Solve Systems of Linear Equations in Three Variables

Transcription

1 TEKS 3.4 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Systems of Linear Equations in Three Variables Before You solved systems of equations in two variables. Now You will solve systems of equations in three variables. Why? So you can model the results of a sporting event, as in Ex. 45. Key Vocabulary linear equation in three variables system of three linear equations solution of a system of three linear equations ordered triple A linear equation in three variables x, y, and z is an equation of the form ax 1 by 1 cz 5 d where a, b, and c are not all zero. The following is an example of a system of three linear equations in three variables. 2x 1 y 2 z 5 5 Equation 1 3x 2 2y 1 z 5 16 Equation 2 4x 1 3y 2 5z 5 3 Equation 3 A solution of such a system is an ordered triple (x, y, z) whose coordinates make each equation true. The graph of a linear equation in three variables is a plane in three-dimensional space. The graphs of three such equations that form a system are three planes whose intersection determines the number of solutions of the system, as shown in the diagrams below. Exactly one solution The planes intersect in a single point. Infinitely many solutions The planes intersect in a line or are the same plane. No solution The planes have no common point of intersection. 178 Chapter 3 Linear Systems and Matrices

2 ELIMINATION METHOD The elimination method you studied in Lesson 3.2 can be extended to solve a system of linear equations in three variables. KEY CONCEPT For Your Notebook The Elimination Method for a Three-Variable System STEP 1 STEP 2 STEP 3 Rewrite the linear system in three variables as a linear system in two variables by using the elimination method. Solve the new linear system for both of its variables. Substitute the values found in Step 2 into one of the original equations and solve for the remaining variable. If you obtain a false equation, such as 0 5 1, in any of the steps, then the system has no solution. If you do not obtain a false equation, but obtain an identity such as 0 5 0, then the system has infinitely many solutions. E XAMPLE 1 Use the elimination method Solve the system. 4x 1 2y 1 3z 5 1 Equation 1 2x 2 3y 1 5z 5214 Equation 2 6x 2 y 1 4z 521 Equation 3 ANOTHER WAY In Step 1, you could also eliminate x to get two equations in y and z, or you could eliminate z to get two equations in x and y. Solution STEP 1 Rewrite the system as a linear system in two variables. 4x 1 2y 1 3z 5 1 Add 2 times Equation 3 12x 2 2y 1 8z 522 to Equation 1. 16x 1 11z 521 New Equation 1 2x 2 3y 1 5z 5214 Add 23 times Equation 3 218x 1 3y 2 12z 5 3 to Equation x 2 7z 5211 New Equation 2 STEP 2 Solve the new linear system for both of its variables. 16x 1 11z 521 Add new Equation 1 216x 2 7z 5211 and new Equation 2. 4z 5212 z 523 Solve for z. x 5 2 Substitute into new Equation 1 or 2 to find x. STEP 3 Substitute x 5 2 and z 523 into an original equation and solve for y. 6x 2 y 1 4z 521 Write original Equation 3. 6(2) 2 y 1 4(23) 521 Substitute 2 for x and 23 for z. y 5 1 Solve for y. c The solution is x 5 2, y 5 1, and z 523, or the ordered triple (2, 1, 23). Check this solution in each of the original equations. 3.4 Solve Systems of Linear Equations in Three Variables 179

3 E XAMPLE 2 Solve a three-variable system with no solution Solve the system. x 1 y 1 z 5 3 Equation 1 4x 1 4y 1 4z 5 7 Equation 2 3x 2 y 1 2z 5 5 Equation 3 REVIEW SYSTEMS For help with solving linear systems with many solutions or no solution, see p Solution When you multiply Equation 1 by 24 and add the result to Equation 2, you obtain a false equation. 24x 2 4y 2 4z 5212 Add 24 times Equation 1 4x 1 4y 1 4z 5 7 to Equation New Equation 1 c Because you obtain a false equation, you can conclude that the original system has no solution. E XAMPLE 3 Solve a three-variable system with many solutions Solve the system. x 1 y 1 z 5 4 Equation 1 x 1 y 2 z 5 4 Equation 2 3x 1 3y 1 z 5 12 Equation 3 Solution STEP 1 Rewrite the system as a linear system in two variables. x 1 y 1 z 5 4 Add Equation 1 x 1 y 2 z 5 4 to Equation 2. 2x 1 2y 5 8 New Equation 1 x 1 y 2 z 5 4 Add Equation 2 3x 1 3y 1 z 5 12 to Equation 3. STEP 2 4x 1 4y 5 16 New Equation 2 Solve the new linear system for both of its variables. 24x 2 4y 5216 Add 22 times new Equation 1 4x 1 4y 5 16 to new Equation STEP 3 Because you obtain the identity 0 5 0, the system has infinitely many solutions. Describe the solutions of the system. One way to do this is to divide new Equation 1 by 2 to get x 1 y 5 4, or y 52x 1 4. Substituting this into original Equation 1 produces z 5 0. So, any ordered triple of the form (x, 2x 1 4, 0) is a solution of the system. GUIDED PRACTICE for Examples 1, 2, and 3 Solve the system. 1. 3x 1 y 2 2z x 1 y 2 z x 1 y 1 z 5 3 6x 2 2y 1 z 522 2x 1 2y 2 2z 5 6 x 1 y 2 z 5 3 x 1 4y 1 3z 5 7 5x 1 y 2 3z 5 8 2x 1 2y 1 z Chapter 3 Linear Systems and Matrices

4 E XAMPLE 4 Solve a system using substitution MARKETING The marketing department of a company has a budget of $30,000 for advertising. A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 per month and have as many radio as television and newspaper combined. How many of each type of ad should the department run each month? Solution STEP 1 Write verbal models for the situation. TV 1 Radio 1 Newspaper 5 Total Equation p TV 1 200p Radio 1 500p Newspaper 5 Monthly budget Equation 2 Radio 5 TV 1 Newspaper Equation 3 STEP 2 STEP 3 Write a system of equations. Let x be the number of TV, y be the number of radio, and z be the number of newspaper. x1 y 1 z 5 60 Equation x 1 200y 1 500z 5 30,000 Equation 2 y5 x 1 z Equation 3 Rewrite the system in Step 2 as a linear system in two variables by substituting x 1 z for y in Equations 1 and 2. x 1 y 1 z 5 60 Write Equation 1. x 1 (x 1 z) 1 z 5 60 Substitute x 1 z for y. 2x 1 2z 5 60 New Equation x 1 200y 1 500z 5 30,000 Write Equation x 1 200(x 1 z) 1 500z 5 30,000 Substitute x 1 z for y. 1200x 1 700z 5 30,000 New Equation 2 STEP 4 Solve the linear system in two variables from Step 3. AVOID ERRORS In Example 4, be careful not to write the ordered triple in the order in which you solved for the variables. (12, 18, 30) (18, 30, 12) 21200x z 5236,000 Add 2600 times new Equation x 1 700z 5 30,000 to new Equation z z 5 12 Solve for z. x 5 18 Substitute into new Equation 1 or 2 to find x. y 5 30 Substitute into an original equation to find y. c The solution is x 5 18, y 5 30, and z 5 12, or (18, 30, 12). So, the department should run 18 TV, 30 radio, and 12 newspaper each month. GUIDED PRACTICE for Example 4 4. WHAT IF? In Example 4, suppose the monthly budget is $25,000. How many of each type of ad should the marketing department run each month? 3.4 Solve Systems of Linear Equations in Three Variables 181

5 3.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 11, 25, and 45 5 TAKS PRACTICE AND REASONING Exs. 23, 24, 34, 45, 47, 49, and VOCABULARY Write a linear equation in three variables. What is the graph of such an equation? 2. WRITING Explain how to use the substitution method to solve a system of three linear equations in three variables. EXAMPLES 1, 2, and 3 on pp for Exs CHECKING SOLUTIONS Tell whether the given ordered triple is a solution of the system. 3. (1, 4, 23) 4. (21, 22, 5) 5. (6, 0, 23) 2x 2 y 1 z 525 4x 2 y 1 3z 5 13 x 1 4y 2 2z x 1 2y 2 2z 5 19 x 1 y 1 z 5 2 3x 2 y 1 4z 5 6 x 2 3y 1 z 525 x 1 3y 2 2z x 1 3y 1 z (25, 1, 0) 7. (2, 8, 4) 8. (0, 24, 7) 3x 1 4y 2 2z x 2 y 1 5z x 1 4y 2 z x 1 y 2 z 5 11 x 1 3y 2 6z 5 2 x 2 5y 2 3z 521 x 1 4y 1 3z x 1 y 2 2z 526 2x 1 y 1 4z 5 24 ELIMINATION METHOD Solve the system using the elimination method. 9. 3x 1 y 1 z x 2 y 1 2z x 2 y 1 2z 5 4 2x 1 2y 2 3z 529 2x 1 2y 2 4z 5 5 6x 2 2y 1 4z 528 5x 2 y 1 5z 5 30 x 1 4y 2 6z 521 2x 2 y 1 3z x 2 y 1 2z x 1 y 2 z x 1 y 2 z 5 9 2x 1 2y 1 z 5 11 x 1 y 1 z 5 2 2x 1 6y 1 2z x 1 3y 2 4z x 1 y 5 4 5x 1 7y 1 z 5 4 EXAMPLE 4 on p. 181 for Exs SUBSTITUTION METHOD Solve the system using the substitution method. 15. x 1 y 2 z x 2 y 2 z x 1 y 1 5z x 1 2y 1 4z x 1 5y 1 2z x 1 2y 1 4z x 1 5y 1 z 5 8 2x 2 4y 1 3z 5220 x 2 y 2 2z x 1 3y 2 z x 2 y 1 z x 1 5y 2 z x 1 4y 2 2z 5 6 6x 1 3y 2 4z 5 8 x 1 y 1 z 5 0 2x 2 2y 1 z x 1 2y 1 3z 526 2x 1 2y 1 2z 5227 ERROR ANALYSIS Describe and correct the error in the first step of solving the system. 2x 1 y 2 2z x 1 2y 1 z 5 11 x 2 y 1 z x 1 y 2 2z x 1 2y 1 2z x 1 3y z x 1 2y 2x 1 y 2 2(11 1 3x 1 2y) x 2 3y Chapter 3 Linear Systems and Matrices

6 23. TAKS REASONING Which ordered triple is a solution of the system? 2x 1 5y 1 3z x 2 y 1 4z 5 8 5x 2 2y 1 7z 5 12 A (7, 1, 23) B (7, 21, 23) C (7, 1, 3) D (27, 1, 23) 24. TAKS REASONING Which ordered triple describes all of the solutions of the system? 2x 2 2y 2 z 5 6 2x 1 y 1 3z 523 3x 2 3y 1 2z 5 9 A (2x, x 1 2, 0) B (x, x 2 3, 0) C (x 1 2, x, 0) D (0, y, y 1 4) CHOOSING A METHOD Solve the system using any algebraic method. 25. x 1 5y 2 2z x 1 5y 1 3z x 1 y 2 z 522 2x 2 2y 1 z 5 6 x 2 3y 1 2z 526 x 1 6y 1 3z x 2 7y 1 3z 5 7 2x 1 2y 2 z 5 3 2x 1 y 1 2z x 1 2y x 2 y 1 2z x 2 8y 1 2z x 2 y 1 4z 5 17 x 1 5y 2 z x 1 y 2 2z x 1 2y 2 3z x 1 2y 1 4z 5 6 2x 2 4y 1 z x 1 5y 2 z x 2 y 1 4z x 1 y 1 z 5 3 2x 1 3y 1 4z x 1 3y 2 2z 527 3x 2 4y 1 2z 5228 x 1 y 2 z 528 4x 1 2y 1 3z x 1 5y 1 z TAKS REASONING Write a system of three linear equations in three variables that has the given number of solutions. a. One solution b. No solution c. Infinitely many solutions SYSTEMS WITH FRACTIONS Solve the system using any algebraic method. 35. x 1 } 1 y 1 } 1 z 5 } } x 1 } 5 y 1 } 2 z 5 } } x 1 } 1 y 1 } 3 z 5 } }6 x 1 } 2 y 1 } 1 z 5 } } x 1 } 3 y 1 } 2 z 5 } }3 x 1 } 1 y 1 } 3 z 5 } REASONING For what values of a, b, and c does the linear system shown have (21, 2, 23) as its only solution? Explain your reasoning. x 1 2y 2 3z 5 a 2x 2 y 1 z 5 b 2x 1 3y 2 2z 5 c CHALLENGE Solve the system of equations. Describe each step of your solution. 38. w 1 x 1 y 1 z w 1 x 2 3y 1 z 5 4 2w 2 x 1 2y 2 z 5 1 w 2 3x 1 y 1 z w 1 2x 2 y 1 2z 522 2w 1 2x 1 2y 2 z w 1 x 1 y 2 z 525 w 1 x 2 y 1 3z w 1 2x 1 5y w 1 7x 2 3y w 1 x 1 4y 1 2z 527 2w 2 2x 1 y 5213 w 1 2x 2 2y 1 5z w 1 4x 1 z w 1 x 521 2w 2 x 1 y Solve Systems of Linear Equations in Three Variables 183

7 PROBLEM SOLVING EXAMPLE 4 on p. 181 for Exs PIZZA SPECIALS At a pizza shop, two small pizzas, a liter of soda, and a salad cost $14; one small pizza, a liter of soda, and three sal cost $15; and three small pizzas and a liter of soda cost $16. What is the cost of one small pizza? of one liter of soda? of one salad? 43. HEALTH CLUB The juice bar at a health club receives a delivery of juice at the beginning of each month. Over a three month period, the health club received 1200 gallons of orange juice, 900 gallons of pineapple juice, and 1000 gallons of grapefruit juice. The table shows the composition of each juice delivery. How many gallons of juice did the health club receive in each delivery? Juice 1st delivery 2nd delivery 3rd delivery Orange 70% 50% 30% Pineapple 20% 30% 30% Grapefruit 10% 20% 40% 44. MULTI-STEP PROBLEM You make a tape of your friend s three favorite TV shows: a comedy, a drama, and a reality show. An episode of the comedy lasts 30 minutes, while an episode of the drama or the reality show lasts 60 minutes. The tape can hold 360 minutes of programming. You completely fill the tape with 7 episodes and include twice as many episodes of the drama as the comedy. a. Write a system of equations to represent this situation. b. Solve the system from part (a). How many episodes of each show are on the tape? c. How would your answer to part (b) change if you completely filled the tape with only 5 episodes but still included twice as many episodes of the drama as the comedy? 45. TAKS REASONING The following Internet announcement describes the results of a high school track meet. a. Write and solve a system of equations to find the number of athletes who finished in first place, in second place, and in third place. b. Suppose the announcement had claimed that the Madison athletes scored a total of 70 points instead of 68 points. Show that this claim must be false because the solution of the resulting linear system is not reasonable WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING

8 46. FIELD TRIP You and two friends buy snacks for a field trip. You spend a total of $8, Jeff spends $9, and Curtis spends $9. The table shows the amounts of mixed nuts, granola, and dried fruit that each person purchased. What is the price per pound of each type of snack? Mixed nuts Granola Dried fruit You 1 lb 0.5 lb 1 lb Jeff 2 lb 0.5 lb 0.5 lb Curtis 1 lb 2 lb 0.5 lb 47. TAKS REASONING A florist must make 5 identical bridesmaid bouquets for a wedding. She has a budget of $160 and wants 12 flowers for each bouquet. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. She wants twice as many roses as the other two types of flowers combined. a. Write Write a system of equations to represent this situation. b. Solve Solve the system of equations. How many of each type of flower should be in each bouquet? c. Analyze Suppose there is no limitation on the total cost of the bouquets. Does the problem still have a unique solution? If so, state the unique solution. If not, give three possible solutions. 48. CHALLENGE Write a system of equations to represent the first three pictures below. Use the system to determine how many tangerines will balance the apple in the final picture. Note: The first picture shows that one tangerine and one apple balance one grapefruit. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. 844; TAKS Workbook REVIEW TAKS Preparation p. 678; TAKS Workbook 49. TAKS PRACTICE What are the vertices of a triangle congruent to npqr shown at the right? TAKS Obj. 6 A (3, 1), (1, 22), (4, 25) B (2, 3), (21, 1), (2, 22) C (0, 2), (22, 21), (23, 24) D (22, 23), (24, 1), (21, 4) 50. TAKS PRACTICE What special type of quadrilateral has the vertices K(24, 3), L(27, 3), M(29, 21), and N(22, 21)? TAKS Obj. 7 F Square H Kite G Trapezoid J Parallelogram Πy 5 P x R EXTRA PRACTICE for Lesson 0 3.4, p ONLINE QUIZ at classzone.com 185

9 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE 1. JEWELRY Melinda is making jewelry to sell at a craft fair. The cost of materials is $3.50 to make one necklace and $2.50 to make one bracelet. She sells the necklaces for $9.00 each and the bracelets for $7.50 each. She spends a total of $121 on materials and sells all of the jewelry for a total of $324. Which system of equations represents the situation, where x is the number of necklaces and y is the number of bracelets? TEKS 2A.3.A A 2.5x 2 3.5y x 2 9y B 2.5x 1 3.5y x 1 7.5y C 3.5x 2 2.5y x 1 9y D 3.5x 1 2.5y x 1 7.5y GIFT BASKETS Mike is making gift baskets. Each basket will contain three different kinds of candles: tapers, pillars, and jar candles. Tapers cost $1 each, pillars cost $4 each, and jar candles cost $6 each. Mike puts 8 candles costing a total of $24 in each basket, and he includes as many tapers as pillars and jar candles combined. How many tapers are in a basket? TEKS 2A.3.B F 1tapers G 2tapers H 4tapers J 5tapers 3. BASEBALL From 1999 through 2002, the average annual salary s (in thousands of dollars) of players on two Major League Baseball teams can be modeled by the equations below, where t is the number of years since Florida Marlins: s 5 320t Kansas City Royals: s 5 440t In what year were the average annual salaries of the two baseball teams equal? TEKS 2A.3.B A 1999 B 2000 C 2001 D 2002 classzone.com 4. RESTAURANT SEATING A restaurant has 20 tables. Each table can seat either 4 people or 6 people. The restaurant can seat a total of 90 people. How many 6 seat tables does the restaurant have? TEKS 2A.3.B F 1 table H 7 tables G 5 tables J 15 tables 5. BUSINESS A store orders rocking chairs, hand paints them, and sells the chairs for a profit. A small chair costs the store $51 and sells for $80. A large chair costs the store $70 and sells for $110. The store wants to pay no more than $2000 for its next order of chairs and wants to sell them all for at least $2750. What is a possible combination of small and large rocking chairs that the store can buy and sell? TEKS 2A.3.B A B C D 10 small chairs and 25 large chairs 12 small chairs and 20 large chairs 15 small chairs and 20 large chairs 24 small chairs and 8 large chairs GRIDDED ANSWER SCHOOL OUTING A school is planning a 5 hour outing at a community park. The park rents bicycles for $8 per hour and in-line skates for $6 per hour. The total budget per student is $34. A student bikes and skates the entire time and uses all of the money budgeted. How many hours does the student spend in-line skating? TEKS 2A.3.B 7. SNACK BOOTH At a snack booth, one soda, one pretzel, and two hot dogs cost $7; two sodas, one pretzel, and two hot dogs cost $8; and one soda and four hot dogs cost $10. What is the price (in dollars) of one hot dog? TEKS 2A.3.B 186 Chapter 3 Linear Systems and Matrices