3-1 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slope-intercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee ur SIS tum Related Words: independent sstem, dependent sstem linear sstem 5 7 1 9 Definition: A linear sstem is a collection of linear equations involving the same set of variables. The sstem above is two equations in two variables. Use Your Vocabular The graphs below show the possible tpes of solutions for a sstem of two equations in two variables. Write T for true or F for false. Intersecting Lines F T F O one solution Consistent Independent Coinciding Lines O infinitel man solutions Consistent Dependent. Inconsistent linear sstems intersect at two points. 3. An independent linear sstem has one solution.. A dependent linear sstem has no solutions. Parallel Lines O no solution Inconsistent T 5. Two unique lines with the same slope form an inconsistent sstem. Chapter 3 58
Problem 1 Using a Graph or Table to Solve a Sstem 3 5 Got It? What is the solution of the sstem? e 3 1 5 5 6. Circle the graph of the equations. O O O 7. Circle the row in the calculator screen that contains the solution of the sstem. X 0 1 3 Y1 1.5 1.5 0 Y 5 1 7 8. The solution of the sstem of equations is Q, 1 R. Problem Using a Table to Solve a Problem Got It? Biolog The equation 1 5 1.5 1 models the length, in centimeters, of a Spin Dogfish shark ears old. The equation 5 0.75 1 37 models the length, in centimeters, of a Greenland shark ears old. If the growth rates continue, how long will each shark be when it is 5 ears old? 9. Reasoning Which shark will be longer at age 5? How do ou know? Answers ma var. Sample: The Spin Dogfish has a faster growth rate, so it will be longer at age 5. 10. Underline the correct phrase to complete the sentence. To solve the problem, I need to find an -value / a -value / two -values / two -values. 11. Complete the table for 5 5. 5 1 1.5 1 0.75 37 59.5 55.75 59 Lesson 3-1
1. Spin Dogfish sharks will be 59.5 cm long and Greenland sharks will be 55.75 cm long when the are 5 ears 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 000) San Diego Detroit 1950 33,387 1,89,568 1960 573, 1,670,1 1970 696,769 1,511,8 1980 875,538 1,03,339 1990 1,110,59 1,07,97 000 1,3,00 951,70 SOURCE: U.S. Census Bureau 13. Circle the first calculator step in solving the problem. Calculate the intersection. Enter the data into lists. Enter 1. 1. Write the name of the list ou will use (L 1, L, or L 3 ) net to the data tpe. L L population of San Diego 3 L population of Detroit 1 ears since 1950 15. Circle the pair of equations ou will graph. 1 5 17816.597 c 1 356896.38 c 51917.551 c 1 18901.619 c 1 517816.597 c 1 356896.38 c 5 1917.551 c 1 18901.619 c 16. Cross out the graph that does NOT show the regression lines. 17. Underline the correct word to complete each sentence. The -ais / -ais corresponds to the number of ears since 1950. The -ais / -ais corresponds to the populations of San Diego and Detroit. 18. The scale of the graphs is 60 # # 60 b 10s and 0 # # 1,500,000 b 100,000s. Use the scale to estimate the coordinates of the point of intersection. Accept reasonable estimates. Eact answers are given. The coordinates are Q 0.3, 1,07,919 R. The populations of San Diego and Detroit were equal sometime during the ear 1990. The population was about 1,075,000. Chapter 3 60
Problem Classifing a Sstem Without Graphing Got It? Without graphing, is the sstem e 3 1 5 1 3 5 1 independent, dependent, or inconsistent? 19. Write each equation in slope-intercept form. 3 1 5 5 3 1 1 3 5 1 5 3 3 0. The slope of 3 1 5 is 3 and the slope of 1 3 5 1 is 3. 1. The -intercept of 3 1 5 is and the -intercept of 1 3 5 1 is 3.. Underline the correct words to complete the sentence. Because the slopes of the lines are equal / not equal and the -intercepts are the same / different, the sstem is inconsistent / independent / dependent. Lesson Check Do ou UNDERSTAND? Vocabular Is it possible for a sstem of equations to be both independent and inconsistent? Eplain. Write T for true or F for false. T T 3. The graphs of an inconsistent sstem are parallel.. The graphs of an independent sstem intersect at one point. 5. Now answer the question. Answers ma var. Sample: No; inconsistent lines are parallel and never intersect. Independent lines have one point of intersection. Therefore, a sstem cannot be both independent and inconsistent. Math Success Check off the vocabular words that ou understand. sstem of equations dependent independent consistent inconsistent Rate how well ou can solve a linear sstem using a graph or table. Need to review 0 6 8 10 Now I get it! 61 Lesson 3-1
3- Solving Sstems Algebraicall Vocabular Review 1. Circle the equations that are in standard form. 1 3 5 5 3 5 3 5 1 5 5 0 Write each equation in standard form.. 5 5 3 3. 5 6. 1 3 1 5 0 5 5 3 6 5 3 5 1 Vocabular Builder solution (noun) suh LOO shun Main Idea: If two numbers are substituted for and in a sstem of equations and the make both equations true, then the ordered pair (, ) is a solution of the sstem. Definition: A solution is an ordered pair that makes an equation in two variables true. Use Your Vocabular 5. Write three ordered pairs that are solutions of the equation 55. Answers ma var. Check students work. Q 0, R Q 5, 3 R Q 5, 7R Samples are given. Use the sstem at the right. Write T for true or F for false. F F T F 6. The sstem has a unique solution. 7. The sstem has infinitel man solutions. 8. The sstem has no solution. 9. The solution is (0, 0). e 1 5 3 1 5 (, 1) is a solution of 3 5 because 1 3() 5. T 10. The solution of a sstem can be found b graphing the equations of the sstem. Chapter 3 6
Problem 1 Solving b Substitution Got It? What is the solution of the sstem of equations? e 1 3 5 5 55 11. Follow the steps to find the solution. 1 Solve the first equation for. 3 5 3 5 Substitute the epression for in the second equation. Then solve for. 3 5 5 5 6 10 5 5.5 3 Substitute the value for in either equation. Solve for. 3 5 3.5 5 7.5 5.5 1. The solution of the sstem is Q.5,.5 R. Problem Using Substitution to Solve a Problem Got It? Music An online music compan offers 15 downloads for $19.75 and 0 downloads for $3.50. 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 sstem of equations. Relate Define Write total cost is number of downloads times cost of one download plus Let c the cost of one download and let r the registration fee. registration fee $19.75 15 c r $ 3.50 0 c r 63 Lesson 3-
1. Circle the equation that epresses r in terms of c in the first equation. r 515c 1 19.75 r 5 15c 1 19.75 r 515c 19.75 15. Substitute the equation ou chose in Eercise 1 into the second equation of the sstem and solve for c. 3.5 5 0c 1 r Write the original equation. 3.5 5 0c 1 15 c 1 19.75 Substitute for r. 3.5 5 5 c 1 19.75 Simplif. 3.75 5 5 c Use the Addition Propert of Equalit. 0.95 5 c Divide. 16. Now substitute the value of c into one of the equations of the sstem and solve for r. Answers ma var. Sample: r 515c 1 19.75 r 515(0.95) 1 19.75 r 51.5 1 19.75 r 5 5.5 17. The cost of each download is $.95 and the registration fee is $ 5.50. Problem 3 Solving b Elimination Got It? What is the solution of the sstem of equations? Problem Solving an Equivalent Sstem Got It? What is the solution of the sstem of equations? 1. Underline the correct values to complete the sentence. 3 1 7 5 15 e 5 1 5 To get additive inverses for the -term, multipl the first equation b / 3 / 5 and the second equation b / 3 / 7. 1 8 58 e 5 8 5 0 18. Add the equations. 19. Now choose one of the original equations. Substitute and solve. 1 8 5 8 5 8 5 0 3 1 0 5 1 5 Simplif. 0. Circle the ordered pair that is the solution of the sstem of equations. (, ) (, ) (, 0) Answers ma var. Sample: () 1 8 58 8 1 8 58 8 5 0 5 0 Chapter 3 6
. Circle the equivalent sstem that shows additive inverses for the -term. 3 1 7 5 15 e 5 1 5 6 1 1 5 3c e 35 1 5 8 15 1 35 5 75 e 15 6 5 1 3. Solve the sstem for.. Then substitute and solve for. 15 1 35 5 75 15 6 5 1 9 5 87 5 3 5. The solution of the sstem is Q, 3 R. 5 1 5 5 1 (3) 5 5 510 5 Problem 5 Solving Sstems Without Unique Solutions 1 5 Got It? What is the solution of the sstem of equations? Eplain. e 5 0 6. Circle the first step in solving the sstem. Multipl 1 5 b 1. Multipl 1 5 b. 7. Add 1 5 and 5 0. 8. What is the solution of the sstem? 1 5 5 0 0 5 Place a in the bo if the response is correct. Place an if it is incorrect. The sstem has no solution. The sstem has infinitel man solutions. Lesson Check Math Success Need to review Do ou UNDERSTAND? Vocabular Give an eample of two equivalent sstems. 9. Cross out the sstem of equations that is NOT equivalent to the others. 1 5 5 13 e 5 3 1 5 5 1 e 5 3 Check off the vocabular words that ou understand. 0 6 8 10 Now I get it! 8 1 0 5 96 e 8 5 6 substitution elimination equivalent equation unique solutions Rate how well ou can solve linear sstems algebraicall. 65 Lesson 3-
3-3 Sstems of Inequalities Vocabular Review Complete each statement with the correct word or phrase from the list below. Use each word or phrase onl once. minimum greater than or equal to at least at most 1. The Florida football team needs 9 two more wins to clinch the division title.. The 9 speed a car ma travel on the Florida freewa is 0 miles per hour. 3. If ou have $5 and sweaters sell for $0 each, ou can bu 9 two sweaters.. The height of a rider must be 9 inches in order to ride the Summit Plummet. at least minimum at most greater than or equal to Vocabular Builder inequalit smbols inequalit (noun) in ee KWAL uh tee,,,, Definition: An inequalit is a mathematical statement indicating that one quantit is less than or less than or equal to a second quantit. Use Your Vocabular 5. Place a net to the math statements that are inequalities. Place an net to the math statements that are not. 15. 3 # r s p 5 t 17 Complete each inequalit with R, S, K, or L. 6., 6, so 6 S 7. b $ 10, so 10 K b 8. r # s 1 t, so s 1 t L r 9. Write an inequalit smbol to represent each verbal epression. 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
Problem 1 Solving a Sstem b Using a Table Got It? Assume that and are whole numbers. What is the solution of the sstem of inequalities? e 1 S 3 1 7 K 1 10. Circle the inequalit that has a finite number of whole-number solutions. 1. 3 1 7 # 1 11. Use the inequalit ou circled in Eercise 10 and whole numbers to complete the table of values. 0 0, 1,, 3 1 0, 1, 0, 1, 3 0, 1 0, 1 5 0 6 0 7 0 1. Write the ordered pairs of values from the table that satisf the first inequalit. Problem Solving a Sstem b Graphing Got It? What is the solution of the sstem of inequalities? e 1 K L 1 13. Circle the equivalent sstem that shows the equations in slope-intercept form. e $0.5 1 $ 1 e # 1 $ 1 (, 1) (5, 0) (6, 0) (7, 0) e #0.5 1 $ 1 $ 1 0.5 e $ 1 67 Lesson 3-3
1. Circle the graph of the solution of the sstem. O Problem 3 Using a Sstem of Inequalities Got It? A pizza parlor charges $1 for each vegetable topping and $ for each meat topping. You want at least five toppings on our pizza. You have $10 to spend on toppings. How man of each tpe of topping can ou get on our pizza? 15. Complete the model to write a sstem of inequalities. Relate Define Write 5 5 number of vegetable toppings cost of vegetable toppings Let v the number of vegetable toppings. Let m the number of meat toppings. v plus plus number of meat toppings cost of meat toppings m v m 10 Meat Toppings 10 8 6 is at least is at most 16. Circle the sstem of inequalities that is equivalent to the sstem in Eercise 15. e v 1 m $ 5 m # 10 1 v e m # 5 v m $ 5 1 v e m $ 5 v 1 m # v 1 5 17. Number Sense Circle the tpes of numbers that can represent the number of toppings ou can get on our pizza. rational numbers integers whole numbers real numbers 18. Use the graph of the sstem at the right. Underline the correct number to complete each sentence. If ou order 0 vegetable toppings, ou can order at most 0 / 5 / 10 meat toppings. If ou order 10 vegetable toppings, ou can order at most 0 / 5 / 10 meat toppings. 5 10 5 0 0 6 8 10 Vegetable Toppings Chapter 3 68
Problem Solving a Linear/Absolute-Value Sstem Got It? What is the solution of the sstem of inequalities? 19. The graph at the right shows the boundaries 1 of the sstem. Shade,3 1 1 verticall. Shade. u 1 u horizontall. Darken the region of overlap. Then label each inequalit. e R 1 3 1 1 S» 1 1 1 3 1 6 O 6 Lesson Check Do ou UNDERSTAND? Reasoning Is the solution of a sstem of linear inequalities a union or an intersection of the solutions of the two inequalities? Justif our answer. Write T for true or F for false. T F 0. Sstems of inequalities are similar to sstems of equations. 1. The solution of a sstem of equations is the union of all points on the graphs of the lines. F Math Success Need to review. The solution of a sstem of inequalities is the union of all points in the graphs of the inequalities. 3. Now answer the question. Answers ma var. Sample: The solution of a sstem of inequalities is the intersection of the solutions of the inequalities. Check off the vocabular words that ou understand. linear inequalities overlap absolute-value sstem Rate how well ou can solve sstems of linear inequalities. 0 6 8 10 Now I get it! 69 Lesson 3-3
3- Linear Programming Vocabular Review 1. Draw a line from each polgon in Column A to the number of vertices it has in Column B. Column A pentagon 5 quadrilateral 8 octagon Column B. Write the letter of each verte of the quadrilateral at the right. (6, ) (3, 5) (, 1) (1, ) C B D A 6 B C A O 6 D Vocabular Builder constraint (noun) kun STRAYNT Related Words: constrain, restrict, limit, feasible region Main Idea: The constraints in a linear programming situation form a sstem of inequalities. The graph of this sstem is the feasible region and contains all the points that satisf the constraints. Definition: A constraint is a restriction or limitation. Use Your Vocabular Complete each statement with a word from the list. Use each word onl once. constraint constrain constrained 3. Weight is one 9 on vehicles allowed on the bridge.. An injur can cause 9 motion. constraint constrained 5. The rules of a game 9 how ou pla. constrain Chapter 3 70
Linear programming is a method for finding the minimum or maimum value of some quantit, given a set of constraints. The constraints form a sstem of linear inequalities. The graph of the solutions is the feasible region. Ke Concept Verte Principle of Linear Programming If there is a maimum 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 maimum or minimum value of a linear objective function could occur. Q 0, 0 R Q 3, 0 R Q 3, 3 R O Problem 1 Testing Vertices Got It? Use the graph and the constraints below. What values of and in the feasible region maimize P for the objective function P 5 1 3? 7. Label the vertices of the feasible region with their coordinates. 6 (0,.5) (0, 0) (3, 1) O (, 0) 8 8. Evaluate P 5 1 3 at each verte. At (0, 0), P 5 1 3 5 0 1 3(0) 5 0 At (0,.5), P 5 1 3 5 0 1 3(.5) 5 7.5 Constraints: 1 # 5 # $ 0, $ 0 At (3, 1), P 5 1 3 5 3 1 3(1) 5 6 At (, 0), P 5 1 3 5 1 3(0) 5 9. P has a maimum value of 7.5 when 5 0 and 5.5. 71 Lesson 3-
Problem Using Linear Programming to Maimize Profit Got It? Business You are screen-printing T-shirts and sweatshirts to sell at the Polk Count Blues Festival and are working with the following constraints. It takes 10 min to make a 1-color T-shirt. It takes 0 min to make a 3-color sweatshirt. You have 0 hours at most to make shirts. Supplies for a T-shirt cost $. Supplies for a sweatshirt cost $0. 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 $0. How man of each tpe of shirt should ou make to maimize our profit? 10. Complete the inequalities that describe the constraints. 10 1 0 # 1 $ 100 50 constraints 1 0 # 600 $ $ 0 0 11. Circle the objective function that models the situation. 8 36 1 P 5 10 1 P 5 10 1 0 P 5 1 0 P 5 6 1 0 1. Shade the feasible region on the graph. O 30 60 90 10 13. The vertices of the feasible region are Q 50, 0R, Q 10, 0R, Q100, 10 R, and Q5, 5 R. Chapter 3 7
1. Evaluate P at each verte. At (50, 0), P 5 6 1 0 5 6(50) 1 0(0) 5 300 At (10, 0), P 5 6 1 0 5 6(10) 1 0(0) 5 70 At (100, 10), P 5 6 1 0 5 6(100) 1 0(10) 5 800 At (5, 5), P 5 6 1 0 5 6(5) 1 0(5) 5 650 15. How man of each tpe of shirt should ou make to maimize our profit? T-shirts 5 100 sweatshirts 5 10 Lesson Check Do ou UNDERSTAND? Write a sstem of constraints whose graphs determine a trapezoid. Write an objective function and evaluate it at each verte. 16. Write the constraints that produce the feasible region below. O $ 0 R 1 R 3 1 R 3 17. Use the objective function P 53 1. Evaluate the function at each verte of the feasible region. ( 3, 0) ( 1, ) (1, ) (3, 0) Math Success Check off the vocabular words that ou understand. constraint linear programming objective function feasible region Rate how well ou can solve linear programming problems. Need to review P 53 1 53(3) 1 (0) 5 9 P 53 1 53(1) 1 () 5 7 0 6 8 10 P 53 1 53(1) 1 () 5 1 Now I get it! P 53 1 53(3) 1 (0) 59 73 Lesson 3-
3-5 Sstems With Three Variables Vocabular Review 1. Circle the number of points that determine a plane. 1 3 Vocabular Builder ordered triple (noun) AWR durd TRIP ul Definition: An ordered triple (,, z) has three coordinates and describes a point in three-dimensional space. Main Idea: The first coordinate of an ordered triple represents the point s location along the -ais, the second coordinate its location along the -ais, and the third coordinate its location along the z-ais. Use Your Vocabular. Circle the ordered triple that has -coordinate 0. Problem 1 (1, 0, 1) (1, 1, 0) (0, 1, 1) (1, 0, 0) Solving a Sstem Using Elimination Got It? What is the solution of the sstem at the right? Use elimination. Check our answer in all three original equations. 3. How can ou combine A and B to eliminate?. How can ou combine B and C to eliminate? add B to A subtract B from A add B to C subtract B from C 5. Combine A and B to eliminate. 6. Combine B and C to eliminate. 1 z 51 1 1 3z 53 A B C 1 1 3z 53 1 z 5 0 1 z 51 1 1 3z 53 1 z 5 0 1 0 1 z 5 3 1 0 1 5 z 5 3 Chapter 3 7
7. Use the equations from Eercises 5 and 6 to write and solve a sstem of two equations in two variables. Answers ma var. Sample: 1 z 5 5z 5z 3 1 5z 53 3(z ) 1 5z 53 6z 6 1 5z 53 z 5 3 z 53 1 z 5 1 (3) 5 1 5 5 8 5 8. Use the solutions from Eercise 7 and 9. The solution of the original sstem is ➀ to substitute and solve for. Q,, 3 R. 1 z 51 1 (3) 51 1 51 5 10. Now check our answer in all three original equations. 1 z 51 1 1 3z 53 1 z 5 0 Check: 1 z 51 1 (3) 0 1 1 51 Check: 1 1 3z 53 1 1 3(3) 0 3 3 53 Check: 1 z 5 0 () 1 (3) 0 0 0 5 0 Problem Solving an Equivalent Sstem Got It? What is the solution of the sstem at the right? Use elimination. 11. First, add a multiple of A to B to eliminate. Circle the number ou will use to multipl A. 1 1 1. Net, add a multiple of B to C to eliminate. Circle the number ou will use to multipl B. 1 1 13. Write the two equations ou get b eliminating. 3 1 8 z 5 19 3 1 1 z 5 1 1. Now find the solution of the two-variable 15. Substitute the values of and z into sstem of equations. A to solve for. 3 8z 519 (3 1 z 5 1) 9z 518 z 5 3 1 z 51 3 1 1() 51 3 53 51 A B C 1 3z 5 1 (1) 1 3() 5 1 1 8 5 1 5 1 3z 5 1 z 5 5 1 z 5 75 Lesson 3-5
Problem 3 Solving a Sstem Using Substitution Got It? What is the solution of the sstem? Use substitution. 16. Reasoning Wh should ou choose A to solve for? Place a in the bo if the response is correct. Place an if it is incorrect. A B C 1 z 5 1 z 5 1 1 z 5 10 The coefficient of is 1. The coefficient of is negative. The coefficients of and are equal. 17. Now solve the sstem. Solve A for. Then substitute for in B and C. Net, solve for, for z, and for. 5 z, so ( z ) 1 z 5 1 8 1 z 1 16 1 z 5 1 7 1 z 515 and ( z ) 1 z 5 10 z 8 1 z 5 10 6 3z 5 18 7 1 z 515 6 3z 5 18 3(7 1 z 515) (6 3z 5 18) 9 5 9 5 1 6 3z 5 18 6(1) 3z 5 18 3z 5 1 z 5 1 z 5 (1) 1 () 5 5 18. The solution of the sstem is Q, 1, R. Problem Solving a Real-World Problem Got It? Business You manage a clothing store and budget $500 to restock 00 shirts. You can bu T-shirts for $1 each, polo shirts for $ each, and rugb shirts for $36 each. If ou want to have the same number of T-shirts as polo shirts, how man of each tpe of shirt should ou bu? 19. Use the information in the problem to complete the model below. Relate Define $1 Let 5 z T-shirts T-shirts T-shirts the number of T-shirts polo shirts polo shirts rugb shirts 00 $ polo shirts $ 36 rugb shirts $ 500 the number of polo shirts the number of rugb shirts Write 1 3 5 z 00 1 36z 500 Chapter 3 76
0. Circle the method(s) ou will use to solve the sstem. 1. Now solve the sstem. elimination equivalent sstem substitution Substitute for 5 in A and C. A 1 1 z 5 00 1 1 z 5 00 1 z 5 00 C 1 1 1 36z 5 500 1 1 1 36z 5 500 36 1 36z 5 500 1 z 5 150 Solve the sstem. 1 5 00 e 1 z 5 150 1 z 5 00 ( 1 z 5 150) 5 50 Solve for and z. 5 50 5 1 z 5 00 (50) 1 z 5 00 100 1 z 5 00 z 5 100. You should bu 50 T-shirts, 50 polo shirts, and 100 rugb shirts. Lesson Check Do ou UNDERSTAND? Writing How man solutions does the sstem at the right have? Eplain our answer in terms of intersecting planes. 3. Solve the sstem b adding A and B and then adding C. A B C 3 1 z 5 5 3 1 z 5 1 6 z 5 10. The sstem has zero / one / infinitel man solution(s). The graphs of the three equations are intersecting / parallel / perpendicular planes. Math Success Check off the vocabular words that ou understand. linear sstem elimination substitution Rate how well ou can solve sstems with three variables. Need to review 3 1 z 5 5 3 1 z 5 6 1 z 5 3 0 6 8 10 6 1 z 5 3 1 6 z 5 10 0 5 13 Now I get it! 77 Lesson 3-5
3-6 Solving Sstems Using Matrices Vocabular Review 1. Underline the correct word to complete the sentence. The partial solution of the sstem of equations at the right uses substitution / elimination / equivalent sstems. 1 3 z 5 1 1 z 5 3 1 5 5 1 Vocabular Builder rref (noun) Definition: The rref (reduced row echelon form) function on a calculator generates the matri that represents the solution of a sstem of equations. 1 0 0 5 Eample: C0 1 0 8 S is a matri in reduced row echelon form representing 0 0 1 the solution (5, 8, ). Use Your Vocabular. Draw a line from each rref matri in Column A to the solution it represents in Column B. Column A 1 0 0 C0 1 0 0 0 1 1 0 0 C0 1 0 0 0 1 1 0 0 C0 1 0 0 0 1 Column B 0S (0,, 3) 3 0 S (, 0, 3) 3 3S (, 3, 0) 0 Chapter 3 78
Problem 1 Identifing a Matri Element Got It? What is element a 13 in matri A? 9 17 1 A 5 C 0 5 8 6 3 10 ` S 0 3. Underline the correct words to complete the sentence. The matri has 3 rows / columns and rows / columns.. The element a 13 is in row 1 and column 3. 5. Use the matri below. Circle element a 13. 9 17 1 A 5 C 0 5 8 6 3 10 ` S 0 Problem Representing Sstems With Matrices Got It? How can ou represent the sstem of equations at the right with a matri? 6. How man rows and columns will the matri have? number of rows 5 number of columns 5 3 5 7 b 3 1 55 7. Write the matri. 7 3 1 5 Problem 3 Writing a Sstem from a Matri Got It? What linear sstem does B 0 6 5 ` 1 R represent? 8. Underline the correct numbers to complete the sentence. The matri represents a sstem of / 3 / equations in / 3 / variables. 9. Complete the sstem of equations. 1 0 5 6 5 1 5 1 79 Lesson 3-6
10. Use the row operation indicated to complete each matri. Switch an two rows. Ke Concept Row Operations Switch Rows 1 and. B 5 3 3 6 R becomes 3 5 6 3 Multipl a row b a constant. Multipl Row b 3. B 5 3 3 6 R becomes 5 3? 3 3? 3? 3 6 5 9 5 6 3 18 Add one row to another. Add Row to Row 1. B 5 3 1 3 5 1 3 1 6 3 6 R becomes 3 6 5 7 3 7 9 6 You can combine an of these steps to solve a sstem using a matri. Problem Solving a Sstem Using a Matri 9 5 5 Got It? What is the solution of the sstem of equations? e 3 1 7 5 17 11. The sstem is solved below. Write a justification for each step. B 9 3 7 ` 5 17 R 9 0 05) 3(3 07 17) 0 3 6) B 9 0 3 ` 5 6 R 1 3 (0 3 6) 0 01 0) B 9 0 1 ` 5 R 9 5) (0 1 ) 9 0 9) B 9 0 0 1 ` 9 1 R 9 (9 0 9) 1 0 1) B 1 0 0 1 ` 1 R 1. Circle the solution of the sstem. Write the matri for the sstem. Multipl Row b 3. Add to Row 1. Replace Row b the sum. Multipl Row b 1 3. Multipl Row b. Add to Row 1. Replace Row 1 b the sum. Multipl Row 1 b 1 9. Replace Row 1. (1, 0) (0, 1) (1, ) Chapter 3 80
Problem 5 Using a Calculator to Solve a Linear Sstem Got It? What is the solution of the sstem of equations? 13. Circle the matri that models the sstem. a 1 b 1 6c 5 1 a b 1 c 5 8b 1 c 51 1 6 C 1 0 8 1 1 S 1 1 6 C 1 0 8 1 1 S 1 1 0 C 8 6 1 1 1 S 1 1. Use the rref() function on our calculator to find the solution. 1 a 5 1 b 5 c 5 3 15. Check the solution. a 1 b 1 6c 5 1 a b 1 c 5 8b 1 c 51 Check: 1 1 ( 1 ) 1 6(3) 0 1 1 1 1 18 5 1 1 5 1 Check: (1) ( 1 ) 1 3 0 1 1 3 5 5 Check: 8( 1 ) 1 3 0 1 1 3 51 1 51 Lesson Check Do ou UNDERSTAND? How man elements are in a 3 matri? 16. There are rows, and each row has elements. 17. A 3 matri has 3, or 16 elements. Math Success Check off the vocabular words that ou understand. matri matri element row operation Rate how well ou can use matrices to solve sstems of equations. Need to review 0 6 8 10 Now I get it! 81 Lesson 3-6