Solve Linear Systems Algebraically

3.2 Solve Linear Systems Algebraically Before You solved linear systems graphically. Now You will solve linear systems algebraically. Why? So you can model guitar sales, as in E. 55. Key Vocabulary substitution method elimination method In this lesson, you will study two algebraic methods for solving linear systems. The first method is called the substitution method. KEY CONCEPT For Your Notebook The Substitution Method STEP STEP 2 STEP 3 Solve one of the equations for one of its variables. Substitute the epression from Step into the other equation and solve for the other variable. Substitute the value from Step 2 into the revised equation from Step and solve. E XAMPLE Use the substitution method Solve the system using the substitution method. 2 5y 5 25 3y 5 3 Equation Equation 2 Solution STEP Solve Equation 2 for. 5 23y 3 Revised Equation 2 STEP 2 Substitute the epression for into Equation and solve for y. 2 5y 5 25 Write Equation. 2(23y 3) 5y 5 25 Substitute 23y 3 for. y 5 Solve for y. STEP 3 Substitute the value of y into revised Equation 2 and solve for. 5 23y 3 Write revised Equation 2. 5 23() 3 Substitute for y. 5 230 Simplify. c The solution is (230, ). CHECK Check the solution by substituting into the original equations. 2(230) 5() 0 25 Substitute for and y. 230 3() 0 3 25 5 25 Solution checks. 3 5 3 60 Chapter 3 Linear Systems and Matrices

ELIMINATION METHOD Another algebraic method that you can use to solve a system of equations is the elimination method. The goal of this method is to eliminate one of the variables by adding equations. KEY CONCEPT For Your Notebook The Elimination Method STEP STEP 2 STEP 3 Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables. Add the revised equations from Step. Combining like terms will eliminate one of the variables. Solve for the remaining variable. Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. E XAMPLE 2 Use the elimination method Solve the system using the elimination method. 3 2 7y 5 0 6 2 8y 5 8 Equation Equation 2 Solution STEP Multiply Equation by 22 so that the coefficients of differ only in sign. SOLVE SYSTEMS In Eample 2, one coefficient of is a multiple of the other. In this case, it is easier to eliminate the -terms because you need to multiply only one equation by a constant. 3 2 7y 5 0 3 22 26 4y 5 220 6 2 8y 5 8 6 2 8y 5 8 STEP 2 Add the revised equations and solve for y. 6y 5 22 y 5 22 STEP 3 Substitute the value of y into one of the original equations. Solve for. 3 2 7y 5 0 Write Equation. 3 2 7(22) 5 0 Substitute 22 for y. 3 4 5 0 Simplify. 5 2 4 } 3 Solve for. c The solution is 2 4 } 3, 22 2. CHECK You can check the solution algebraically using the method shown in Eample. You can also use a graphing calculator to check the solution. at classzone.com Intersection X=-.333333 Y=-2 GUIDED PRACTICE for Eamples and 2 Solve the system using the substitution or the elimination method.. 4 3y 5 22 2. 3 3y 5 25 3. 3 2 6y 5 9 5y 5 29 5 2 9y 5 3 24 7y 5 26 3.2 Solve Linear Systems Algebraically 6

E XAMPLE 3 Standardized Test Practice To raise money for new football uniforms, your school sells silk-screened T-shirts. Short sleeve T-shirts cost the school $5 each and are sold for $8 each. Long sleeve T-shirts cost the school $7 each and are sold for $2 each. The school spends a total of $2500 on T-shirts and sells all of them for $4200. How many of the short sleeve T-shirts are sold? A 50 B 00 C 50 D 250 Solution STEP Write verbal models for this situation. Equation Short sleeve cost (dollars/shirt) p Short sleeve shirts (shirts) Long sleeve cost (dollars/shirt) p Long sleeve shirts (shirts) 5 Total cost (dollars) Equation 2 5 p 7 p y 5 2500 Short sleeve selling price (dollars/shirt) p Short sleeve shirts (shirts) Long sleeve selling price (dollars/shirt) p Long sleeve shirts (shirts) 5 Total revenue (dollars) 8 p 2 p y 5 4200 STEP 2 Write a system of equations. Equation 5 7y 5 2500 Total cost for all T-shirts Equation 2 8 2y 5 4200 Total revenue from all T-shirts sold STEP 3 Solve the system using the elimination method. Multiply Equation by 28 and Equation 2 by 5 so that the coefficients of differ only in sign. 5 7y 5 2500 3 28 240 2 56y 5 220,000 8 2y 5 4200 3 5 40 60y 5 2,000 AVOID ERRORS Choice D gives the number of long sleeve T-shirts, but the question asks for the number of short sleeve T-shirts. So you still need to solve for in Step 3. Add the revised equations and solve for y. 4y 5 000 y 5 250 Substitute the value of y into one of the original equations and solve for. 5 7y 5 2500 Write Equation. 5 7(250) 5 2500 Substitute 250 for y. 5 750 5 2500 Simplify. 5 50 Solve for. The school sold 50 short sleeve T-shirts and 250 long sleeve T-shirts. c The correct answer is C. A B C D 62 Chapter 3 Linear Systems and Matrices

GUIDED PRACTICE for Eample 3 4. WHAT IF? In Eample 3, suppose the school spends a total of $375 on T-shirts and sells all of them for $660. How many of each type of T-shirt are sold? CHOOSING A METHOD In general, the substitution method is convenient when one of the variables in a system of equations has a coefficient of or 2, as in Eample. If neither variable in a system has a coefficient of or 2, it is usually easier to use the elimination method, as in Eamples 2 and 3. E XAMPLE 4 Solve linear systems with many or no solutions Solve the linear system. a. 2 2y 5 4 b. 4 2 0y 5 8 3 2 6y 5 8 24 35y 5 228 Solution a. Because the coefficient of in the first equation is, use the substitution method. Solve the first equation for. 2 2y 5 4 Write first equation. 5 2y 4 Solve for. Substitute the epression for into the second equation. 3 2 6y 5 8 Write second equation. 3(2y 4) 2 6y 5 8 Substitute 2y 4 for. 2 5 8 Simplify. c Because the statement 2 5 8 is never true, there is no solution. AVOID ERRORS When multiplying an equation by a constant, make sure you multiply each term of the equation by the constant. b. Because no coefficient is or 2, use the elimination method. Multiply the first equation by 7 and the second equation by 2. 4 2 0y 5 8 3 7 28 2 70y 5 56 24 35y 5 228 3 2 228 70y 5 256 Add the revised equations. 0 5 0 c Because the equation 0 5 0 is always true, there are infinitely many solutions. GUIDED PRACTICE for Eample 4 Solve the linear system using any algebraic method. 5. 2 2 3y 5 29 6. 6 5y 5 22 7. 5 3y 5 20 24 y 5 3 22 2 5y 5 9 2 2 3 } 5 y 5 24 8. 2 2 2y 5 2 9. 8 9y 5 5 0. 5 5y 5 5 3 2y 5 24 5 2 2y 5 7 5 3y 5 4.2 3.2 Solve Linear Systems Algebraically 63

3.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 29, and 59 5 STANDARDIZED TEST PRACTICE Es. 2, 40, 50, 57, 58, and 60. VOCABULARY Copy and complete: To solve a linear system where one of the coefficients is or 2, it is usually easiest to use the? method. 2. WRITING Eplain how to use the elimination method to solve a linear system. EXAMPLES and 4 on pp. 60 63 for Es. 3 4 SUBSTITUTION METHOD Solve the system using the substitution method. 3. 2 5y 5 7 4. 3 y 5 6 5. 6 2 2y 5 5 4y 5 2 2 2 3y 5 24 23 y 5 7 6. 4y 5 7. 3 2 y 5 2 8. 3 2 4y 5 25 3 2y 5 22 6 3y 5 4 2 3y 5 25 9. 3 2y 5 6 0. 6 2 3y 5 5. 3 y 5 2 2 4y 5 22 22 y 5 25 2 3y 5 8 2. 2 2 y 5 3. 3 7y 5 3 4. 2 5y 5 0 8 4y 5 6 3y 5 27 23 y 5 36 EXAMPLES 2 and 4 on pp. 6 63 for Es. 5 27 ELIMINATION METHOD Solve the system using the elimination method. 5. 2 6y 5 7 6. 4 2 2y 5 26 7. 3 2 4y 5 20 2 2 0y 5 9 23 4y 5 2 6 3y 5 242 8. 4 2 3y 5 0 9. 5 2 3y 5 23 20. 0 2 2y 5 6 8 2 6y 5 20 2 6y 5 0 5 3y 5 22 2. 2 5y 5 4 22. 7 2y 5 23. 3 4y 5 8 3 2 2y 5 236 22 3y 5 29 6 8y 5 8 24. 2 5y 5 3 25. 4 2 5y 5 3 26. 6 2 4y 5 4 6 2y 5 23 6 2y 5 48 2 8y 5 2 27. ERROR ANALYSIS Describe and correct the error in the first step of solving the system. 3 2y 5 7 5 4y 5 5 26 2 4y 5 7 5 4y 5 5 2 5 22 5 222 CHOOSING A METHOD Solve the system using any algebraic method. 28. 3 2y 5 29. 2 2 3y 5 8 30. 3 7y 5 2 4 y 5 22 24 5y 5 20 2 3y 5 6 3. 4 2 0y 5 8 32. 3 2 y 5 22 33. 2y 5 28 22 5y 5 29 5 2y 5 5 3 2 4y 5 224 34. 2 3y 5 26 35. 3 y 5 5 36. 4 2 3y 5 8 3 2 4y 5 25 2 2y 5 29 28 6y 5 6 37. 4 2 y 5 20 38. 7 5y 5 22 39. 2 y 5 2 6 2y 5 2 3 2 4y 5 24 6y 5 6 64 Chapter 3 Linear Systems and Matrices

40. MULTIPLE CHOICE What is the solution of the linear system? 3 2y 5 4 6 2 3y 5 227 A (22, 25) B (22, 5) C (2, 25) D (2, 5) GEOMETRY Find the coordinates of the point where the diagonals of the quadrilateral intersect. 4. y 42. y 43. (3, 7) (, 4) (4, 4) y (, 3) (5, 5) (0, 2) (5, 0) (, 6) (7, 4) (6, ) (, 2) (7, 0) SOLVING LINEAR SYSTEMS Solve the system using any algebraic method. 44. 0.02 2 0.05y 5 20.38 45. 0.05 2 0.03y 5 0.2 46. 2 } 3y 5 234 3 0.03 0.04y 5.04 0.07 0.02y 5 0.6 2 } 2 y 5 2 47. } 2 } 2 3 y 5 } 5 6 5 } 2 } 7 2 y 5 } 3 4 48. 3 } } y 2 5 49. 2 } } y 2 5 4 4 3 2 3 2 2 y 5 2 2 2y 5 5 50. OPEN-ENDED MATH Write a system of linear equations that has (2, 4) as its only solution. Verify that (2, 4) is a solution using either the substitution method or the elimination method. SOLVING NONLINEAR SYSTEMS Use the elimination method to solve the system. 5. 7y 8y 5 30 52. y 2 5 4 53. 2y y 5 44 3y 2 8y 5 90 5 2 y 5 2 32 2 y 5 3y 54. CHALLENGE Find values of r, s, and t that produce the indicated solution(s). 23 2 5y 5 9 r sy 5 t a. No solution b. Infinitely many solutions c. A solution of (2, 23) PROBLEM SOLVING EXAMPLE 3 on p. 62 for Es. 55 59 55. GUITAR SALES In one week, a music store sold 9 guitars for a total of $36. Electric guitars sold for $479 each and acoustic guitars sold for $339 each. How many of each type of guitar were sold? 56. COUNTY FAIR An adult pass for a county fair costs $2 more than a children s pass. When 378 adult and 24 children s passes were sold, the total revenue was $2384. Find the cost of an adult pass. 3.2 Solve Linear Systems Algebraically 65

57. SHORT RESPONSE A company produces gas mowers and electric mowers at two factories. The company has orders for 2200 gas mowers and 400 electric mowers. The production capacity of each factory (in mowers per week) is shown in the table. Factory A Factory B Gas mowers 200 400 Electric mowers 00 300 Describe how the company can fill its orders by operating the factories simultaneously at full capacity. Write and solve a linear system to support your answer. 58. MULTIPLE CHOICE The cost of gallons of regular gasoline and 6 gallons of premium gasoline is $58.55. Premium costs $.20 more per gallon than regular. What is the cost of a gallon of premium gasoline? A $2.05 B $2.25 C $2.29 D $2.55 59. TABLE TENNIS One evening, 76 people gathered to play doubles and singles table tennis. There were 26 games in progress at one time. A doubles game requires 4 players and a singles game requires 2 players. How many games of each kind were in progress at one time if all 76 people were playing? 60. EXTENDED RESPONSE A local hospital is holding a two day marathon walk to raise funds for a new research facility. The total distance of the marathon is 26.2 miles. On the first day, Martha starts walking at 0:00 A.M. She walks 4 miles per hour. Carol starts two hours later than Martha but decides to run to catch up to Martha. Carol runs at a speed of 6 miles per hour. a. Write an equation to represent the distance Martha travels. b. Write an equation to represent the distance Carol travels. c. Solve the system of equations to find when Carol will catch up to Martha. d. Carol wants to reduce the time she takes to catch up to Martha by hour. How can she do this by changing her starting time? How can she do this by changing her speed? Eplain whether your answers are reasonable. 6. BUSINESS A nut wholesaler sells a mi of peanuts and cashews. The wholesaler charges $2.80 per pound for peanuts and $5.30 per pound for cashews. The mi is to sell for $3.30 per pound. How many pounds of peanuts and how many pounds of cashews should be used to make 00 pounds of the mi? 62. AVIATION Flying with the wind, a plane flew 000 miles in 5 hours. Flying against the wind, the plane could fly only 500 miles in the same amount of time. Find the speed of the plane in calm air and the speed of the wind. 63. CHALLENGE For a recent job, an electrician earned $50 per hour, and the electrician s apprentice earned $20 per hour. The electrician worked 4 hours more than the apprentice, and together they earned a total of $550. How much money did each person earn? 5 WORKED-OUT SOLUTIONS 66 Chapter 3 Linear on p. Systems WS and Matrices 5 STANDARDIZED TEST PRACTICE

MIXED REVIEW Solve the equation. 64. 25 4 5 29 (p. 8) 65. 6(2a 2 3) 5 230 (p. 8) 66..2m 5 2.3m 2 2.2 (p. 8) 67. 3 5 4 (p. 5) 68. 2 5 3 (p. 5) 69. 2 7 5 3 (p. 5) Tell whether the lines are parallel, perpendicular, or neither. (p. 82) 70. Line : through (2, 0) and (, 5) 7. Line : through (4, 5) and (9, 22) Line 2: through (3, 27) and (8, 28) Line 2: through (6, 26) and (22, 2) Write an equation of the line. (p. 98) 72. y 73. y 74. 2 (2, 4) (3, ) 4 4 y (2, 4) (5, ) (2, 27) (22, 22) 2 PREVIEW Prepare for Lesson 3.3 in Es. 75 80. Graph the inequality in a coordinate plane. (p. 32) 75. < 23 76. y 2 77. 2 y > 78. y 2 4 79. 4 2 y 5 80. y < 23 2 QUIZ for Lessons 3. 3.2 Graph the linear system and estimate the solution. Then check the solution algebraically. (p. 53). 3 y 5 2. 2 y 5 25 3. 2 2y 5 22 2 2y 5 28 2 3y 5 6 3 y 5 220 Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent. (p. 53) 4. 4 8y 5 8 5. 25 3y 5 25 6. 2 2y 5 2 2y 5 6 y 5 } 5 2 2 y 5 25 3 Solve the system using the substitution method. (p. 60) 7. 3 2 y 5 24 8. 5y 5 9. 6 y 5 26 3y 5 228 23 4y 5 6 4 3y 5 7 Solve the system using the elimination method. (p. 60) 0. 2 2 3y 5 2. 3 2 2y 5 0 2. 2 3y 5 7 2 3y 5 29 26 4y 5 220 5 8y 5 20 3. HOME ELECTRONICS To connect a VCR to a television set, you need a cable with special connectors at both ends. Suppose you buy a 6 foot cable for $5.50 and a 3 foot cable for $0.25. Assuming that the cost of a cable is the sum of the cost of the two connectors and the cost of the cable itself, what would you epect to pay for a 4 foot cable? Eplain how you got your answer. EXTRA PRACTICE for Lesson 3.2, p. 02 ONLINE 3.2 Solve QUIZ Linear Systems at classzone.com Algebraically 67