shows the steps. Write the thinking and the steps in the correct order.. u u # Think Cards Clearl shade the overlapping region. Graph the second inequalit with a solid line. Shade. Graph the first inequalit with a dotted line. Shade. Pick a point in the overlapping region and check it in both inequalities. Think Write Cards (, ). u u #. # O O O Write First, graph the first inequalit with a dotted line. Shade. Step O Second, graph the second inequalit with a solid line. Shade. Step O Third, clearl shade the overlapping region. Fourth, pick a point in the overlapping region and check it in both inequalities. Step Step O (, ). u u #. # Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Sstems of Inequalities College Admissions An entrance eam has two sections, a verbal section and a mathematics section. You can score a maimum of points. For admission, the school of our choice requires a math score of at least. Write a sstem of inequalities to model scores that meet the school s requirements. Then solve the sstem b graphing. Know. The sum of the verbal score and the mathematics score must be less than or equal to.. Each of the scores must be L. The mathematics score must be greater than or equal to. Need. To solve the problem, ou need to find the possible scores that meet the school s requirements. Plan. What sstem of inequalities models this situation? Answers ma var. Sample: Let the verbal score, let the math score; L, L, K, L.. Graph our sstem of inequalities on the grid at the right.. How do ou know which region in our graph represents the solution? Answers ma var. Sample: Test a point, such as (, ). It satisfies all of the inequalities, so the (, ) points in its region represent solutions of the problem. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Sstems of Inequalities Find all whole number solutions of each sstem using a table.. e # (, ), (, ), (, ), (, ), (, ), (, ), (, ). e # (, ), (, ), # (, ), (, ), $. e #. e. # Solve each sstem of inequalities b graphing.. e. # 9. e # $. e, # (, ), (, ), (, ), (, ), (, ). e, $ (, ), (, ), (, ), (, ), (, ), (, ) (, ), (, ), (, ), (, ), (, ), (, ), (, ) (, ), (, ), (, ). The dr cleaner charges $ to clean a pair of pants and $ to clean a shirt. You want to get at least items cleaned. You have $ to spend on dr cleaning. a. Write a sstem of inequalities to model the situation. K, L where b. Solve the sstem b using a table. (, ), (, 9), number of pairs of pants and number of shirts (, ), (, ), (, ), (, 9), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ) O O. e,, O (, ), (, ), (, ), (, ), (, ), (, ),,. e.. e $, O O. e,.. e, # O O.. e.. e. $ O O. (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (9, ), (9, ), (, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Sstems of Inequalities. Suppose ou are buing two kinds of notebooks for school. A spiral notebook costs $, and a three-ring notebook costs $. You must have at least notebooks. The cost of the notebooks can be no more than $. L a. Write a sstem of inequalities to model the situation. b. Graph and solve the sstem. K d L L O. A camp counselor needs no more than campers to sign up for two mountain hikes.the counselor needs at least campers on the low trail and at least campers on the high trail. a. Write a sstem of inequalities to model the situation. b. Graph and solve the sstem. K c L L O Solve each sstem of inequalities b graphing. 9. e,. e. $ u u. u u O O. e,,u u. e $ #u u O. e,. u u O. e., u u O. Error Analsis Your homework assignment is to solve the sstem e $ using a graph. You turn in the graph at the right. What $ u u did ou do wrong? Draw a correct graph. You graphed L u u as L. O O. Open-Ended Write a sstem of inequalities that has no solution. Answers ma var, but should be an two inequalities where there is no overlap. Sample: S, R. A doctor needs at least adults for a medical stud. He cannot use more than men in the stud. Write a sstem of inequalities to model the situation and solve the sstem b graphing. L, K K, L, where represents the number of men and represents the number of women Number of men Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. Number of women
- Practice Form K Sstems of Inequalities Find all whole number solutions of each sstem using a table. To start, make a table of values for and that satisf the first inequalit.. e #.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, #. e #,,,,,,,,,,,,,,,,,,,,, 9,,,,,,,,, $. e,9,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9,,,,,,,,,, 9,,,,,,,,, Solve each sstem of inequalities b graphing. To start, graph the first inequalit.. e #,. e #.. e $, O O O. You want to bake at least loaves of bread for a bake sale. You want at least twice as man loaves of banana bread as nut bread. Write and graph a sstem of inequalities to model the situation. Variables ma var. Sample: L, L, where the number of loaves of nut bread and the number of loaves of banana bread O Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Sstems of Inequalities. Writing Eplain how ou would test whether (, 9) is a solution of the sstem e. #. Substitute the - and -coordinates into BOTH inequalities. If BOTH inequalities are true, then the ordered pair is a solution of the sstem. Since 9 S and 9 K are both true, (, 9) is a solution of the given sstem. 9. An eam has two sections: a multiple choice section and an essa. You can score a maimum of points. To pass the test, ou must get at least points on the essa. Write a sstem of inequalities to model passing scores. Then graph the sstem. Variables ma var. Sample: K, L, where the score on the multiple choice section and the score on the essa O. For our rock collection displa, ou want to have at most samples. You want to have at least times as man sedimentar samples as metamorphic samples. Write and graph a sstem of inequalities to model the situation. Variables ma var. Sample: L, K, where the number of metamorphic samples and the number of sedimentar samples O Solve each sstem of inequalities b graphing. #. c $ $.. c $ $,. c $ $ O O O Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Sstems of Inequalities Multiple Choice For Eercises, choose the correct letter.. Which sstem of inequalities is shown in the graph? e #. e. # e $, e, $. Which of the following graphs shows the solution of the sstem of inequalities? e $ G # u u O O A O O O. Which point lies in the solution set for the sstem? e, C $ (, ) (, ) (, ) (, ). How man of the ordered pairs in the data table provided are solutions of the sstem? e # G $,,,, 9,,,,,,. Is (, ) a solution of the sstem? e., Short Response Eplain how ou made our determination. [] No; substitute the ordered pair into each inequalit and simplif. Both inequalities are false, so the ordered pair is not a solution of the sstem. [] correct answer, without work shown [] incorrect answer and no work shown OR no answer given Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Enrichment Sstems of Inequalities The regions below were formed b graphing several inequalities. Write the coordinates of each verte of the figure. Stud each graph, and determine the inequalities used.. Vertices: (,), (, ), (, ), (,); O Inequalities: L, K, L, + K. Vertices: (, ), (, ), (,), (, ), (, ), (, ); O Inequalities: L, K, L, K, L, K a. For each graph, choose one point in the region. Eplain how that point satisfies each of the inequalities. b. For each graph, choose a point outside the region that satisfies one or more of the inequalities but not all of them. a. Answers ma var. Sample: (, ); Substituting the values and in each inequalit gives a true statement. a. Answers ma var. Sample: (, ); Substituting the values and in each inequalit gives a true statement. b. Answers ma var. Sample: (, ); () () S. However, substituting and in each of the three remaining inequalities gives a true statement. b. Answers ma var. Sample: (, ); () () S. However, substituting and in each of the five remaining inequalities gives a true statement. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Sstems of Inequalities Solving a Sstem b Using a Table Problem An English class has computers for at most students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have students and each performance group must have students. In how man was can ou set up the computer groups? Step Relate the unknowns and define them with variables. number of research groups, number of performance groups number of research groups number of performance groups #? number of research groups? number of performance groups # Step Make a table of values for and that satisf the first inequalit. The replacement values for and must be whole numbers. # #,,,,,,,,,, Step In the table, check each pair of values to see which satisf the other inequalit. Highlight these pairs. These are the solutions of the sstem. You can have: groups doing research and,,, or groups watching performances or group doing research and,, or groups watching performances or groups doing research and,, or groups watching performances or groups doing research and or group watching performances or groups doing research and groups watching performances,,,,,,,,,, Eercises Find the whole number solutions of each sstem using tables.. e, # (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ) $. e # (, ), (, ), (, ), (, ), (, ). e $, (, ), (, ), (, ), (, ), (, ), (, ). e,, (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Sstems of Inequalities Solving a Sstem b Graphing Problem What is the solution of the sstem of inequalities? e. $ Step Solve each inequalit for.. $. and $, Step Graph the boundar lines. Use a solid line for $ or # inequalities. Use a dotted line for. and, inequalities. O Step Shade on the appropriate side of each boundar line. The overlap is the solution to the sstem. Eercises O Solve each sstem of inequalities b graphing.. e #. e. $, O O. e., O # 9. e. O 9. e, $ O #. e # O. e #. O. e,, O. e $ # O Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- ELL Support Linear Programming Choose the word from the list that best matches each sentence. constraints feasible region linear programming objective function vertices. The limits or restrictions are also called constraints.. Linear programming is a method for finding a minimum or maimum value of some quantit, given a set of constraints.. On a graph, the feasible region contains all of the points that satisf the constraints.. The objective function models a quantit that is related to the constraint variables.. The vertices of the feasible region are the points where the least and greatest values for the objective function occur. Use a word from the list above to complete each sentence.. The inequalities # and # are the constraints.. The equation P is an eample of an objective function where P stands for profit.. The intersections of the constraint graphs are the vertices of the feasible region. 9. You have to graph the inequalities to find the feasible region for a linear programming problem. Verte is the singular form of vertices. Verte means one of the vertices. Multiple Choice. Which of the following is a verte of the feasible region on the graph at the right? C (, ) (, ) (, ) (, ). How man constraints were graphed to form the feasible region on the graph? H Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Linear Programming Cooking Baking a tra of corn muffins takes cups of milk and cups of wheat flour. Baking a tra of bran muffins takes cups of milk and cups of wheat flour. A baker has cups of milk and cups of wheat flour. He makes $ profit per tra of corn muffins and $ profit per tra of bran muffins. How man tras of each tpe of muffin should the baker make to maimize his profit? Understanding the Problem. Organize the information in a table. Corn Muffin Tras, Bran Muffin Tras, Total Milk (cups) Flour (cups) Profit. What are the constraints and the objective function? Constraints: K K e L L Objective Function: P Planning the Solution. Graph the constraints on the grid at the right.. Label the vertices of the feasible region on our graph. Getting an Answer (, ) (, ) (, ) (, ). What is the value of the objective function at each verte? P(, ) ; P(, ) ; P(, ) ; P(, ) Corn Muffin Tras Bran Muffin Tras. At which verte is the objective function maimized? (, ). How can ou interpret the solution in the contet of the problem? The baker should make tras of corn muffins and tras of bran muffins to maimize his profit. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Linear Programming Find the values of and that maimize or minimize the objective function for each graph.. Maimum for. Minimum for. Maimum for P (, ) P (, ) P (, ) D(,) C(,) A(,) O B(,) D(,) C(,) A(,) O B(,) Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. #. c $ $ C(,) A(,) B(,) O C(,) A(, ) B(,) O #. c # $, $ D(,) C(,) A(,) B(,) O. Maimum for. Minimum for. Minimum for P (, ) P 9 (, ) P (, ) C(,) B(,) # 9. c # $, $ Minimum for Maimum for Maimum for C P P ; (, ) (, ) (, ) O (, ) (, ) (, ) (, ) (, ) O ; (, ) O (, ) A(,) (, ) (, ) (, ) O ; (, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Linear Programming. You are going to make and sell baked goods. A loaf of Irish soda bread is made with c flour and c sugar. Kugelhopf cake is made with c flour and c sugar. You will make a profit of $. on each loaf of Irish soda bread and a profit of $ on each Kugelhopf cake. You have c flour and c sugar. a. How man of each kind of baked goods should ou make to maimize the profit? b. What is the maimum profit? $ Four loaves of Irish soda bread and two Kugelhopf cakes. Suppose ou make and sell skin lotion. A quart of regular skin lotion contains c oil and c cocoa butter. A quart of etra-rich skin lotion contains c oil and c cocoa butter. You will make a profit of $/qt on regular lotion and a profit of $/qt on etra-rich lotion. You have c oil and c cocoa butter. a. How man quarts of each tpe of lotion should ou make to maimize our profit?. qt regular, qt etra-rich b. What is the maimum profit? $ Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. Find the maimum or minimum value. #. c # $, $ #. c # $, $ Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. #. c # $, $ Maimum for Maimum for Minimum for P (, ); P (, ); C (, );. Writing Eplain wh solving a sstem of linear equations is a necessar skill for linear programming. See below. See graphs below.. A doctor allots minutes for routine office visits and minutes for full phsicals. The doctor cannot do more than phsicals per da. The doctor has 9 available hours for appointments each da. A routine office visit costs $ and a full phsical costs $. How man routine office visits and full phsicals should the doctor schedule to maimize her income for the da? What is the maimum income? routine office visits; full phsicals; $... (, ) (, ) (, ) (, ),, O (, ) O (, ) O (, ). Answers ma var. Sample: You must know how to find the point of intersection to be able to determine the critical points of a sstem of constraints. The vertices enable ou to find the and values that maimize or minimize an objective function.
- Practice Form K Linear Programming Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. #. c # $, $ #. c # $, $ #. c # $, $ Maimum for Minimum for Maimum for P P P B(, ) C(, ) A(, ) O D(, ) (, ) O (, ) (, ) (, ) B(, ) A(, ) O C(, ) ma P at (, ) min P at (, ) ma P at (, ) D(, ). Teams chosen from forest rangers and trainees are planting trees. An eperienced team consisting of two rangers can plant trees per week. A training team consisting of one ranger and two trainees can plant trees per week. Eperienced Teams Training Teams Total Number of Teams Number of Rangers Number of Trainees Number of Trees Planted a. Write an objective function and constraints for a linear program that models the problem. P ; constraints: K, K, L, L b. How man of each tpe of team should be formed to maimize the number of trees planted? How man trainees are used in this solution? How man trees are planted in a week? eperienced teams and training teams; trees planted Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Linear Programming Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. Find the maimum or minimum value. See graphs below. #. c # 9 $, $ #. c # $, $ # 9. c # $, $ Minimum for Maimum for Maimum for P P P. Reasoning Wh are $ and $ part of the constraints in man linear programs? In most real-life situations, ou cannot have negative resources. 9. Error Analsis Your friend sas the graph to the right can be used to determine the maimum for P with the # constraints c #. What mistake did our friend make? $, $ O He shaded the graph to show the constraints as greater than instead of less than... C(, 9). (, ½) (, ) B(, ) (, ) (, ) A(, ) D(, ) (, ) (, ) O O (, ) O min P at (, ) ma P at (, ) ma P at (, ) Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Linear Programming Multiple Choice For Eercises, choose the correct letter.. The vertices of a feasible region are (, ), (, ), (, ), and (, ). For which objective function is the maimum cost C found at the verte (, )? C C C C C. A feasible region has vertices at (, ), (, ), Q, R, and (, ). What are the maimum and minimum values for the objective function P? F minimum (, ) minimum (, ) maimum Q, R maimum Q, R minimum (, ) minimum (, ) maimum (, ) maimum (, ). Which values of and minimize N for the objective function N? A $ Constraints c $ $, $ (, ) (, ) (, ) (, ). Which of the following sstems has the vertices (, ), (, ), (, ), and (, )? G $ c $ $, $ # c # $, $ # c # $, $ # c # $, $ Short Response. The figure at the right shows the feasible region for a sstem of constraints. This sstem includes $ and $. What are the remaining constraints? Show our work. [] Work should show the use of two points of each line L to find slope and linear inequalities, b L [] correct inequalities, without work shown OR correct process with one computational error [] no answer given and no work shown OR no answer given C(,) B(,) A(,) O Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Enrichment Linear Programming The Supreme Shipping Compan can load its trucks with both rectangular and clindrical containers. A rectangular container has a volume of ft and weighs lb. A clindrical container has a volume of ft and weighs lb. Let denote the number of rectangular containers carried b a truck, and let denote the number of clindrical containers.. What constraint must be satisfied if each truck has room for at most cubic ft of containers? K. What constraint must be satisfied if each truck can carr a maimum of lb? K. What additional constraints must be satisfied because the problem involves real objects? L, L, where and are whole numbers. Graph the feasibilit set on the grid and label its vertices. Call the verte on the -ais A, the verte on the -ais B, and the verte in Quadrant I C. Label the origin O. B. Suppose that Supreme Shipping charges $ to ship either a C rectangular or a clindrical container and wishes to maimize its income. A a. What is the objective function? I O b. What is the value of the objective function at verte A? $ c. At verte B? $ d. At verte C? $ e. What combination of containers should Supreme Shipping use to maimize its income? rectangular containers, clindrical containers Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Linear Programming Problem What point in the feasible region maimizes P for the objective function P? What point minimizes P? # # Constraints d $ $ Step Step Step Graph the constraints and shade the feasible region. D A O Find the coordinates for each verte of the region. VERTEX Evaluate P at each verte. P A (, ) P () () B (, ) P () () C (, ) P () () D (, ) P () () The maimum value of the objective function is. It occurs when and. The minimum value of the objective function is. It occurs when and. Eercises C B Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. vertices: Q, 9 R, Q, R, # $ Q, R, Q, R $ $ $ vertices: (, ), (, ),. d. d (, ), (, ). c # # # vertices: (, ), Q, 9 # # $ R, ma: at (, ); $ Q P P, R, (, ) min: at (, ) P ma: at (, ); min: ma: 9 at Q, R at Q, R; min: at Q, R ( /, ) (, ) (, ) (, ) (, ) (, /) (, ) O (, 9 /) (, ) (, 9 /) (, /) O O Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Linear Programming Problem Your school band is selling calendars as a fundraiser. Wall calendars cost $ per case of. You sell them at $ per calendar. Pocket calendars cost $ per case of. You sell them at $ per calendar. You make a profit of $ per case of wall calendars and $9 per case of pocket calendars. If the band can bu no more than total calendars and spend no more than $, how can ou maimize our profit if ou sell ever calendar? What is the maimum profit? Relate Organize the information in a table. Wall Calendars Pocket Calendars Total Number of Cases Number of Units Cost Profit 9 9 Define Let number of cases of wall calendars Let number of cases of pocket calendars Write Use the information in the table and the definitions of and to write the constraints and the objective function. Simplif the inequalities if necessar. # # # # # c # $, $ Objective function: P 9 Step Step Step Graph the constraints and shade to see the feasible region. Find the coordinates for each verte of the region. Evaluate the objective function using the verte coordinates. A(, ) P () 9() B(, ) P () 9() C(, ) P () 9() D(, ) P () 9() O You can maimize our profit b selling cases of wall calendars and cases of pocket calendars. The maimum profit is $. Eercises. Your band decides to sell the wall calendars for $9 each. a. How man of each tpe of calendar should ou now bu to maimize our profit? cases of wall calendars and no cases of pocket calendars b. What is the maimum profit? $ Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- ELL Support Sstems With Three Variables z A student solved the sstem of equations z and wrote a description of z 9 each step on note cards. Write each description net to the correct step. Write the solution as an ordered triple. Pair the equations. Eliminate in each sstem. Write the two new equations as a sstem. Solve for. Check our solution b substituting values for,, and z into the other two original equations. Substitute the value for into one of the equations and solve for z. Substitute the values for and z into one of the original equations and solve for. z z z z z 9 z z z 9 Pair the equations. Eliminate in each sstem. Write the two new equations as a sstem. Solve for. () z z z () () () () () () () 9 9 9 The solution is (,, ). Substitute the value for into one of the equations and solve for z. Substitute the values for and z into one of the original equations and solve for. Check our solution b substituting values for,, and z into the other two original equations. Write the solution as an ordered triple. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Sstems With Three Variables Sports A stadium has 9, seats. Seats sell for $ in Section A, $ in Section B, and $ in Section C. The number of seats in Section A equals the total number of seats in Sections B and C. Suppose the stadium takes in $,, from each sold-out event. How man seats does each section hold? Understanding the Problem. Define a variable for each unknown in this problem. Let the number of seats in Section A Let the number of seats in Section B Let z the number of seats in Section C. What sstem of equations represents this situation? z 9, z z,, Planning the Solution. Can ou write a simpler equivalent equation for one of the equations in our sstem? If so, write the equivalent equation. es; z,. What method of solving looks easier for this problem? Eplain. Answers ma var. Sample: Substitution; one of the equations is alread solved for. Getting an Answer. Solve the sstem of equations. (,,,,,). How can ou interpret the solution in the contet of the problem? Section A holds, seats, Section B holds, seats, and Section C holds, seats. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Sstems With Three Variables Solve each sstem b elimination. Check our answers. z. z z No unique solution z. z z z. z z z. z z z. z z z. z z. z z z. z z Solve each sstem b substitution. Check our answers. z. z z z 9. z z (,, ) z. z z (,, ). z z z 9. z z (,, ) z. z 9 z 9. z z (,, ) No solution (,, ) (,, ) z. z z (,, ) (,, ) (,, ) z 9. z z z. z (,, ) (,, ) (,, ) z. z z (,, ) (,, ) Q,, R z. z 9 9 z z. z z z. z z Q,, R (,, ) (,, ) z. z z (,, ) z. z (,, ) No unique solution (,, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Sstems With Three Variables Write and solve a sstem of equations for each problem.. The sum of three numbers is. The sum of three times the first number, twice the second number, and the third number is 9. The difference between the second number and half the third number is. Find the numbers. z where represents the first number, represents z 9, the second number, and z represents the third z number;,,. Monica has $, $, and $ bills in her wallet that are worth $9. If she had one more $ bill, she would have just as man $ bills as $ and $ bills combined. She has bills total. How man of each denomination does she have? z 9 where represents the number of $ bills, represents z, the number of $ bills, and z represents the number of z $ bills; eleven $ bills, seven $ bills, five $ bills. Writing How do ou decide whether substitution is the best method to solve a sstem in three variables? Answers ma var. Sample: If one equation can be solved easil for one variable, then substitution is the best method to use. z. Error Analsis A student solves the sstem of equations. z z The student gets a solution of (,, ). Is the solution correct? How can ou be sure? Show our work. Yes; check the solution b substituting it in the original equations; show the values substituted into the equations. z 9. Reasoning Wh is there no solution to the sstem? z Answers ma var. Sample: All three equations have z the same relative coefficients and variables, but end up equaling different values. This is a model for parallel planes. z cannot equal,, and at the same time. Therefore, there is no solution to this sstem.. The first number plus the third number is equal to the second number. The sum of the first number and the second number is si more than the third number. Three times the first number minus two times the second number is equal to the third number. What is the sum of the three numbers?. Which of the following is a sstem with the solution (,, )? D z z z z 9 z z z z z z z z Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Sstems With Three Variables Solve each sstem b elimination. Check our answers. To start, pair the equations to eliminate one variable and add. z. c z z (,, ) a b c 9. ca b c a b c (,, ) ( z) (? ) z z Solve each sstem b substitution. Check our answers. z. c z z (,, ) z. c z z (,, ) z ( z ) z z z z z. You have coins in pennies, nickels, and dimes in our pocket. The value of the coins is $.. There are four times the number of pennies as nickels. How man of each tpe of coin do ou have? pennies, nickels, dimes. Writing When ou solve a sstem of equations, eplain how ou can determine if our solution is correct. Substitute our solution back into the original equations. If all of the equations are true, the solution is correct. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Sstems With Three Variables Solve each sstem using an method. z. c z z 9 (,, ) z. c z z (,, ) z 9. c z z (,, ). For a part, ou are cooking a large amount of stew that has meat, potatoes, and carrots. The meat costs $ per pound, the potatoes cost $ per pound, and the carrots cost $ per pound. You spend $. on. pounds of food. You bu twice as man carrots as potatoes. a. Write a sstem of three equations that represent how much Variables ma var. food ou bought. z. b. How much of each ingredient did ou bu? Sample: c z. lb meat,. lb potatoes, lb carrots z. Multiple Choice What is the value of z in the solution of the sstem? z c z C z Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. Name Class Date Gridded Response Solve each eercise and enter our answer in the grid provided.. A change machine contains nickels, dimes, and quarters. There are coins in the machine, and the value of the coins is $.. There are times as man nickels as dimes. How man quarters are in the machine?. The sum of three numbers is. The first number is equal to twice the second number minus. The third number is equal to one more than the sum of the first and second numbers. What is the first number?. A fish s tail weighs 9 lb. Its head weighs as much as its tail plus half its bod. Its bod weighs as much as its head and tail. How man pounds does the fish weigh?. You are training for a triathlon. In our training routine each week, ou bike times as far as ou run and ou run times as far as ou swim. One week ou trained a total of miles. How man miles did ou swim that week?. Three multiplied b the first number is equal to the second number plus. The second number is equal to one plus two multiplied b the third number. The third number is one less than the first number. What is the sum of all three numbers? Answers..... 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 - Standardized Test Prep Sstems With Three Variables
- Enrichment Sstems With Three Variables The Italian Navigator Has Landed The above phrase was one of the most important coded messages that has ever been sent. It referred to the fact that a team of phsicists had managed to achieve the first successful controlled nuclear chain reaction. The phsicist who directed these efforts was an accomplished theorist and eperimenter, whose work in producing artificial radioactive elements won him the Nobel Prize in Phsics in 9. E N R I C O F E R M I 9 First solve each of the following sets of equations. For each letter with a value between and, write that letter in its corresponding location in the puzzle. A C E A C A C E L M N L M N L N A B O A B O 9 A B O A I F A I F A I F B I K I K B I D R D E R D E R A U I A U I A U L R T L R T L R T Solution: A, C, E 9 Solution: L, M, N Solution: A, B, O Solution: A, I, F Solution: No unique solution Solution: D, E, R Solution: A, U, I Solution: L, R, T Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Sstems With Three Variables Problem What is the solution of the sstem? A B C z z 9 z 9 Use elimination. The equations are numbered to make the process eas to follow. A C B C B C z z 9 D z z 9 z 9 z 9 z E z D E (z) (z) z) z) D A () Substitute z into equation and solve for. Substitute the values of and z into one of the original equations. Solve for. The solution is (,, ). Eercises Solve each sstem b elimination. Check our answers. z. z z z. z z Pair the equations to eliminate. Pair a different set of equations. Multipl equation b to eliminate. Then add the two equations. Equations and form a sstem. Multipl equation b and add to equation to eliminate and solve for z. z. z z (,, ) (,, ) (,, ) z. z 9 z z. z z (,, ) (,, ) (,, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 z. z z
- Reteaching (continued) Sstems With Three Variables Problem What is the solution of the sstem? Use substitution. A B C z 9 z z Step Choose an equation that can be solved easil for one variable. Choose equation and solve for. A z 9 z 9 Step Substitute the epression for into equations and and simplif. ( z 9) z ( z 9) z z z 9 z z D z E z D z Step Write the two new equations as a sstem. Solve for and z. e E z z Multipl D b. z Substitute into D. z 9 Multipl E b. () z z z Step Use one of the original equations to solve for. z 9 Substitute and z into A. () () 9 The solution of the sstem is (,, ). Eercises Solve each sstem b substitution. Check our answers. z. z z (,, ) z. z z (,, ) z. z z (,, ) z. z z (,, ) z 9. z z (,, ) z. z z (,, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- ELL Support Solving Sstems Using Matrices Choose the word from the list that best matches each sentence. coefficient column dimensions matri matri element row. A rectangular arra of numbers is called a matri.. The numbers that are written horizontall in a matri form a row.. Each number in a matri is called a matri element.. The dimensions of a matri are the number of rows and columns in the arra.. A column in a matri is formed b the numbers written verticall.. You can use a matri to represent a sstem of equations.. Each coefficient and constant from a sstem of equations are used to represent the sstem within a matri.. Each matri row represents an equation. 9. The last matri column shows the constants to the right of the equal signs. The size of a matri is described b listing the dimensions of the matri. This means listing the number of rows and columns in the matri. rows cc S The size of this matri is. 9 s columns The number of rows is alwas the first number. The number of columns is alwas the second number. Write the dimensions of each matri below.. c d 9 9. C S. c 9 d Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems Using Matrices Paint A hardware store mies paints in a ratio of two parts red to si parts ellow to make two gallons of pumpkin orange. A ratio of five parts red to three parts ellow makes two gallons of pepper red. A gallon of pumpkin orange sells for $, and a gallon of pepper red sells for $. Find the cost of quart of red paint and the cost of quart of ellow paint. Know. There are z z quarts in gallon.. z zqt red z zqt ellow z zqt pumpkin orange z zqt red z zqt ellow z zqt pepper red. z zquarts of pumpkin orange cost z z. z zquarts of pepper red cost z $ z. $ Need. To solve the problem ou need to define: Variables ma var. Sample: cost of qt of red paint, cost of qt of ellow paint. To solve the problem ou need to find: a sstem of two equations Plan. What sstem of equations represents this situation? Variables ma var. Sample:,. How can ou represent the sstem of equations with a matri?. Solve the sstem of equations using the matri. (.,.) c ` d 9. How can ou interpret the solutions in the contet of the problem? qt of red paint costs $. and qt of ellow paint costs $.. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems Using Matrices Identif the indicated element. A c ` d B. A. B. B. A. B. A. B. A Write a matri to represent each sstem. 9. e c ` d. e c ` d Write the sstem of equations represented b each matri. z. c z z D T. D T. c ` 9 d. D e 9 c z z Solve the sstem of equations using a matri. T z c z z. e (, ) z. c 9 z z (,, ) (,, ) z. c z z Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems Using Matrices Graphing Calculator Solve each sstem. z. c z z 9 z 9. c z z z. c z 9 9z (,, ) (,, ) (,, ). Suppose the movie theater ou work at sells popcorn in three different sizes. A small costs $, a medium costs $, and a large costs $. On our shift, ou sold total containers of popcorn and brought in $. You sold twice as man large containers as small ones. a. How man of each popcorn size did ou sell? small, medium, large b. How much mone did ou bring in from selling small size containers? $. Open Ended Write a matri for a sstem of equations that does not have a unique solution. Answers ma var. Sample: c ` d.. The following matri shows the prices passengers on an airline flight paid for a recent ticket and how man passengers were on that flight. Some passengers paid full price for their tickets, and some bought their tickets during a halfprice sale. How man passengers bought each price of ticket? c `, d people bought $ tickets; people bought $ tickets.. Error Analsis Your friend sas that the matri below represents the sstem of equations. What error did our friend make? What is the correct sstem of equations? D T z c z z Your friend incorrectl converted the matri entr to a coefficient of when writing the first equation; c z z z Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems Using Matrices Identif the indicated element. A C S. a. a. a. a row, column Write a matri to represent each sstem.. b B ` R. b Write the sstem of equations represented b each matri.. B ` b B ` R R 9. B ` b z 9. c z z C R. C 9 S S c z z z Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form K Solving Sstems Using Matrices. Error Analsis Your classmate sas that in the matri to the right, a is. What mistake did our classmate make? What is the correct answer? A She used column, row instead of row, column. The correct answer is. C S Solve the sstem of equations using a matri.. b B ` R (, ). b. b (, ) (, ). You work at a fruit stand that sells apples for $ per pound, oranges for $ per pound, and bananas for $ per pound. Yesterda ou sold pounds of fruit and made $. You sold more pounds of apples than bananas. a. Write a matri to show the sstem of equations for this situation. b. How man pounds of each kind of fruit did ou sell esterda? c. What kind of fruit did ou sell the most? apples C lb apples, lb oranges, lb bananas S. Open-Ended Write a matri for a sstem of two equations that does not have a solution. Answers ma var. Sample: an matri containing the equations of two parallel lines. Writing Eplain how to write the sstem b as a matri. First, write each equation in the same variable order. Line up the variables. Leave space where a coefficient is. Write the matri using the coefficients and constants. Enter zeros for an missing variables. The matri is B ` R. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Solving Sstems Using Matrices Multiple Choice For Eercises, choose the correct letter.. Which sstem of equations is equivalent to D T? B z c z z z c z z z c z z z c z z. What is the solution of the sstem represented b the matri D T? H (,, ) (,, ) (,, ) (,, ). How man elements are in a matri? D Short Response. A clothing store is having a sale. A pair of jeans costs $ and a shirt costs $. You spend $ and bu a total of items. Using a matri, how man pairs of jeans and shirts do ou bu? Show our work. [] Matrices ma var. Sample: c ` d ; pairs of jeans; shirts [] correct solution, without work shown OR correct process with one computational error [] incorrect answer and no work shown OR no answer given Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Enrichment Solving Sstems Using Matrices Well-Conditioned Sstems of Linear Equations A sstem of linear equations is said to be well-conditioned if a small change in the values of the coefficients produces a small change in the values of the solutions. A sstem is said to be ill-conditioned if a small change in the values of the coefficients produces a large change in the values of the solutions. To determine whether a sstem of linear equations is well-conditioned or ill-conditioned, change each coefficient b one percent. Each time, write and solve the new sstem, finding the values of and to two decimal places b writing and solving a matri. Then compare the new values of and to the solutions of the original sstem. A change of less than one percent in the values of and can be considered small.. Determine whether Sstem is well-conditioned or ill-conditioned b completing the following steps. Sstem : e a. Find the values of and. b. Change the coefficient of in the first equation b one percent, from to.. Write the new sstem. Find the values of and. c. Change the coefficient of in the first equation b one percent, from to.99. Write the new sstem. Find the values of and. d. Change the coefficient of in the second equation b one percent, from to.. Write the new sstem. Find the values of and. e. Change the coefficient of in the second equation b one percent, from to.. Write the new sstem. Find the values of and. f. Is the sstem well-conditioned or ill-conditioned?. Determine whether Sstem is well-conditioned or ill-conditioned b completing the same steps as in Eercise. Sstem : e 999 a., a.,. b. e ; 9.,.. b. e ;., 999.99 999.99 c. e ; 9.,.9.99 c. e ;.,. 999 d. e. ;.9,. d. e ;.99,.. 999 e. e. ; 9., 9. e. e ;.9, 9.. 999 f. well-conditioned f. ill-conditioned Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems Using Matrices Problem z How can ou represent the sstem of equations with a matri? c z Step Write each equation in the same variable order. Line up the like variables. Write in variables that have a coefficient of. z c z z Step Write the matri using the coefficients and constants. Remember to enter a for variables with no numeric coefficient. D T Eercises Write a matri to represent each sstem.. e c ` d z 9. c z z D 9 T z. c z z D T z. c z z. c z z. c z z D T D T D T Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Solving Sstems Using Matrices Problem What is the solution of the sstem? e Step Write the matri for the sstem. c ` d Step Multipl Row b. Add to Row. Replace Row with the sum. Write the new matri. ) ( ) 9 9) 9 c ` 9 d Step Divide Row b 9. Write the new matri. 9 ( 9 9) c ` d Step Multipl Row b. Add to Row. Replace Row with the sum. Write the new matri. ( ) ) ) c ` d Step Multipl Row b. Write the new matri. ( ) c ` d This matri is equivalent to the sstem e. The solution is (, ). Eercises Solve each sstem of equations using a matri.. e (, ). e 9. e (, ) (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Quiz Lessons - through - Form G Do ou know HOW? Solve each sstem of equations b graphing.. e (, ). b (, ) Solve each sstem of equations b substitution or elimination.. b (, ). b (, ). b (, ) Solve each sstem of inequalities b graphing., 9. b. O. b $ # O. Pump Up Gm has an initial joining fee of $ and monthl membership dues of $. Universe Gm has an initial joining fee of $ and monthl membership dues of $9. a. When will the costs to join and maintain membership at the gms be equal? at months b. If ou planned on continuing our gm membership for onl ears, which gm would ou join? Eplain. Pump Up Gm costs less over ears, but it has a much higher initial fee. Do ou understand? 9. Open-Ended Write and solve a sstem of two inequalities b graphing. Answers ma var. Sample: S ; K O. Error Analsis Your teacher asked the class to graph the sstem of equations to find the solution. You turned in Graph A and said there were no solutions. Your friend turned in Graph B and said there were infinite solutions. Which of ou is correct? What mistake was made? e Answers ma var. Sample: You are correct. Your friend forgot to change the negative sign on the second equation s -intercept when dividing b. Graph A O Graph B O Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Quiz Lessons - through - Form G Do ou know how? Graph the sstem of constraints. Find the values of and that maimize or minimize the objective function for each graph. #. c # $, $ $. c # $ #. c # $, # Maimum for Minimum for Maimum for C P C (, ); maimum value is (, ); minimum value is (, ); maimum value is 9 Solve each sstem b substitution or elimination. z 9. c z z. c z 9 z z. c z z (,, ) (,, ) (,, ) Write a matri to represent each sstem.. e z. c z z C S C Do ou understand? S z 9. c z z C. Open Ended Give an eample of a sstem of three equations in three variables that has (,, ) as a solution. Check our answers to show the ordered triple is a solution for all three equations. An three equations that all check as equal when (,, ) is substituted for the variables. S. A fruit market is selling oranges in a lb bag for $ and a lb bag for $. You spend $ and bu a total of bags of oranges. Using a matri, how man lb bags and lb bags of oranges did ou bu? How man total pounds of oranges did ou bu? three lb bags; five lb bags; lb Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Chapter Test Form G Do ou know HOW? Without graphing, classif each sstem as independent, dependent, or inconsistent.. e dependent. e inconsistent Solve each sstem b substitution or elimination.. e Graph the solutions of each sstem.. e. # (9, ). e (, ). e (, ). e #.. You have fewer quarters than dimes and fewer nickels than quarters. The total value of the coins is $.. How man quarters, nickels, and dimes do ou have? quarters, dimes, nickels Graph each sstem of constraints. Name all vertices. Then find the values of and that maimize or minimize the objective function. # 9. c # $, $ Maimum for P (, ), (, ), (, ), (, ); (, ), O O #. c # $, $ Minimum for C (, ), (, ), (, ), (, ); (, ), O O. Which point gives the minimum value for P and lies within the sstem of restrictions? # # c # # # (, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Chapter Test (continued) Form G. What is the solution of the sstem represented b the matri? C S (,, ) Solve each sstem of equations. z. c z z Do ou UNDERSTAND? z. c z z (,, ) No unique solution. c z z (,, ). List three methods used to solve sstems of equations. Describe the strengths of each method. Graphing, substitution, elimination; Answers ma var. Sample: Graphing allows ou to visualize the solution. Substitution enables ou to reduce the number of variables quickl. Elimination is applicable to an sstem and efficient with two variables.. You burn Cal/min walking and Cal/min running. You walk to min each da and run to min each da. You never spend more than an hour running and walking together. How much time should ou spend on each activit to maimize the number of Calories ou burn? Will ou have eercised enough to burn off a Calorie meal? min running, min walking; es, ou will have burned off Calories.. Plumber A charges $ for a house call and $ for each hour spent on the job. Plumber B charges $ for a house call and $ for each hour spent on the job. If our job will take hours to complete, which plumber should ou use? How much will it cost ou? Plumber B; $ 9. Open Ended Write a sstem of inequalities that has infinite solutions. Answers ma var. Sample: K, L. Error Analsis A student sas that the sstem of equations is represented b the matri. What error did the student make? What is the correct matri? z c z C S C The student used a for the z-coefficient in the second equation instead of a. S Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Quiz Lessons - through - Form K Do ou know HOW? Solve each sstem b graphing.. e 9. e O O no solution (, ) Solve each sstem b substitution or elimination.. e O (, ). e (, ). e (, ). e (, ) Do ou UNDERSTAND?. The communit theater is selling tickets to its pla. An adult ticket costs $ and a child ticket costs $. The theater wants to take in at least $ from ticket sales and has onl seats. Variables ma var. Sample: e K a. Write a sstem of inequalities to model the situation. L b. What is one possible combination of ticket sales that would satisf the theater s goal? Answers ma var. Sample: adult tickets and child tickets. Reasoning Is it possible for a dependent linear sstem to consist of two lines with different slopes? No; a dependent sstem has two lines whose graphs are the same. If the lines have different slopes, then their graphs are not the same. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Quiz Lessons - through - Form K Do ou know HOW? Find the values of and that maimize or minimize the objective function for each graph. # 9. # $, $ #. # $, $ #. # $, $ Maimum for Minimum for Maimum for P P P (., ) (, ) (, ) Solve each sstem b substitution or elimination. z. z z (,, ) Do ou UNDERSTAND? z. z z (,, ) z. z z (,, ) z. Error Analsis To represent the sstem z, ou picked Matri A. z Your friend picked Matri B. Which of ou is correct? What mistake was made? Matri A C S Matri B C S You are correct; our friend used as the coefficient of a missing variable instead of.. The sum of three numbers is. The second number is two more than twice the first number. The second number is three times the third number. What are the numbers? first number:, second number:, third number: Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Test Form K Do ou know HOW? Solve each sstem b substitution or elimination.. e. e no solution (, ) (, ). e Graph the solution of each sstem.. e $,. e # #. e. #. You have bills in our wallet in $, $, and $ bills. There are twice as man $ bills as $ bills. The number of $ bills is one more than the number of $ bills. How man of each bill do ou have? How much mone do ou have? si $ bills, three $ bills, four $ bills; $ Graph the sstem of constraints. Identif all vertices. Find the values of and that maimize or minimize the objective function. Then find the maimum or minimum value. #. # C(, ) B(, ) $, $ Maimize for P A(, ) D(, ) ma P at (, ) Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Test (continued) Form K 9. What is the solution of the sstem represented b the matri? C S (,, ) (,, ) (,, ) (,, ) C Do ou UNDERSTAND?. Writing Eplain how ou determine whether a sstem of linear equations is independent, dependent, or inconsistent without graphing the lines. Rewrite both equations in slope-intercept form. If the lines have the same slope and same -intercept, then the are equations of the same line, and the sstem is dependent. If the lines have the same slope but different -intercepts, the are parallel lines, and the sstem is inconsistent. If the lines have different slopes, then the sstem is independent.. Mechanic A charges $ for car repairs and $ for each hour spent on our car. Mechanic B charges $ for repairs and $ for each hour spent on our car. a. If our car takes hours to repair, which mechanic charges the least mone? Mechanic B b. How much will it cost ou to have the work done b the less epensive mechanic? $. At a bookstore, ou spend $ on books and magazines. Books cost $ each and magazines cost $ each. Write a matri that represents this sstem. How man books and how man magazines did ou bu? B ` R ; books, magazines. Reasoning The sum of three numbers is. The second number is twice the third number. Do ou have enough information to determine the three numbers? If so, what are the three numbers? If not, what information do ou still need? No; ou need a third equation that defines another relationship between two or three of the numbers. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Performance Tasks Task Make three sstems of two equations/inequalities from an in the bo. $ # $ Use each method to solve one sstem. Show our work. a. graphing b. substitution c. elimination Then find a sstem that has each of the following. d. coincident lines e. intersecting lines f. parallel lines g. perpendicular lines Eplain our reasoning. Your models should present situations in which ou make comparisons and draw conclusions. [] Check students work. [] Student writes, solves, and graphs sstems and clearl demonstrates an in-depth understanding of the mathematical principles involved. A comparison is made, and the answer full supports the conclusion. [] Student shows a solid understanding of the mathematical principles. A comparison is made, but further detail or more clarit is needed. [] Student shows a limited understanding of the mathematical principles involved. Situation does not present a comparison or draw a conclusion. [] Student makes no attempt, or no response is given. Task Your art club wants to sell greeting cards using members drawings. Small blank cards cost $ per bo of. Large blank cards cost $ per bo of. You make a profit of $. per bo of small cards and $ per bo of large cards. The club can bu no more than total cards and spend no more than $. a. How can the art club maimize its profit? b. The card compan has a minimum order requirement of boes of each size per order. Does the art club meet this minimum requirement when it maimizes its profit? c. What is the maimum profit the art club can make and meet the minimum order requirement? [] a. ", ", #, #, maimize profit for P., boes of large cards, boes of small cards b. No, this does not meet the minimum order requirements c. An order of boes of small cards and boes of large cards meets the minimum requirements and gives a maimum profit of $. [] correct equations, but with one computational error [] incorrect equations OR multiple computational errors [] student began writing inequalities but did not solve or show understanding [] no attempt was made to solve the problem Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
Chapter Performance Tasks (continued) Task Eplain in detail how to solve a sstem of equations with three variables b each method. a. graphing b. substitution c. elimination Using each method, solve the following sstem. c z z Task [] Check students work; (,, ) [] Student s eplanation of each method is accurate and contains sufficient detail to indicate a clear understanding of the method. Student solves sstem correctl using each method. Student shows steps with enough detail to indicate understanding. [] Student s eplanation of each method is accurate and contains sufficient detail to indicate a clear understanding of the method. Student solves sstem correctl using all three methods. Steps could be eplained more clearl or have more detail. [] Student s eplanation does not eplain in detail each method. Student solves the sstem correctl using onl two methods. Eplanation of the steps is not clear or missing important details. [] Student makes no attempt, or no response is given. You want to put together kits to make beaded necklaces using kinds of beads: silver, black, and crstal. Each necklace requires a total of beads. Silver beads cost $ for a package of beads. Black beads cost $. for a package of beads. Crstal beads cost $ for a package of beads. Each necklace has times as man black beads as crstal beads. You have $ to spend on beads. a. How do the equations at the right relate to the problem? What does each variable represent? z c...z z b. Write a matri to represent the sstem and solve. What does this solution tell ou? c. How man packages of each kind of bead should ou bu to make necklace kits? Is $ enough mone to bu this man packages of beads? [] a. The first equation shows the number of beads needed for each necklace; the second equation shows the cost per bead for each kind of bead and the amount to spend for each necklace; the third equation shows that there are times as man black beads as crstal beads in each necklace. b. C....9S; silver beads, black beads, z crstal beads c. You need 9 packages of silver beads, packages of black beads, packages of crstal beads; no, this man packages costs $.. [] Showed understanding of equation relations and variables in problem; set up correct matri, but with one computational error. [] incorrect relations OR incorrect matri OR multiple computational errors [] Student began eplaining relations OR began writing matri, but did not solve or show understanding. [] Student made no effort to solve the problem. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Chapter Cumulative Review Multiple Choice For Eercises, choose the correct letter.. Which line is parallel to? A.. Which is a solution of the sstem of inequalities? e H (, ) (, ) (, ) (, ). What point in the feasible region maimizes C for the objective function C? A (9, ) (, ) (, ) (, 9) ". negative vertical. Which number is rational but not an integer? I,,. The data in a scatter plot can be modeled b the equation. What kind of correlation is this? B horizontal positive. What is the value of the epression a a evaluated at a? H 9. Which of the following best describes the transformations used to obtain the graph of from the graph of? B Reflect across the -ais, shift right one unit and down three units. Reflect across the -ais, shift left one unit and down three units. Reflect across the -ais, shift left one unit and up three units. Reflect across the -ais, shift right one unit and down three units.. What is the solution of the inequalit ( ) $ ( )? H all real numbers no real numbers $ # 9. What is the slope of the line through the points (, ) and (, )? B. Which graph shows the sstem represented b the matri? c O O O Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 ` d G O
Chapter Cumulative Review (continued) Short Response. What is the slope of each line? a. b.. For each function what is f ()? a. f () b. f (). A store sells pens and notepads in packages. The price of a package of pens and notepads is $.. The price of a package of pens and notepads is $.. The store is also running a sale on a package of pen and notepad for $.. Is the sale price a better deal than the other packages? Show our work. No; in the original packages a pen is $. and a notepad is $.. Together the are $. In the sale package, the are $... Suppose varies directl with, and when. What is when? Show our work. k; k(); k ; ; ; Q R (, ). Solve the sstem of equations b substitution. Show our work. e ; ( ) ; ; ; ; () ; (, ) Etended Response. A musician wants to sell CDs of her music and DVDs of her concert videos. It costs $. to make a CD and $ to make a DVD. The profit for each CD is $9. and the profit for each DVD is $. The musician can spend no more than $ and she wants to have at least items to sell. a. How man CDs and DVDs should the musician sell to maimize her profit? b. The musician wants to bu a new amplifier sstem that costs $. Will she have enough mone to make this purchase after selling everthing? c. If the musician is able to make this purchase, how much mone will she have left over? If she is not able to make this purchase, how much more mone does she need? [] a.. ", #, #, #, maimize profit for P 9., 9 CDs, DVDs to maimize profit b. Yes, she can make the purchase c. She will have $,. left over. [] correct equations, but with one computational error [] incorrect equations OR multiple computational errors [] correct answers, without work shown [] no answers given Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Teacher Instructions Chapter Project Teacher Notes: Hot! Hot! Hot! About the Project The Chapter Project gives students an opportunit to simulate a successful business b minimizing costs, maimizing profits, and establishing a process for filling orders promptl. Students research costs, stud profit margins, and establish selling prices. Students track inventor b designing a spreadsheet. Introducing the Project Encourage students to keep all project-related materials in a separate folder. Ask students to graph a sstem of inequalities. Investigate how solutions to this sstem could be used to maimize profits. Discuss the information needed to keep track of a compan s filled orders and available stock, and the best wa to organize that information on a spreadsheet. Activit : Graphing Students write and graph a sstem of inequalities. Then the determine how man pints of each sauce the can make. Activit : Analzing Students use their answers from Activit to determine how much of each sauce the should produce to maimize profit. Students use this information to find the maimum profit. Activit : Researching Students visit a grocer store to estimate the cost of each ingredient of the sauces. Students use this information to find the cost to produce one pint of each tpe of sauce. Then students determine the selling price that will maintain the same profit margin. Activit : Organizing Students determine a production schedule to meet demand given that the compan will produce the same amount of sauce each week. Students prepare a spreadsheet to keep track of filled orders and available stock. Finishing the Project You ma wish to plan a project da on which students share their completed projects. Encourage students to eplain their processes as well as their results. Have students review their project work and update their folders. Encourage students to review the inequalities, graphs, and eplanations that are needed for the project. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Project: Hot! Hot! Hot! Suppose ou are the owner of the Sizzlin Sauce Compan. Your compan makes two different kinds of sauce, Red Hot Sauce and Scorchin Hot Sauce. Red Hot Sauce Yield: pint pt tomato sauce with onions green peppers, diced hot chili peppers, seeded and diced Scorchin Hot Sauce Yield: pint pt tomato sauce with onions green peppers, diced hot chili peppers, seeded and diced As the owner of a successful business, ou want to minimize costs, maimize profit, and keep customers satisfied b filling orders promptl. List of Materials Calculator Graph paper Activities Activit : Graphing To fill an order for Sizzlin Sauce sauces, ou bought green peppers and hot chili peppers. Write and graph a sstem of inequalities to represent the number of pints of each kind of sauce ou can make. Refer to the recipes above. Select one solution of the sstem and determine how man peppers ou will have left over. the number of pints of Red Hot Sauce, the number of pints of Scorchin Hot Sauce; ", ", #, # ; Answers ma var. Sample: If pt of each sauce are produced, there will be green peppers left over. Activit : Analzing Suppose ou make $. profit on pint of Red Hot Sauce and $. profit on pint of Scorchin Hot Sauce. Using the restrictions from Activit, decide how much of each sauce ou should make and sell to maimize our profit. What is the maimum profit? Red Hot Sauce: pt, Scorchin Hot Sauce: pt; $ Number of Pints of Scorchin' Hot Sauce Number of Pints of Sauce Number of Pints of Red Hot Sauce Activit : Researching Visit a local grocer store to estimate the cost of each sauce ingredient. Remember that buing in large quantities can save ou mone. Find the cost of producing pt of each tpe of sauce. What selling price will ou set for each sauce to maintain our profit? Check students work. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Project: Hot! Hot! Hot! (continued) Activit : Organizing You can sell our sauce to a supermarket chain, a local grocer, and a specialt store. The supermarket chain will bu pt ever weeks, the grocer will bu pt ever weeks, and the specialit store will bu pt each week. How man pints of sauce should ou produce each week to fill these orders? Presume that ou want to produce the same number of pints each week, and that the tpe of sauce is not a factor in filling these orders. Design a spreadsheet to track our stock of sauce after each week. Use cell formulas. Answers ma var. Sample: Spreadsheet with columns A: Week No.; B: Sauce Made (pt); C: Supermarket Orders (pt); D: Local Grocer orders (pt); E: Specialit Store Orders (pt); F: Sauce Available After Orders (pt) Finishing the Project The activities should help ou to complete our project. Your report should include our analsis of the cost of producing Sizzlin Sauces. Include our profit analsis and production spreadsheet. Illustrate our reasoning and decisions with graphs. Reflect and Revise Present our analsis to a small group of classmates. After ou have heard their analses and presented our own, decide if our work is complete, clear, and convincing. If needed, make changes to improve our presentation. Etending the Project Are there other epenses ou could epect in addition to those ou have alread considered? Estimate them. Modif our recommendations if necessar. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Chapter Project Manager: Hot! Hot! Hot! Getting Started Read the project. As ou work on the project, ou will need a calculator, graph paper, materials to record our calculations, and materials to make accurate and attractive graphs. Keep all of our work for the project in a folder. Checklist Activit : writing and graphing inequalities Activit : maimizing profit Activit : determining selling price Suggestions Make sure answers are reasonable. Check our work with a graphing calculator. Document our information. Activit : designing production schedule profit analsis Scoring Rubric Remember that ou will produce the same amount of sauce ever week. Besides the cost of ingredients, what information would need to be taken into consideration to accuratel determine the profit? How would ou go about finding this information? Correct inequalities are written and graphed. Calculations and graphs are accurate. The folder is well organized and provides useful information. Minor errors are made. Reasoning and eplanations are essentiall correct. Graphs contain minor errors in scale or are labeled incorrectl. The folder provides useful information, but needs to be better organized. Inaccurate inequalities are written and graphed. The folder lacks organization. Graphs could be neater and more accurate. Eplanations lack detail. Major concepts are misunderstood. Project satisfies few of the requirements and shows poor organization and effort. Major elements of the project are incomplete or missing. Your Evaluation of Project Evaluate our work, based on the Scoring Rubric. Teacher s Evaluation of the Project Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.