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7-2 Solving Systems Using Substitution 7-2 1. Plan Objectives 1 To solve systems using substitution Examples 1 2 3 Using Substitution Using Substitution and the Distributive Property Real-World Problem Solving What You’ll Learn Check Skills You’ll Need • To solve systems using Solve each equation. substitution 1. m - 6 = 4m + 8 –423 . . . And Why GO for Help Lessons 3-3 and 7-1 3. 13 t + 5 = 10 15 2. 4n = 9 - 2n 112 For each system, is the ordered pair a solution of both equations? To solve problems involving transportation, as in Example 3 y = -x + 4 y = x - 6 no 4. (5, 1) 5. (2, 2.4) 4x + 5y = 20 2x + 6y = 10 no New Vocabulary • substitution method Math Background Solving systems using substitution begins with the concept of evaluating variable expressions. Here a variable is replaced with its value in terms of the other variable. More Math Background: p. 372C 1 UsingSubstitution Substitution 1 1 Using Part You can solve a system of equations by graphing when the solution contains integers or when you have a graphing calculator. Another method for solving systems of equations is the substitution method. By replacing one variable with an equivalent expression containing the other variable, you can create a onevariable equation that you can solve using methods shown in Chapter 2. Vocabulary Tip Substitution means one value or expression is used in place of another. Lesson Planning and Resources 1 See p. 372E for a list of the resources that support this lesson. Using Substitution EXAMPLE y = -4x + 8 y=x+7 Solve using substitution. Step 1 Write an equation containing only one variable, and solve it. PowerPoint y = -4x + 8 Bell Ringer Practice x + 7 = -4x + 8 5x + 7 = 8 Check Skills You’ll Need Divide each side by 5. Step 2 Solve for the other variable in either equation. Lesson 3-3: Example 1 Extra Skills and Word Problem Practice, Ch. 3 y = 0.2 + 7 Substitute 0.2 for x in y ≠ x ± 7. y = 7.2 Simplify. Since x = 0.2 and y = 7.2, the solution is (0.2, 7.2). Solving Systems by Graphing Lesson 7-1: Example 1 Extra Skills and Word Problem Practice, Ch. 7 Check 7.2 0 -4(0.2) + 8 7.2 = 7.2 ✓ Quick Check 382 Since y ≠ x ± 7 was used in step 2, see if (0.2, 7.2) solves y ≠ -4x ± 8. Simplify. 1 Solve using substitution. Check your solution. y = 2x 7x - y = 15 (3, 6) Chapter 7 Systems of Equations and Inequalities Special Needs Below Level L1 If some students have difficulty distinguishing the colors of two different lines on a graph, redraw the lines in two different thicknesses or have them work with a partner. 382 Subtract 7 from each side. x = 0.2 Equations with Variables on Both Sides Substitute x ± 7 for y. Add 4x to each side. 5x = 1 For intervention, direct students to: Start with one equation. learning style: visual L2 Remind students that they must check that a solution makes all equations in the system true in order for it to be a solution of the entire system. learning style: verbal 2. Teach To use the substitution method, you must have an equation that has already been solved for one of the variables. 2 EXAMPLE Using Substitution and the Distributive Property Solve using the substitution method. Guided Instruction 3y + 2x = 4 -6x + y = -7 3 -6x + y = -7 Add 6x to each side. Step 2 Write an equation containing only one variable and solve. 3y + 2x = 4 Substitute 6x – 7 for y. Use parentheses. 18x - 21 + 2x = 4 Use the Distributive Property. 20x = 25 x = 1.25 PowerPoint Start with the other equation. 3(6x - 7) + 2x = 4 Additional Examples 1 Solve using substitution. y = 2x + 2 y = -3x + 4 (0.4, 2.8) Combine like terms and add 21 to each side. Divide each side by 20. Step 3 Solve for the other variable in either equation. -6 (1.25) + y = -7 -7.5 + y = -7 y = 0.5 2 Solve using substitution. -2x + y = -1 4x + 2y = 12 (1.75, 2.5) Substitute 1.25 for x in -6x + y ≠ -7. Simplify. Add 7.5 to each side. Since x = 1.25 and y = 0.5, the solution is (1.25, 0.5). Quick Check 2 Solve using substitution. Check your solution. 3 EXAMPLE Real-World 6y + 8x = 28 3 = 2x - y (2.3, 1.6) Problem Solving Transportation Your school is planning to bring 193 people to a competition at another school. There are eight drivers available and two types of vehicles, school buses and minivans. The school buses seat 51 people each, and the minivans seat 8 people each. How many buses and minivans will be needed? Let b = number of school buses. drivers b + m = 8 Step 1 b + m = 8 51b - 8b + 64 = 193 According to the U.S. Department of Transportation, every year about 460,000 public school buses transport 24 million students to and from school. Quick Check Substitute -b + 8 for m in the second equation. Solve for b. 43b + 64 = 193 43b = 129 b=3 Step 3 (3) + m = 8 Ask: Why it is sometimes easier to solve equations using substitution rather than graphing? Sometimes the solution is a very large number or a decimal. Substitute 3 for b in b + m = 8. m=5 Three school buses and five minivans will be needed to transport 193 people. 3 Geometry A rectangle is 4 times longer than it is wide. The perimeter of the rectangle is 30 cm. Find the dimensions of the rectangle. 3 cm by 12 cm Lesson 7-2 Solving Systems Using Substitution Advanced Learners Resources • Daily Notetaking Guide 7-2 L3 • Daily Notetaking Guide 7-2— L1 Adapted Instruction Closure Subtract b from each side. Step 2 51b + 8(-b + 8)= 193 Connection Let m = number of minivans. people 51b + 8m = 193 3 A youth group with 26 members is going to the beach. There will also be five chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed? 3 vans and 2 cars Solve the first equation for m. m = -b + 8 Real-World Error Prevention Some students may equate 51b with 51 buses. Point out that b represents the number of buses, and 51 represents the number of people each bus can hold. Step 1 Solve the second equation for y because it has a coefficient of 1. y = 6x - 7 EXAMPLE 383 English Language Learners ELL L4 Part of the graphs of some systems form geometric figures. Have students write a system of three equations whose graph forms a triangle. learning style: visual Some students may be confused by words such as chaperone (as in Additional Example 3). Explain, and discuss the word problems to verify that the students understand what they need to do. learning style: verbal 383 3. Practice EXERCISES For more exercises, see Extra Skill and Word Problem Practice. Practiceand andProblem ProblemSolving Solving Practice Assignment Guide A 1 A B 1-39 C Challenge 40-41 Test Prep Mixed Review 42-45 46-54 Practice by Example GO for Help Mental Math Match each system with its solution at the right. Example 1 1. y = x + 1 D (page 382) y = 2x - 1 2. y = 12 x + 4 C 2y + 2x = 2 3. 2y = x + 3 B that they first substitute the coordinate into the equation in each system that is easiest to work with, such as x = y in Exercise 3. Example 2 (page 383) Example 3 (page 383) Name Apply Your Skills 5. y = 4x - 8 y = 2x + 10 6. C(n) = -3n - 6 C(n) = n - 4 8. y = -4x + 1212 y = 14 x + 4 9. h = 6g - 4 7. m = 5p + 8 m = -10p + 3 10. a = 25 b - 3 a = 2b - 18 h = -2g + 28 11. y = x - 2 2x + 2y = 4 12. c = 3d - 27 4d + 10c = 120 13. 3x - 6y = 30 y = -6x + 34 14. m = 4n + 11 -6n + 8m = 36 15. 7x - 8y = 112 y = -2x + 9 16. t = 0.2s + 10 4s + 5t = 35 17. Geometry The length of a rectangle is 5 cm more than twice the width. The perimeter of the rectangle is 34 cm. Find the dimensions of the rectangle. 4 cm by 13 cm 18. Suppose you have $28.00 in your bank account and start saving $18.25 every week. Your friend has $161.00 in his account and is withdrawing $15 every week. When will your account balances be the same? 4 wk Solve each system by substitution. Check your solution. 19. a - 1.2b = -3 20. 0.5x + 0.25y = 36 21. y = 0.8x + 7.2 0.2b + 0.6a = 12 (15, 15) y + 18 = 16x (9, 126) 20x + 32y = 48 (–4, 4) L2 L1 Adapted Practice Practice B L3 L4 Reteaching D. (2, 3) Solve each system using substitution. Check your solution. 5–16. See margin. Exercises 1–4 Suggest to students GPS Guided Problem Solving C. (-2, 3) x = 12 y + 2 x=y Enrichment B. (3, 3) 4. x - y = 1 A Homework Quick Check To check students’ understanding of key skills and concepts, go over Exercises 6, 18, 22, 24, 32. A. (3, 2) Class Practice 7-2 Date For Exercises 22–24, define variables and write a system of equations for each situation. Solve using substitution. L3 Solving Systems Using Substitution Solve each system using substitution. Write no solution or infinitely many solutions where appropriate. 1. y = x y = -x + 2 2. y = x + 4 y = 3x 3. y = 3x - 10 y = 2x - 5 4. x = -2y + 1 x=y-5 5. y = 5x + 5 y = 15x - 1 6. y = x - 3 y = -3x + 25 7. y = x - 7 2x + y = 8 9. x + 2y = 200 x = y + 50 10. 3x + y = 10 y = -3x + 4 11. y = 2x + 7 y = 5x + 4 12. 3x - 2y = -0 3x + 2y = -5 13. 4x + 2y = 8 y = -2x + 4 14. 6x - 3y = 6 y = 2x + 5 15. 2x + 4y = -6 2x - 3y = -7 16. 5x - 3y = -4 17. y 5 223 x 1 4 2x + 3y = -6 18. 2x + 3y = 8 3y 5 4 2 x 2 20. x + y = 0 21. 5x + 2y = 6 y 5 2 5x 1 1 2 24. y 5 2 32 x 1 1 5x + 3y = -4 19. 3x - y = 14 2x + y = 16 © Pearson Education, Inc. All rights reserved. 8. y = 3x - 6 -3x + y = -6 x=y+4 22. 2x + 5y = -6 23. 4x + 3y = -3 4x + 5y = -12 2x + 3y = -1 4x + 6y = 6 25. 5x - 6y = 19 4x + 3y = 10 26. 2x + 2y = 6.6 5x - 2y = 0.3 27. 2x - 4y = 13.8 3x - 4y = 17.7 28. 4.3x + 4y = 18 29. 3x - 4y = -5 30. y 5 31 x 1 10 x = 3y + 6 4.5x + 6y = 12 31. 2x + 5y = 62 3x - 5y = 23.3 34. 5x + 6y = -76 5x + 2y = -44 x=y+2 32. -5x + 3y = 06 -2x - 3y = 60 35. 3x - 2y = 10 y 5 3x 2 1 2 22. Agriculture A farmer grows only sunflowers and flax on his 240-acre farm. This year he wants to plant 80 more acres of sunflowers than of flax. How many acres of each crop does the farmer need to plant? 80 acres flax, 160 acres sunflowers 23. Renting Videos Suppose you want to join a video store. Big Video offers a special discount card that costs $9.99 for one year. With the discount card, each video rental costs $2.49. A discount card from Main Street Video costs $20.49 for one year. With the Main Street Video discount card, each video rental costs $1.79. After how many video rentals is the cost the same? 15 video rentals 33. x 5 34 y 2 6 y 5 4x 1 8 3 Real-World Connection 36. -3x + 2y = -6 37. At an ice cream parlor, ice cream cones cost $1.10 and sundaes cost $2.35. One day, the receipts for a total of 172 cones and sundaes were $294.20. How many cones were sold? 38. You purchase 8 gal of paint and 3 brushes for $152.50. The next day, you purchase 6 gal of paint and 2 brushes for $113.00. How much does each gallon of paint and each brush cost? -2x + 2y = -6 Sunflower seeds are sold as snacks and as bird food, and they are a source of cooking oil. 384 Chapter 7 Systems of Equations and Inequalities pages 384–386 Exercises 5– 16. Coordinates given in alphabetical order. 5. (9, 28) 6. –21 , –4 21 ( 384 24. Buying a Car Suppose you are thinking about buying one of two cars. Car A GPS will cost $17,655. You can expect to pay an average of $1230 per year for fuel, maintenance, and repairs. Car B will cost about $15,900. Fuel, maintenance, and repairs for it will average about $1425 per year. After how many years are the total costs for the cars the same? 9 yr ) 7. (6 31 , –31 8. 2, 4 12 9. (4, 20) 10. 43 , 9 38 11. (2, 0) ( ) ( ) ) 12. 13. 14. 15. 16. (7 177 , 11 178 ) (6, –2) (3, –2) (8, –7) (–3, 9.4) GO nline Homework Help Visit: PHSchool.com Web Code: ate-0702 25. Multiple Choice You have 28 coins that are all nickels n and dimes d. The value of the coins is $2.05. Which system of equations can be used to find the number of nickels and the number of dimes? D n + d = 28 n + d = 205 10n + 5d = 2.05 n + d = 28 10n + 5d = 205 n + d = 28 n + d = 28 5n + 10d = 205 Estimation Graph each system to estimate the solution. Then use substitution to find the exact solution of the system. 26–31. See back of book. 32. Answers may vary. Sample: y ≠ x and y ≠ –3x ± 2; (12 , 12 ) 26. y = 2x y = -6x + 4 27. y = 12 x + 4 y = - 4x - 5 28. x + y = 0 5x + 2y = -3 29. y = 2x + 3 y = 0.5x - 2 30. y = -x + 4 y = 2x + 6 31. y = 0.7x + 3 y = -1.5x - 7 32. Open-Ended Write a system of linear equations with exactly one solution. Use substitution to solve your system. See left. 33. a. Solve each system below using substitution. (x, y) such that y ≠ 0.5x ± 4; no solution y = 0.5x + 4 6x - 2y = 10 -x + 2y = 8 Error Prevention! Exercise 17 Some students may use p = 34 for the second equation. Explain to students that the expression for finding perimeter, 2/ + 2w, must be used so that the equations have the same variables. Alternative Method Exercise 18 Students may want to solve this problem by keeping two running balances like those used in bank account registers. Have them make side-by-side vertical lists to find when the numbers are the same. Week $28 ± 18.25 $161 – 15 1 2 46.25 64.50 etc. 101.00 146.00 131.00 etc. 101.00 y = 3x + 1 b. Solve each system by graphing. See back of book. c. Critical Thinking Make a general statement about the solutions you get when solving by graphing and the results you get when solving by substitution. See margin. 46. 12 Solve each system using substitution. C Challenge 6 34. y = 2x 6x - y = 8 (2, 4) 35. y = 3x + 1 x = 3y + 1 (–21 , –12 ) 36. x - 3y = 14 x - 2 = 0 (2, –4) 37. 2x + 2y = 5 38. 4x + y = -2 39. 3x + 5y = 2 y = 14 x (2, ) 1 2 -2x - 3y = 1 ( –12 , 0) 6 O 40. There are 1170 students in a school. The ratio of girls to boys is 23 i 22. The system below describes relationships between the number of girls and the number of boys. g 23 g + b = 1170 b = 22 47. 12 x 18 y 4 2 x O 41. Sprinting The graph at the left represents the start of a 100-meter race between Joetta and Gail. The red line and blue line represent Joetta’s and Gail’s time and distance. Joetta averages 8.8 m/s. Gail averages 9 m/s but started 0.2 s after Joetta. At time 0.2, Gail’s distance is 0 m. You can use pointslope form to write an equation that relates Gail’s time t to her distance d. Distance (meters) 6 x + 4y = -4 (4, –2) a. Solve the proportion for g. g ≠ 23 22 b b. Solve the system. (b, g) ≠ (572, 598) c. How many more girls are there than boys? 26 48. d = 9t - 1.8 2 y 2 O y - y1 = m(x - x1 ) d - 0 = 9(t - 0.2) Time (seconds) y 49. Since Joetta started at t = 0, the equation d = 8.8t relates her time and distance. a. Solve the system using substitution. (t, d) ≠ (9, 79.2) b. Will Gail overtake Joetta before the finish line? yes 2 2 4 x y x O 2 2 lesson quiz, PHSchool.com, Web Code: ata-0702 Lesson 7-2 Solving Systems Using Substitution 33. c. When there is exactly one solution, the graph will show two lines that intersect in a single point, and the substitution method will give the coordinates of the point. When there are infinitely many solutions, the graphs coincide, and substitution leads to an equation that is always true. When there are no solutions, the graphs are parallel, and substitution results in an equation that is never true. 385 45.[2] 7(–7) – 4(–2) 29 –49 ± 8 29 –49 29 No, (–2, –7) must satisfy both equations to be a solution of the system. [1] no explanation given 50. f(x) 2 x O 2 385 4. Assess & Reteach Standardized Test PrepTest Prep Gridded Response PowerPoint Lesson Quiz Solve each system using substitution. 42. Find the value of the y-coordinate of the solution to the given system. 5x + 5y = 179 x = 5y - 143 29.8 43. Find the value of the y-coordinate of the solution to the given system. y = 9x + 3480 y = 81x - 7104 4803 44. Tina has $220 in her account. Cliff has $100 in his account. Starting in July, Tina adds $25 to her account on the first of each month, while Cliff adds $35 to his. How many dollars will they have in their accounts when the amounts are the same? 520 1. 5x + 4y = 5 y = 5x (0.2, 1) 2. 3x + y = 4 2x - y = 6 (2, –2) 45. Is (-2, -7) the solution of the following system? Justify your answer. 7y - 4x = 29 See margin. x=y-5 Short Response 3. 6m - 2n = 7 3m + n = 4 (1.25, 0.25) Alternative Assessment Group students in pairs. Give each student a system of equations. One partner solves the system using a graphing calculator. The other student uses substitution to solve the system. They compare answers to check. Have partners switch roles. Mixed Review GO for Help Solve each system by graphing. 46–48. See margin for graphs. Lesson 7-1 46. y = x - 2 y = 12 x + 4 (12, 10) Test Prep y = 14 x + 1 (4, 2) 49. y = 3x - 2 50. f(x) = x + 1 51. f(x) = -2x 52. y = Δx« + 5 53. y = -2Δx« 54. y = Δx + 3« - 1 Checkpoint Quiz 1 Checkpoint Quiz 1 Lessons Lessons 7-1 7-1 through through 7-2 7-2 Solve each system by graphing. 1–3. See margin for graphs. y = 23 x (3, 2) y = -2x + 1 (1, –1) For additional practice with a variety of test item formats: • Standardized Test Prep, p. 425 • Test-Taking Strategies, p. 420 • Test-Taking Strategies with Transparencies 3. y = 14 x - 1 2. y = 43 x - 2 1. y = 3x - 4 Resources y = -2x - 10 (–4, –2) Solve each system using substitution. 4. y = 3x - 14 y = x - 10 (2, –8) 5. y = 2x + 5 y = 6x + 1 (1, 7) 7. 3x + 4y = 12 y = -2x + 10 ( 8. 4x + 9y = 24 28 5 , –65 ) 6. x = y + 7 y = 8 + 2x (–15, –22) y = 2 13 x + 2 (6, 0) In Exercises 9 and 10, write and solve a system of equations for each situation. Use this Checkpoint Quiz to check students’ understanding of the skills and concepts of Lessons 7-1 and 7-2. 9. A rectangle is 3 times longer than it is wide. The perimeter is 44 cm. Find the dimensions of the rectangle. 2L ± 2W ≠ 44, 3W ≠ L; 5.5 cm by 16.5 cm 10. A farmer grows only pumpkins and corn on her 420-acre farm. This year she wants to plant 250 more acres of corn than of pumpkins. How many acres of each crop does the farmer need to plant? p ± c ≠ 420, c ≠ p ± 250; 85 acres pumpkins, 335 acres corn Resources Grab & Go • Checkpoint Quiz 1 386 Chapter 7 Systems of Equations and Inequalities Checkpoint Quiz 1. y x O 2 386 y = -x + 3 (2, 1) Graph each function. 49–50. See margin. 51–54. See back of book. Lesson 5-3 A sheet of blank grids is available with the Test-Taking Strategies booklet. Give this sheet to students for practice with filling in the grids. 48. y = 34 x - 1 47. y = -2x + 5 2 2. 3. y 2 y x 4 x O 2 4 2 O 2