Solving Systems Using Tables and Graphs

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1 - 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.

2 - 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 \$ each. The class spent a total \$. How man people rented canoes and how man people rented kaaks?, ; 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 Male Times (s) Female Times (s) SOURCE: Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

3 - 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. e zero solutions infinitel man solutions one solution Graph and solve each sstem.. e (, ). e (, ). e no solution O O O. 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.

4 - 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 rows. When will ou both have knitted the same number of rows? G. minutes minutes. 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.

5 - 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.

6 - 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.. e ( ),. e. e,,. Your internet provider offers two different plans. One plan costs \$. per plus a \$ monthl service charge. The other plan costs \$. per with no service charge. Variable ma var. Sample:.,. a. Write a sstem of equations to model the cost of the two internet plans. b. For how man messages will both plans cost the same? messages c. If ou send and receive about messages per month, which plan should ou use? the plan that costs \$. per plus the \$ 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.

7 - 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,. 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.

8 - 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? D. What is the value of in the solution? G. Which of the following sstems of equations has the solution (, )? e e e 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 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 \$ 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? [].,. 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.

9 - 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.

10 - 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 #,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, \$. e,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 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.

11 - Practice (continued) Form K Sstems of Inequalities. Writing Eplain how ou would test whether (, ) 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 S and K are both true, (, ) is a solution of the given sstem.. 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.

12 - 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 \$,,,,,,,,,,. 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.

13 - 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.

14 - 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.

15 - 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 # \$, \$ #. c # \$, \$ #. 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.. 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(, ). (, 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.

16 - 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.

17 - Think About a Plan Sstems With Three Variables Sports A stadium has, 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, 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.

18 - 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. 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.

19 - Practice (continued) Form K Sstems With Three Variables Solve each sstem using an method. z. c z z (,, ) z. c z z (,, ) z. 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.

20 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 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 Standardized Test Prep Sstems With Three Variables

21 - 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. 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.

22 - 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. B ` b z. c z z C R. C S S c z z z Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.

23 - 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.

24 - 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.

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