Solving Systems Using Tables and Graphs


 Philip Rose
 2 years ago
 Views:
Transcription
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 onetime 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 slopeintercept 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.. OpenEnded 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 reallife 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 soldout 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. OpenEnded 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.
Solving Systems Using Tables and Graphs. Use the chart below to review vocabulary. These vocabulary words will help you complete this page.
 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
More informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationSolving Special Systems of Linear Equations
5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest
More informationAnytime plan TalkMore plan
CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations
More informationSolution of the System of Linear Equations: any ordered pair in a system that makes all equations true.
Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationEssential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)
5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bedandbreakfast.
More informationName Class Date. Additional Vocabulary Support
 Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More information9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES
86 CHAPTER 9 LINEAR PROGRAMMING 9. LINEAR PROGRAMMING INVOLVING TWO VARIABLES Figure 9.0 Feasible solutions Man applications in business and economics involve a process called optimization, in which we
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationExample 1: Model A Model B Total Available. Gizmos. Dodads. System:
Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve realworld
More informationChapter 3 & 8.18.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x  4y = 10 16x + 8y = 10
Chapter 3 & 8.18.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair
More informationGraphing Linear Equations in SlopeIntercept Form
4.4. Graphing Linear Equations in SlopeIntercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More information{ } Sec 3.1 Systems of Linear Equations in Two Variables
Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationIntroduction  Algebra I
LIFORNI STNRS TEST lgebra I Introduction  lgebra I The following released test questions are taken from the lgebra I Standards Test. This test is one of the alifornia Standards Tests administered as part
More information2 Solving Systems of. Equations and Inequalities
Solving Sstems of Equations and Inequalities. Solving Linear Sstems Using Substitution. Solving Linear Sstems Using Elimination.3 Solving Linear Sstems Using Technolog.4 Solving Sstems of Linear Inequalities
More informationPreAP Algebra 2 Lesson 26 Linear Programming Problems
Lesson 26 Linear Programming Problems Objectives: The students will be able to: use sstems of linear inequalities to solve real world problems. set up constraints & objective functions for linear programming
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationDirect Variation. COMPUTERS Use the graph at the right that shows the output of a color printer.
95 Direct Variation MAIN IDEA Use direct variation to solve problems. New Vocabular direct variation constant of variation Math nline glencoe.com Etra Eamples Personal Tutor SelfCheck Quiz CMPUTERS Use
More informationRELEASED. North Carolina READY EndofGrade Assessment Mathematics. Grade 8. Student Booklet
REVISED 7/4/205 Released Form North Carolina READY EndofGrade Assessment Mathematics Grade 8 Student Booklet Academic Services and Instructional Support Division of Accountabilit Services Copright 203
More informationSOLVING SYSTEMS OF EQUATIONS
SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationIntroduction. Introduction
Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation
More information1) (3) + (6) = 2) (2) + (5) = 3) (7) + (1) = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = 7) (5)(4) = 8) (3)(6) = 9) (1)(2) =
Extra Practice for Lesson Add or subtract. ) (3) + (6) = 2) (2) + (5) = 3) (7) + () = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = Multiply. 7) (5)(4) = 8) (3)(6) = 9) ()(2) = Division is
More informationSolving Systems. of Linear Equations
Solving Sstems 8 MODULE of Linear Equations? ESSENTIAL QUESTION How can ou use sstems of equations to solve realworld problems? LESSON 8.1 Solving Sstems of Linear Equations b Graphing 8.EE.8, 8.EE.8a,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
More informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
 NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
More informationThe SlopeIntercept Form
7.1 The SlopeIntercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph
More informationSolving Systems of Linear Equations by Graphing
. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bedandbreakfast. It
More informationWord Problems Involving Systems of Linear Equations
Word Problems Involving Systems of Linear Equations Many word problems will give rise to systems of equations that is, a pair of equations like this: 2x+3y = 10 x 6y = 5 You can solve a system of equations
More informationMath 1324 Exam 3 Review (Spring 2011) (covers sections 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 3.3)
c Dr. Patrice Poage, March 30, 011 1 Math 134 Exam 3 Review (Spring 011) (covers sections.1,.,.3,.4, 3.1, 3., 3.3) 1. A = [ 15  138 B= 41 5 81 C= [ 3 h 4 p 6 Using the above matrices, find the value
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationInequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,
Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page 7577 Es. 420. 2328, 2939 odd, 4043, 4952, 5973 odd
More informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationSYSTEMS OF LINEAR EQUATIONS
SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
More informationSystems of Equations and Matrices
Sstems of Equations and Matrices A sstem of equations is a collection of two or more variables In this chapter, ou should learn the following How to use the methods of substitution and elimination to solve
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationM122 College Algebra Review for Final Exam
M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant
More informationCHAPTER 5: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)...
Table of Contents CHAPTER 5: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)... 11 5.0 ANCHOR PROBLEM: CHICKENS AND PIGS... 14 SECTION 5.1: UNDERSTAND SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS... 15 5.1a Class
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationGASOLINE The graph represents the cost of gasoline at $3 per gallon.
96 SlopeIntercept Form MAIN IDEA Graph linear equations using the slope and intercept. New Vocabular slopeintercept form intercept Math Online glencoe.com Etra Eamples Personal Tutor SelfCheck Quiz
More informationSystems of Linear Equations
Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing. Solving Sstems of Equations b Using the Substitution Method. Solving Sstems of Equations b Using the Addition Method. Applications
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More informationClassifying Solutions to Systems of Equations
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationAlgebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview
Algebra 1 Topic 8: Student Activity Sheet 1; use with Overview 1. A car rental company charges $29.95 plus 16 cents per mile for each mile driven. The cost in dollars of renting a car, r, is a function
More informationTHE POINTSLOPE FORM
. The PointSlope Form () 67. THE POINTSLOPE FORM In this section In Section. we wrote the equation of a line given its slope and intercept. In this section ou will learn to write the equation of a
More informationThe Rectangular Coordinate System
6.2 The Rectangular Coordinate Sstem 6.2 OBJECTIVES 1. Give the coordinates of a set of points on the plane 2. Graph the points corresponding to a set of ordered pairs In Section 6.1, we saw that ordered
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1100 Overview: This unit models realworld situations by using one and twovariable linear equations. This unit will further expand upon pervious
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationDirect Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship
6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationLINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0
LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )
More informationHCC COLLEGE PLACEMENT TEST MATHEMATICS
HCC COLLEGE PLACEMENT TEST MATHEMATICS PLEASE READ BEFORE TAKING THE MATHEMATICS PORTION OF THE COLLEGE PLACEMENT TEST It is highl recommended that ou review this packet BEFORE ou take the College Placement
More informationSAINT JOHN PAUL II CATHOLIC ACADEMY. Entering Grade 6 Summer Math
SAINT JOHN PAUL II CATHOLIC ACADEMY Summer Math 2016 In Grade 5 You Learned To: Operations and Algebraic Thinking Write and interpret numerical expressions. Analyze patterns and relationships. Number and
More informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationVARIABLES AND COMBINING LIKE TERMS 2.1.1 and 2.1.2
VARIABLES AND COMBINING LIKE TERMS 2.. and 2..2 Algebraic epressions can be represented b the perimeters and areas of algebra tiles 2 (rectangles and squares) and combinations of algebra tiles. The dimensions
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationDIAMOND PROBLEMS 1.1.1
DIAMOND PROBLEMS 1.1.1 In every Diamond Problem, the product of the two side numbers (left and right) is the top number and their sum is the bottom number. Diamond Problems are an excellent way of practicing
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationLearning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations
Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationAlgebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:
Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More informationAlgebra 1 EndofCourse Exam Practice Test with Solutions
Algebra 1 EndofCourse Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fillin Response Items, write your answer in the box provided, placing one digit in
More informationSystems of Linear Equations in Three Variables
5.3 Systems of Linear Equations in Three Variables 5.3 OBJECTIVES 1. Find ordered triples associated with three equations 2. Solve a system by the addition method 3. Interpret a solution graphically 4.
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More information2. System of linear equations can be solved by graphing, substitution, or eliminating a variable.
1 Subject: Algebra 1 Grade Level: 9 th Unit Plan #: 6 UNIT BACKGROUND Unit Title: Systems of Equations and Inequalities Grade Level: 9 Subject/Topic: Algebra 1 Key Words: Graphing, Substitution, Elimination
More informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationXIV. Mathematics, Grade 8
XIV. Mathematics, Grade 8 Grade 8 Mathematics Test The spring 0 grade 8 Mathematics test was based on standards in the five domains for grade 8 in the Massachusetts Curriculum Framework for Mathematics
More informationSolving Systems of Equations. 11 th Grade 7Day Unit Plan
Solving Sstems of Equations 11 th Grade 7Da Unit Plan Tools Used: TI83 graphing calculator (teacher) Casio graphing calculator (teacher) TV connection for Casio (teacher) Set of Casio calculators (students)
More information