Solving Systems Using Tables and Graphs 3-1 Vocabulary Review 1. Cross out the equation that is NOT in slope-intercept form. y 5 217 x r5s a 5 !3b 1 5 3x 1 7y 5 13 Vocabulary Builder linear system linear system (noun) LIN ee ur SIS tum 5x 7y 1 4x y 9 Related Words: independent system, dependent system Use Your Vocabulary The graphs below show the possible types of solutions for a system of two equations in two variables. Write T for true or F for false. Intersecting Lines y Coinciding Lines y x x O one solution Consistent Independent Parallel Lines y O O infinitely many solutions Consistent Dependent no solution Inconsistent F 2. Inconsistent linear systems intersect at two points. T 3. An independent linear system has one solution. F 4. A dependent linear system has no solutions. T 5. Two unique lines with the same slope form an inconsistent system. Chapter 3 58 x Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Definition: A linear system is a collection of linear equations involving the same set of variables. The system above is two equations in two variables. Problem 1 Using a Graph or Table to Solve a System Got It? What is the solution of the system? e 3x 2 2y 5 4 3x 1 2y 5 5 6. Circle the graph of the equations. y y 4 2 4 2 O y 2 x 2 4 2 x O 4 2 2 4 2 4 4 2 2 O x 2 4 2 4 7. Circle the row in the calculator screen that contains the solution of the system. X Y1 Y2 0 1 2 3 4 –2 –1.5 –1 –.5 0 5 2 –1 –4 –7 8. The solution of the system of equations is Q 2 , 21 R . Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 2 Using a Table to Solve a Problem Got It? Biology The equation y1 5 1.5x 1 22 models the length, in centimeters, of a Spiny Dogfish shark x years old. The equation y2 5 0.75x 1 37 models the length, in centimeters, of a Greenland shark x years old. If the growth rates continue, how long will each shark be when it is 25 years old? 9. Reasoning Which shark will be longer at age 25? How do you know? Answers may vary. Sample: The Spiny Dogfish has a faster growth _______________________________________________________________________ rate, so it will be longer at age 25. _______________________________________________________________________ 10. Underline the correct phrase to complete the sentence. To solve the problem, I need to find an x-value / a y-value / two x-values / two y-values . 11. Complete the table for x 5 25. x y1 1.5x 22 y1 0.75x 37 25 59.5 55.75 59 Lesson 3-1 12. Spiny Dogfish sharks will be 59.5 cm long and Greenland sharks will be 55.75 cm long when they are 25 years old. Problem 3 Using Linear Regression Got It? The table shows the populations of the San Diego and Detroit metropolitan regions. When were the populations of these regions equal? What was that population? Populations of San Diego and Detroit (1950–2000) San Diego Detroit 1950 1960 1970 1980 1990 2000 334,387 573,224 696,769 875,538 1,110,549 1,223,400 1,849,568 1,670,144 1,511,482 1,203,339 1,027,974 951,270 SOURCE: U.S. Census Bureau 13. Circle the first calculator step in solving the problem. Calculate the intersection. Enter the data into lists. Enter y1 . 14. Write the name of the list you will use (L1 , L2 , or L3 ) next to the data type. L2 population of San Diego L3 population of Detroit L1 years since 1950 y1 5 17816.597 cx 1 356896.238 c y1 5 217816.597x c 1 356896.238 c y2 5 219217.551 cx 1 1849401.619 c y2 5 19217.551x c 1 1849401.619 c 16. Cross out the graph that does NOT show the regression lines. 17. Underline the correct word to complete each sentence. The x-axis / y-axis corresponds to the number of years since 1950. The x-axis / y-axis corresponds to the populations of San Diego and Detroit. 18. The scale of the graphs is 260 # x # 60 by 10s and 0 # y # 1,500,000 by 100,000s. Use the scale to estimate the coordinates of the point of intersection. Accept reasonable estimates. Exact answers are given. 1,074,919 R. The coordinates are Q 40.3 , The populations of San Diego and Detroit were equal sometime during the year 1990 . The population was about Chapter 3 1,075,000 . 60 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 15. Circle the pair of equations you will graph. Problem 4 Classifying a System Without Graphing 23x 1 y 5 4 Got It? Without graphing, is the system e x 2 1 y 5 1 3 independent, dependent, or inconsistent? 19. Write each equation in slope-intercept form. x 2 13 y 5 1 y 5 3x 1 4 23x 1 y 5 4 y 5 3x 2 3 20. The slope of 23x 1 y 5 4 is 3 and the slope of x 2 13 y 5 1 is 3 . 21. The y-intercept of 23x 1 y 5 4 is 4 and the y-intercept of x 2 13 y 5 1 is 23 . 22. Underline the correct words to complete the sentence. Because the slopes of the lines are equal / not equal and the y-intercepts are the same / different , the system is inconsistent / independent / dependent . Lesson Check • Do you UNDERSTAND? Vocabulary Is it possible for a system of equations to be both independent and inconsistent? Explain. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Write T for true or F for false. T 23. The graphs of an inconsistent system are parallel. T 24. The graphs of an independent system intersect at one point. 25. Now answer the question. Answers may vary. Sample: No; inconsistent lines are parallel and _______________________________________________________________________ never intersect. Independent lines have one point of intersection. _______________________________________________________________________ Therefore, a system cannot be both independent and inconsistent. _______________________________________________________________________ Math Success Check off the vocabulary words that you understand. system of equations dependent independent consistent inconsistent Rate how well you can solve a linear system using a graph or table. Need to review 0 2 4 6 8 Now I get it! 10 61 Lesson 3-1 3-2 Solving Systems Algebraically Vocabulary Review 1. Circle the equations that are in standard form. 4x 1 3y 5 2 y 5 3x 2 5 4x 2 3 5 2y 2x 1 5y 5 0 Write each equation in standard form. 2. y 5 5x 2 3 3. y 2 4 5 6x 5x 2 y 5 3 4. 2y 1 3x 2 12 5 0 6x 2 y 5 24 3x 2 y 5 12 Vocabulary Builder LOO shun Main Idea: If two numbers are substituted for x and y in a system of equations and they make both equations true, then the ordered pair (x, y) is a solution of the system. Definition: A solution is any ordered pair that makes an equation in two variables true. Use Your Vocabulary 5. Write three ordered pairs that are solutions of the equation y 5 25x 2 2. Answers may vary. Check students’ work. Q 0 , 22 R Q 25 , 23 R Q 5 , 227 R Samples are given. Use the system at the right. Write T for true or F for false. F 6. The system has a unique solution. F 7. The system has infinitely many solutions. T 8. The system has no solution. F 9. The solution is (0, 0). T y 2x 1 2 5 3 e 4x 1 y 5 2 10. The solution of a system can be found by graphing the equations of the system. Chapter 3 62 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. solution (noun) suh (2, 1) is a solution of y 3x 5 because 1 3(2) 5. Problem 1 Solving by Substitution Got It? What is the solution of the system of equations? e x 1 3y 5 5 22x 2 4y 5 25 11. Follow the steps to find the solution. Solve the first equation for x. x 3y 5 1 2 x 5 3y Substitute the expression for x in the 2x 4y 5 second equation. Then solve for y. 2+ 3y 5 , 4y 5 4y 5 6y 10 2 3 Substitute the value for y in either equation. Solve for x. y 5 y 2.5 x 3y 5 x 3+ ,5 2.5 x 5 7.5 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. x 2.5 12. The solution of the system is Q 22.5 , R. 2.5 Problem 2 Using Substitution to Solve a Problem Got It? Music An online music company offers 15 downloads for $19.75 and 40 downloads for $43.50. Each price includes a one-time registration fee. What is the cost of each download and the registration fee? 13. Complete the model to write a system of equations. Relate Define Write total cost is number of downloads times cost of one download plus registration fee Let c the cost of one download and let r the registration fee. $19.75 15 r c r $ 43.50 40 r c r 63 Lesson 3-2 14. Circle the equation that expresses r in terms of c in the first equation. r 5 215c 1 19.75 r 5 15c 1 19.75 r 5 215c 2 19.75 15. Substitute the equation you chose in Exercise 14 into the second equation of the system and solve for c. 43.5 5 40c 1 r Write the original equation. 43.5 5 40c 1 215 c 1 19.75 Substitute for r. 43.5 5 25 c 1 19.75 Simplify. 23.75 5 25 c Use the Addition Property of Equality. 0.95 5 c Divide. 16. Now substitute the value of c into one of the equations of the system and solve for r. Answers may vary. Sample: r 5 215c 1 19.75 r 5 215(0.95) 1 19.75 r 5 214.25 1 19.75 r 5 5.5 17. The cost of each download is $ and the registration fee is $ 5.50 . Solving by Elimination Got It? What is the solution of the system of equations? 18. Add the equations. 22x 5x 1 2 3x 1 22x 1 8y 5 28 5x 2 8y 5 20 19. Now choose one of the original equations. Substitute and solve. 5 5 28 20 0y 5 12 x 5 4 8y 8y e Answers may vary. Sample: 22(4) 1 8y 5 28 28 1 8y 5 28 8y 5 0 y50 Simplify. 20. Circle the ordered pair that is the solution of the system of equations. ( 24, 22) Problem 4 (4, 22) (4, 0) Solving an Equivalent System Got It? What is the solution of the system of equations? 21. Underline the correct values to complete the sentence. e 3x 1 7y 5 15 5x 1 2y 5 24 To get additive inverses for the x-term, multiply the first equation by 2 / 3 / 5 and the second equation by 22 / 23 / 27. Chapter 3 64 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 3 .95 22. Circle the equivalent system that shows additive inverses for the x-term. e 3x 1 7y 5 15 5x 1 2y 5 24 e 6x 1 14y 5 3c 235x 2 14y 5 28 23. Solve the system for y. e 15x 1 35y 5 75 215x 2 6y 5 12 24. Then substitute and solve for x. 15x 1 35y 5 75 215x 2 6y 5 12 29y 5 87 y5 3 5x 1 2y 5 24 5x 1 2(3) 5 24 5x 5 210 x 5 22 25. The solution of the system is Q 22 , 3 R . Problem 5 Solving Systems Without Unique Solutions Got It? What is the solution of the system of equations? Explain. e 2x 1 y 5 22 2x 2 2y 5 0 26. Circle the first step in solving the system. Multiply 2x 1 y 5 22 by 21. Multiply 2x 1 y 5 22 by 2. 27. Add 22x 1 2y 5 24 and 2x 2 2y 5 0. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 22x 2x 1 2 2y 2y 5 5 24 0 0 5 24 28. What is the solution of the system? Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect. ✓ The system has no solution. ✗ The system has infinitely many solutions. Lesson Check • Do you UNDERSTAND? Vocabulary Give an example of two equivalent systems. 29. Cross out the system of equations that is NOT equivalent to the others. e 4y 1 5x 5 13 4y 2 x 5 3 e y 1 5x 5 12 4y 2 x 5 3 e 8y 1 40x 5 96 8y 2 2x 5 6 Math Success Check off the vocabulary words that you understand. substitution elimination equivalent equation unique solutions Rate how well you can solve linear systems algebraically. Need to review 0 2 4 6 8 Now I get it! 10 65 Lesson 3-2 3-3 Systems of Inequalities Vocabulary Review Complete each statement with the correct word or phrase from the list below. Use each word or phrase only once. greater than or equal to at least at most 1. The Florida football team needs 9 two more wins to clinch the division title. at least 2. The 9 speed a car may travel on the Florida freeway is 40 miles per hour. minimum 3. If you have $45 and sweaters sell for $20 each, you can buy 9 two sweaters. at most 4. The height of a rider must be 9 42 inches in order to ride the Summit Plummet. greater than or equal to Vocabulary Builder inequality symbols inequality (noun) in ee KWAL uh tee ., /, ≤, ≥, U Definition: An inequality is a mathematical statement indicating that one quantity is less than or less than or equal to a second quantity. Use Your Vocabulary 5. Place a ✓ next to the math statements that are inequalities. Place an ✗ next to the math statements that are not. ✓ 15x . 3y ✓ ✓ y#x ✗ r2s p 5 t 2 17 Complete each inequality with R, S, K, or L. 6. y , 6, so 6 S y 7. b $ 10, so 10 K b 8. r # s 1 t , so s 1 t L r 9. Write an inequality symbol to represent each verbal expression. p is at most 10 u is greater than m z is at least 9 p K 10 u S m z L 9 Chapter 3 66 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. minimum Problem 1 Solving a System by Using a Table Got It? Assume that x and y are whole numbers. What is the solution of the system of inequalities? e x1yS4 3x 1 7y K 21 10. Circle the inequality that has a finite number of whole-number solutions. x1y.4 3x 1 7y # 21 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 11. Use the inequality you circled in Exercise 10 and whole numbers to complete the table of values. x y 0 0, 1, 2, 3 1 0, 1, 2 2 0, 1 , 2 3 0 , 1 4 0, 1 5 0 6 0 7 0 12. Write the ordered pairs of values from the table that satisfy the first inequality. (4, 1) (5, 0) (6, 0) (7, 0) Problem 2 Solving a System by Graphing Got It? What is the solution of the system of inequalities? e x 1 2y K 4 y L 2x 2 1 13. Circle the equivalent system that shows the equations in slope-intercept form. e y $ 20.5x 1 2 y $ 2x 2 1 e 2y # 4 1 x y $ 2x 2 1 e y # 20.5x 1 2 y $ 2x 2 1 67 e y $ 22 1 0.5x y $ 2x 2 1 Lesson 3-3 14. Circle the graph of the solution of the system. y y 4 y 4 4 2 x 2 4 2 4 x O 4 x 4 4 2 2 4 Using a System of Inequalities Problem 3 Got It? A pizza parlor charges $1 for each vegetable topping and $2 for each meat topping. You want at least five toppings on your pizza. You have $10 to spend on toppings. How many of each type of topping can you get on your pizza? 15. Complete the model to write a system of inequalities. 5 cost of vegetable toppings plus number of meat toppings is at least 5 is at most 10 5 cost of meat toppings Let v the number of vegetable toppings. Let m the number of meat toppings. Define 5 Write plus v m v 2m Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Relate number of vegetable toppings 10 16. Circle the system of inequalities that is equivalent to the system in Exercise 15. e v1m$5 2m # 10 1 v e m#52v m$51v e m$52v m # 212 v 1 5 17. Number Sense Circle the types of numbers that can represent the number of toppings you can get on your pizza. rational numbers integers whole numbers real numbers 18. Use the graph of the system at the right. Underline the correct number to complete each sentence. 0 / 5 / 10 meat toppings. If you order 10 vegetable toppings, you can order at most 0 / 5 / 10 meat toppings. Meat Toppings If you order 0 vegetable toppings, you can order at most 10 8 6 4 2 0 0 2 4 6 8 Vegetable Toppings Chapter 3 68 10 Problem 4 Solving a Linear/Absolute-Value System Got It? What is the solution of the system of inequalities? e y R 2 13 x 1 1 y S 2»x 2 1… y 19. The graph at the right shows the boundaries of the system. Shade y , 213 x 1 1 vertically. Shade y . 2u x 2 1 u horizontally. Darken the region of overlap. Then label each inequality. y 1 3 x 1 4 y 2Ux 1U 2 6 4 2 O x 2 6 Lesson Check • Do you UNDERSTAND? Reasoning Is the solution of a system of linear inequalities a union or an intersection of the solutions of the two inequalities? Justify your answer. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Write T for true or F for false. T 20. Systems of inequalities are similar to systems of equations. F 21. The solution of a system of equations is the union of all points on the graphs of the lines. F 22. The solution of a system of inequalities is the union of all points in the graphs of the inequalities. 23. Now answer the question. Answers may vary. Sample: The solution of a system of inequalities is the _______________________________________________________________________ intersection of the solutions of the inequalities. _______________________________________________________________________ Math Success Check off the vocabulary words that you understand. linear inequalities overlap absolute-value system Rate how well you can solve systems of linear inequalities. Need to review 0 2 4 6 8 Now I get it! 10 69 Lesson 3-3 3-4 Linear Programming Vocabulary Review 1. Draw a line from each polygon in Column A to the number of vertices it has in Column B. Column A Column B pentagon 5 quadrilateral 8 6 octagon 4 4 y (4, 21) (3, 5) C B C 2 A 2. Write the letter of each vertex of the quadrilateral at the right. (6, 4) B (1, 2) D O A x 2 D 6 constraint (noun) kun STRAYNT Related Words: constrain, restrict, limit, feasible region Main Idea: The constraints in a linear programming situation form a system of inequalities. The graph of this system is the feasible region and contains all the points that satisfy the constraints. Definition: A constraint is a restriction or limitation. Use Your Vocabulary Complete each statement with a word from the list. Use each word only once. constraint constrain constrained 3. Weight is one 9 on vehicles allowed on the bridge. constrained 4. An injury can cause 9 motion. constrain 5. The rules of a game 9 how you play. Chapter 3 constraint 70 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Linear programming is a method for finding the minimum or maximum value of some quantity, given a set of constraints. The constraints form a system of linear inequalities. The graph of the solutions is the feasible region. Key Concept Vertex Principle of Linear Programming If there is a maximum or a minimum value of a linear objective function, it occurs at one or more vertices of the feasible region. 6. The graph at the right shows a feasible region. Write the coordinates at which a maximum or minimum value of a linear objective function could occur. Q 0 , 0 R Q 3 , 0 R 4 y 2 4 2 O x 2 4 4 Q 3 , 3 R 4 Problem 1 Testing Vertices Got It? Use the graph and the constraints below. What values of x and y in the feasible region maximize P for the objective function P 5 x 1 3y? 7. Label the vertices of the feasible region with their coordinates. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 6 y Constraints: x 1 2y # 5 •x 2 y # 2 x $ 0, y $ 0 4 (0, 2.5) 2 (0, 0) 2 O (3, 1) (2, 0) 4 x 8 4 8. Evaluate P 5 x 1 3y at each vertex. At (0, 0), P 5 x 1 3y 5 0 1 3(0) 50 At (0, 2.5), P 5 x 1 3y 5 0 1 3(2.5) 5 7.5 At (3, 1), P 5 x 1 3y 5 3 1 3(1) 56 At (2, 0), P 5 x 1 3y 5 2 1 3(0) 52 9. P has a maximum value of 7.5 when x 5 0 and y 5 2.5 . 71 Lesson 3-4 Problem 2 Using Linear Programming to Maximize Profit Got It? Business You are screen-printing T-shirts and sweatshirts to sell at the Polk County Blues Festival and are working with the following constraints. • It takes 10 min to make a 1-color T-shirt. • It takes 20 min to make a 3-color sweatshirt. • You have 20 hours at most to make shirts. • Supplies for a T-shirt cost $4. • Supplies for a sweatshirt cost $20. • You want to spend no more than $600 on supplies. • You want to have at least 50 items to sell. The profit on a T-shirt is $6. The profit on a sweatshirt is $20. How many of each type of shirt should you make to maximize your profit? 10. Complete the inequalities that describe the constraints. x constraints 1200 x1y$ 50 4 x 1 20y # 600 x$ 0 y$ 0 11. Circle the objective function that models the situation. P 5 10x 1 y P 5 10x 1 20y P 5 4x 1 20y 12. Shade the feasible region on the graph. y 48 36 24 12 x O 30 60 90 120 13. The vertices of the feasible region are Q 50 Q 100, Chapter 3 , 0R, Q 10 120 , 0R, R , and Q 25, 25 R. 72 P 5 6x 1 20y Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 10x 1 20y # 14. Evaluate P at each vertex. At (50, 0), P 5 6x 1 20y 5 6(50) 1 20(0) 5 300 At (120, 0), P 5 6x 1 20y 5 6(120) 1 20(0) 5 720 At (100, 10), P 5 6x 1 20y 5 6(100) 1 20(10) 5 800 At (25, 25), P 5 6x 1 20y 5 6(25) 1 20(25) 5 650 15. How many of each type of shirt should you make to maximize your profit? T-shirts 5 sweatshirts 5 100 10 Lesson Check • Do you UNDERSTAND? Write a system of constraints whose graphs determine a trapezoid. Write an objective function and evaluate it at each vertex. 16. Write the constraints that produce the feasible region below. 4 y$0 y yR2 2 4 2 2x 1 y R 3 x O 2 4 x1yR3 2 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 4 17. Use the objective function P 5 23x 1 2y . Evaluate the function at each vertex of the feasible region. (–3, 0) (–1, 2) P 5 23x 1 2y 5 23(23) 1 2(0) 59 (1, 2) P 5 23x 1 2y 5 23(21) 1 2(2) 57 P 5 23x 1 2y 5 23(1) 1 2(2) 51 (3, 0) P 5 23x 1 2y 5 23(3) 1 2(0) 5 29 Math Success Check off the vocabulary words that you understand. constraint linear programming objective function feasible region Rate how well you can solve linear programming problems. Need to review 0 2 4 6 8 Now I get it! 10 73 Lesson 3-4 Systems With Three Variables 3-5 Vocabulary Review 1. Circle the number of points that determine a plane. 1 2 3 4 Vocabulary Builder ordered triple (noun) AWR durd TRIP ul Definition: An ordered triple (x, y, z) has three coordinates and describes a point in three-dimensional space. Use Your Vocabulary 2. Circle the ordered triple that has x-coordinate 0. (1, 0, 1) Problem 1 (1, 1, 0) (0, 1, 1) (1, 0, 0) Solving a System Using Elimination Got It? What is the solution of the system at the right? Use elimination. Check your answer in all three original equations. 3. How can you combine A and B to eliminate y? add B to A 4. How can you combine B and C to eliminate y? subtract B from A add B to C 5. Combine A and B to eliminate y. x2 x1 y1 y1 2 x1 0 y1 Chapter 3 A x 2 y 1 z 5 21 B • x 1 y 1 3z 5 23 C 2x 2 y 1 2z 5 0 subtract B from C 6. Combine B and C to eliminate y. z 5 21 3z 5 23 4 z 5 24 74 x1 2x 2 y1 y1 3 x1 0 y1 3z 5 23 2z 5 20 5 z 5 23 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Main Idea: The first coordinate of an ordered triple represents the point’s location along the x-axis, the second coordinate its location along the y-axis, and the third coordinate its location along the z-axis. 7. Use the equations from Exercises 5 and 6 to write and solve a system of two equations in two variables. Answers may vary. Sample: 2x 1 4z 5 24 2x 5 24z 2 4 x 5 22z 2 2 3x 1 5z 5 23 3(22z 2 2) 1 5z 5 23 26z 2 6 1 5z 5 23 2z 5 3 z 5 23 8. Use the solutions from Exercise 7 and ➀ to substitute and solve for y. 2x 1 4z 5 24 2x 1 4(23) 5 24 2x 2 12 5 24 2x 5 8 x54 9. The solution of the original system is Q 4 , 2 , 23 R . x 2 y 1 z 5 21 4 2 y 1 (23) 5 21 1 2 y 5 21 y52 10. Now check your answer in all three original equations. x 2 y 1 z 5 21 x 1 y 1 3z 5 23 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Check: x 2 y 1 z 5 21 4 2 2 1 (23) 0 21 21 5 21 Problem 2 2x 2 y 1 2z 5 0 Check: x 1 y 1 3z 5 23 4 1 2 1 3(23) 0 23 23 5 23 Check: 2x 2 y 1 2z 5 0 2(4) 2 2 1 2(23) 0 0 050 Solving an Equivalent System Got It? What is the solution of the system at the right? Use elimination. 11. First, add a multiple of A to B to eliminate x. Circle the number you will use to multiply A. 21 22 1 A x 2 2y 1 3z 5 12 B • 2x 2 y 2 2z 5 5 C 2x 1 2y 2 z 5 4 2 12. Next, add a multiple of B to C to eliminate x. Circle the number you will use to multiply B. 21 22 1 2 13. Write the two equations you get by eliminating x. 3 y 1 28 z 5 219 3 y1 14. Now find the solution of the two-variable system of equations. 3y 2 8z 5 219 2 (3y 1 z 5 21) 29z 5 218 z52 1 z 5 21 15. Substitute the values of y and z into A to solve for x. 3y 1 z 5 21 3y 1 1(2) 5 21 3y 5 23 y 5 21 x 2 2y 1 3z 5 12 x 2 2(21) 1 3(2) 5 12 x 1 8 5 12 x54 75 Lesson 3-5 Solving a System Using Substitution Problem 3 Got It? What is the solution of the system? Use substitution. A x 2 2y 1 z 5 24 B • 24x 1 y 2 2z 5 1 C 2x 1 2y 2 z 5 10 16. Reasoning Why should you choose A to solve for x? Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect. ✓ The coefficient of x is 1. ✗ The coefficient of x is negative. ✗ The coefficients of x and y are equal. 17. Now solve the system. Solve A for x. Then substitute for x in B and C. Next, solve for y, for z, and for x. x 5 2y 2 z 2 4 , so 24(2y 2 z 2 4) 1 y 2 2z 5 1 28y 1 4z 1 16 1 y 2 2z 5 1 27y 1 2z 5 215 and 2(2y 2 z 2 4) 1 2y 2 z 5 10 4y 2 2z 2 8 1 2y 2 z 5 10 6y 2 3z 5 18 27y 1 2z 5 215 6y 2 3z 5 18 3(27y 1 2z 5 215) 2(6y 2 3z 5 18) 29y 5 29 y51 6y 2 3z 5 18 6(1) 2 3z 5 18 23z 5 12 z 5 24 x 2 2y 1 z 5 24 x 2 2(1) 1 (24) 5 24 x52 Solving a Real-World Problem Problem 4 Got It? Business You manage a clothing store and budget $5400 to restock 200 shirts. You can buy T-shirts for $12 each, polo shirts for $24 each, and rugby shirts for $36 each. If you want to have the same number of T-shirts as polo shirts, how many of each type of shirt should you buy? 19. Use the information in the problem to complete the model below. Relate T-shirts polo shirts T-shirts polo shirts $12 r T-shirts $ 24 r polo shirts Let y the number of polo shirts 1 Write z the number of rugby shirts Chapter 3 $ 2 3 76 36 5 5 x the number of T-shirts Define rugby shirts r rugby shirts 12x 24 200 x y $ 5400 z x y 36z 200 y 5400 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 18. The solution of the system is Q 2 , 1 , 24 R . 20. Circle the method(s) you will use to solve the system. elimination equivalent system substitution 21. Now solve the system. Substitute for y 5 x in A and C. A x 1 y 1 z 5 200 x 1 x 1 z 5 200 2x 1 z 5 200 Solve the system. 2x 1 y 5 200 e x 1 z 5 150 Solve for y and z. x5y 50 5 y 2x 1 z 5 200 2 (x 1 z 5 150) x 5 50 2x 1 z 5 200 2(50) 1 z 5 200 100 1 z 5 200 z 5 100 C 12x 1 24y 1 36z 5 5400 12x 1 24x 1 36z 5 5400 36x 1 36z 5 5400 x 1 z 5  150 22. You should buy 50 T-shirts, 50 100 polo shirts, and rugby shirts. Lesson Check • Do you UNDERSTAND? Writing How many solutions does the system at the right have? Explain your answer in terms of intersecting planes. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 23. Solve the system by adding A and B and then adding C. A 2x 2 3y 1 z 5 5 B • 2x 2 3y 1 z 5 22 C 24x 1 6y 2 2z 5 10 4x 2 6y 1 2z 5 3 24x 1 6y 2 2z 5 10 0 5 13 2x 2 3y 1 z 5 5 2x 2 3y 1 z 5 22 4x 2 6y 1 2z 5 3 24. The system has zero / one / infinitely many solution(s). The graphs of the three equations are intersecting / parallel / perpendicular planes. Math Success Check off the vocabulary words that you understand. linear system elimination substitution Rate how well you can solve systems with three variables. Need to review 0 2 4 6 8 Now I get it! 10 77 Lesson 3-5 3-6 Solving Systems Using Matrices Vocabulary Review 1. Underline the correct word to complete the sentence. The partial solution of the system of equations at the right uses substitution / elimination / equivalent systems. 2x 1 3y 2 z 5 4 2x 1 2y 1 z 5 23 x 1 5y 5 1 Vocabulary Builder rref (noun) 1 0 0 5 Example: C 0 1 0 † 28 S is a matrix in reduced row echelon form representing 0 0 1 4 the solution (5, 28, 4). Use Your Vocabulary 2. Draw a line from each rref matrix in Column A to the solution it represents in Column B. Column A Column B 1 0 0 2 C 0 1 0 † 0 S 0 0 1 3 (0, 2, 3) 1 0 0 0 C 0 1 0 † 2 S 0 0 1 3 (2, 0, 3) 1 0 0 2 C 0 1 0 † 3 S 0 0 1 0 (2, 3, 0) Chapter 3 78 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Definition: The rref (reduced row echelon form) function on a calculator generates the matrix that represents the solution of a system of equations. Problem 1 Identifying a Matrix Element Got It? What is element a13 in matrix A? 4 29 17  1 ` A5 C 0 5 8 `  6 S ` 23 22 10  0 3. Underline the correct words to complete the sentence. The matrix has 3 rows / columns and 4 rows / columns . 4. The element a13 is in row 1 and column 3 . 5. Use the matrix below. Circle element a13 . 4 29 17  1 ` A5 C 0 5 8 `  6 S ` 23 22 10  0 Problem 2 Representing Systems With Matrices Got It? How can you represent the system of equations at the right with a matrix? b 24x 2 2y 5 7 3x 1 y 5 25 6. How many rows and columns will the matrix have? number of rows 5 2 number of columns 5 3 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 7. Write the matrix. 24 22 7 3 1 25 Problem 3 Writing a System from a Matrix Got It? What linear system does B 2 0  6 `` R represent? 5 22  1 8. Underline the correct numbers to complete the sentence. The matrix represents a system of 2 / 3 / 4 equations in 2 / 3 / 4 variables. 9. Complete the system of equations. 2 x1 0 y5 6 5 x 1 22 y 5 1 79 Lesson 3-6 Key Concept Row Operations 10. Use the row operation indicated to complete each matrix. Switch any two rows. Switch Rows 1 and 2. B 4 5 3 R becomes 3 2 6 3 2 6 4 5 3 Multiply a row by a constant. Multiply Row 2 by 3. B 4 5 3 R becomes 3 2 6 4 3 5 4 5 3 9 6 18 5 3? 3 3? 3? 2 6 Add one row to another. Add Row 2 to Row 1. B 4 5 3 R becomes 3 2 6 413 51 2 3 3 1 2 6 7 7 9 3 2 6 5 6 You can combine any of these steps to solve a system using a matrix. Problem 4 Solving a System Using a Matrix 9x 2 2y 5 5 3x 1 7y 5 17 11. The system is solved below. Write a justification for each step. B B e Write the matrix for the system. 9 22 5  `  R 3 7 17 Multiply Row 2 by 23. Add to Row 1. 9 022 205) 23(3 207 217) 0 223 246) 5 9 22  `  R 0 223 246 Replace Row 2 by the sum. 1 (0 223 246) 223 1 . Multiply Row 2 by 223 0 201 202) B 9 22 5  `  R 0 1 2 Replace Row 1 by the sum. Multiply Row 1 by 19 . 1 9 (9 0 9) 9 0 9 B  `  R 0 1 2 B Multiply Row 2 by 2. Add to Row 1. 9 22 5) 2(0 21 2) 9 20 9) 1 0 1) Replace Row 1. 1 0 1  `  R 0 1 2 12. Circle the solution of the system. (1, 0) Chapter 3 (0, 1) (1, 2) 80 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Got It? What is the solution of the system of equations? Problem 5 Using a Calculator to Solve a Linear System Got It? What is the solution of the system of equations? 13. Circle the matrix that models the system. 1 4 6 21 C 2 2 1 † 4 S 0 8 1 21 1 4 6 21 C 2 22 1 † 4 S 0 28 1 21 a 1 4b 1 6c 5 21 • 2a 2 2b 1 c 5 4 28b 1 c 5 21 1 2 0 21 C 4 22 28 † 4 S 6 1 1 21 14. Use the rref() function on your calculator to find the solution. a5 1 b5 1 2 c5 3 15. Check the solution. a 1 4b 1 6c 5 21 2a 2 2b 1 c 5 4 Check: 1 1 4(12) 1 6(3) 0 21 1 1 2 1 18 5 21 21 5 21 28b 1 c 5 21 Check: 2(1) 2 2(12) 1 3 0 4 2211354 454 Check: 28(12) 1 3 0 21 24 1 3 5 21 21 5 21 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Lesson Check • Do you UNDERSTAND? How many elements are in a 4 3 4 matrix? 16. There are 4 rows, and each row has 4 elements. 17. A 4 3 4 matrix has 4 3 4 , or 16 elements. Math Success Check off the vocabulary words that you understand. matrix matrix element row operation Rate how well you can use matrices to solve systems of equations. Need to review 0 2 4 6 8 Now I get it! 10 81 Lesson 3-6
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