Math 803 Unit 1: Solving Equations in One Variable (8.EE.7) Part 2 1.4 Variables on both sides (2.4 text) 1.5 Solve multi-step equations (2.5 text) Name: Period: Teacher s Name: 1
Lesson 1.4 Equations with Variables on Each Side (Textbook 2.4) Learning Target: I can solve equations with variables on each side of the equals sign. 2
Some equations like 8y 3 = 6y + 17 have variables on each side of the equal sign. To solve equations that have variables on each side of the equation, you must eliminate the variable from one side and then solve the equation. Remember to keep the equation balanced. Solve 8y 3 = 6y + 17. Check your solution. 8y 3 6y 17 6 y 6 y 2y 3 17 3 3 2y 20 2y 20 2 2 y 10 Write the equation. Subtraction Property of Equality Simplify. Addition Property of Equality Simplify. Division Property of Equality Simplify. To check the solution, replace y with 10 in the original equation. 8y 3 6y 17? 8 10 3 6 10 17? 80 3 60 17 Write the original equation. Replace y with 10. Multiply. 77 = 77 The sentence is true. Notes 3
More examples/summary : 4
803 1.4a Skills Practice (8.ee.7ab) Solve Equations with Variables on Each Side Solve each equation. Check your solution. 1. 3w + 6 = 4w 2. a + 18 = 7a 3. 8c = 5c + 21 4. 11d + 10 = 6d 5. 2e = 4e 16 6. 7v = 2v 20 7. 4n 6 = 10n 8. 2y + 27 = 5y 9. 8h = 6h 14 10. 18 2g = 4g 5
11. 4x 9 = 6x 13 12. 5c 15 = 2c + 6 13. t + 10 = 7t 14 14. 8z + 6 = 7z + 4 15. 2e 12 = 7e + 8 16. 9k + 6 = 8k + 13 17. 2d + 10 = 6d 10 18. 2a 9 = 6a + 15 19. 8 3k = 3k + 2 20. 7t 4 = 10t + 14 6
21. 3c 15 = 17 c 22. 14 + 3n = 5n 6 23. 3y + 5.2 = 2 5y 24. 10b 2 = 7b 7.4 25. 2m 2 = 6m 4 26. 3g + 5 = 7g + 4 27. 4s 1 = 8 2s 28. 9w + 3 = 4w 9 29. 6z 7 = 2z 2 30. 3 a = 4a + 12 7
More Practice 1. 4n + 7 = 10 + n 2. 6n 1 = 4n 5. 3. 3n 8 = -2n + 7 4. 3n = 2n + 7 5.. Greens Gym charges a one-time fee of $50 plus $30 per session for a personal trainer. A new fitness center charges a yearly fee of $250 plus $10 for each session with a trainer. For how many sessions is the cost of the two plans the same? 8
MORE Equations with Variables on Each Side In some equations, the coefficients of the variables are rational numbers. Remember when working with fractions, you need to have common denominator before you add or subtract. Example: 2 3 x 1 = 9 1 6 x 4 6 x 1 = 9 1 6 x find the common denominator of the coefficients + 1 x + 1 x 6 6 add 1 x to each side 6 Notes 5 6 5 6 x 1 = 9 simplify +1 +1 add 1 to each side x = 10 simplify 6 5 5 6 x = 10 6 5 multiply by the reciprical x = 12 2 3 (12) 1 = 9 1 6 (12) 8 1 = 9 2 7 = 7 9
Practice: Solve each equation. Then check your solution 1) 2x + 1 = x + 11 2) 3x 7 = 8x + 23 3) 5a 3 = 8a + 6 4) 3n 2 = 5n + 12 5) 1 3 b 7 = 3 2 9 b 6) 15 1 6 n = 1 6 n 1 7) Three times a number equals 40 more than five times the number. What is the number? 8) A number equals four less than three times the number. What is the number? 10
9) Write an equation to find the value of x so that each pair of polygons has the same perimeter. Then solve to determine what the perimeter is. x + 4 x + 2 x + 1 x + 5 x + 3 10) One cellular phone carrier charges $36.25 a month plus $0.10 a minute for local calls. Another carrier charges $24.50 a month and $0.20 a minute for local calls. For how many minutes is the cost of the plans the same? 11) Suppose a video store charges nonmembers $4 to rent each video. A store membership costs $21 and members pay only $2.50 to rent each video. For what number of videos is the cost the same? 11
803 1.4b Homework Practice (8.ee.7ab) Solve Equations with Variables on Each Side Solve each equation. Check your solution. 1. 9m + 14 = 2m 2. 13x = 32 + 5x 3. 8d 25 = 3d 4. t 27 = 4t 5. 7p 5 = 6p + 8 6. 11z 5 = 9z + 7 7. 12 5h = h + 6 8. 4 7f = f 12 9. 6y + 17 = 3y 10 10. 3x 32 = 7x + 28 11. 3.2a 16 = 4a 12. 16.8 v = 6v 12
Define a variable, write an equation, and solve to find each number. 13. Fourteen less than five times a number is three times the number. 14. Twelve more than seven times a number equals the number less six. Write an equation to find the value of x so that each pair of polygons has the same perimeter. Then solve. 15. 16. 13
Write and solve an equation to solve each exercise. 17. GOLF For an annual membership fee of $500, Mr. Bailey can join a country club that would allow him to play a round of golf for $35. Without the membership, the country club charges $55 for each round of golf. How many rounds of golf would Mr. Bailey have to play for the cost to be the same with and without a membership? 18. MUSIC Marc has 45 CDs in his collection, and Corinna has 61. If Marc buys 4 new CDs each month and Corinna buys 2 new CDs each month, after how many months will Marc and Corinna have the same number of CDs? 14
Lesson 1.5 Solving Multi-step Equations (Textbook 2.5) (8.EE.7ab) Learning Target: I can solve Multi-Step equations. To solve multi- step equations 1. expand the expression using the Distributive Property 2. Collect like terms if needed 3. solve the equation using the Properties of Equality Example: 15(20 + d) = 420 Check 15(20 + 8) = 420 300 + 15d = 420 15(28) = 420-300 -300 420 = 420 15d = 120 15 15 d = 8 Practice: 1) 4(n 2) = 16 2) 2(n + 3) = 36 15
3) 5(a 7) = 25 4) - 3(9 + x) = 33 5) 8(4 2x) = 4(3 5x) + 4 6) 8(3a + 6) = 9(2a 4) 7) 3(3n 2) = 2(3n + 3) 8) 7-3n = n 4(2 + n) 16
803 1.5a Skills Practice (8.EE.7ab) Solve Multi-Step Equations Solve each equation. Check your solution. 1. 4(2 + 3c) = 56 2. 63 = 3(1 2n) 3. 29 = 5(2a 1) + 2a 4. 2(3 + g) = 4(g + 3) 5. r r 6. 3(t + 5) + (4t + 2) = 8 17
7. -5(3m + 6) = - 3(4m 2) 8. 18 = 3(3x 6) 9. 12(x + 3 ) = 4(2x + 8) 10. 4(2 y) + 3y = 3(y 4) 11. HEALTH CLUB Currently, 96 members participate in the morning workout, and this number has been increasing by 2 people per week. Currently, 80 members participate in the afternoon workout, and this number has been decreasing by 3 people per week. In how many weeks will the number of people working out in the morning be double the number of people working out in the afternoon? 12. DISTANCE Two cyclists leave town at the same time on the same road going in the same direction. Cyclist A is going 6 miles per hour faster than cyclist B. After 8 hours, cyclist A has traveled three times the distance as cyclist B. Use the equation 24x = 8(x + 6) to find how fast cyclist B is traveling. 18
803 1.5b Homework Practice (8.EE.7ab) Solve Multi Step Equations Solve each equation. Check your solution. 1. 5(x 3) + 2x = 41 2. 4a 3(a 2) = 2(3a 2) 3. (7t 2) ( 3t + 1) = 3(1 3t) 4. 14 2(3p + 1) = 6(4 + p) 5. 2 (14q + 7 ) 3q = 9 6. x (4x 7) = 5x (x + 21) 7 2 7. LAWNS Luisa mows lawns during the summer. She charges $15 if she cuts the grass but charges $5 more if she also trims the grass. Last week she trimmed 5 more yards than she cut. If she made $415 last week, how many yards did she trim? 19
803 1.5c (8.ee.7ab) 1. 6(m 2) = 12 2. 4(x 3) = 4 3. 5(2d + 4) = 35 4. w + 6 = 2(w 6) 5. 3(b + 1) = 4b 1 6. 7w 6 = 3(w + 6) 7. 4(k 6) = 6(k + 2) 8. 3(x 0.8) = 4x + 4 9. 5 9(g + 18) = 1 6g + 3 10. 4(c + 12) = 2c + 18 11. 7(d 2) = 5(d + 2) 12. 5p 17 = 2(2p 7) 20
13. 4(3z 2) = 9z 7 14. 7s + 2 = 4(s + 1) 15. 6(k + 1) = 2k + 7 16. 6(n 1) = 2(n + 1) 24 17. 14y 3 = 5 2y 18. 3(3q + 6) = 8 21
Example: 2(2x + 3) = 5x + 6 x Practice: 1) 6(x 3) + 10 = 2(3x 4) 2) 8(4 2x) = 4(3 5x) + 4x 22
3) 3(6 4x) = -2(6x 9) 4) 2(3x + 5) = 5(2x 4) 4x 5) 3(6 4x) = -2(6x 9) 6) 2(3x + 5) = 5(2x 4) 4x 7) 8z 22 = 3(3z + 11) z 8) 8(c 9) = 6(2c 12) 4c 23
803 1.2-1.5 Review 1. Susan is 5 years older than her sister. The sum of their ages is 51. Then write an equation that could be used to find their ages. 2. Mark is 6 years older than his brother. If the sum of their ages is 86, how old is each person? 3. At a concert, you purchase 3 T-shirts and a concert program for a total cost of $90. The program costs $15 and the T-shirts all cost the same. Write and solve an equation to find the cost of one T-shirt. Solve each equation. 4. d = - 5 5. 2 4 3 m + 2 = 12 6. 6(n 3) + 10 = 2(3n 4) 7. 6w = 10 + 4w 8. 8(4 2n) = 4(3 5n) + 4n 9. 4(5 + 2x) 5 = 3(3x + 7) 24
10. An online movie streaming plan charges an annual fee of $45 plus $2.50 per movie watched. Another plan has no annual fee but charges $3.75 per movie watched. For how many movies is the cost of the plans the same? 11. Find the value of x so that the polygons have the same perimeter. Solve each equation. 12. 50 = 2(a + 3) 13. 4(x 2) = 2(x 4) + 2x 14. 5(y 2) 2 = 2(y + 1) 5 15. 4(p + 1) = 2(8 2p) 18. Tony and some friends went to the movies. They bought 4 drinks and 2 tubs of popcorn and spent a total of $32.50 on the food. Each drink costs $3.50 less than a tub of popcorn. a. Define a variable. Write an equation that can be used to find the cost of one tub of popcorn. b. Solve the equation to find the cost of a tub of popcorn. 25
803 1.4-1.5 word problems (8.EE.7ab) 1. BACKPACKING Guido and Raoul each went backpacking in Glacier National Park. The expressions 4(d + 2) - 2d and 3(2 + d) represent the respective distances Guido and Raoul hiked each day. On what day number d will their distance hiking be the same? 2. SAVINGS The table below shows the savings account balance of each of the Alvarez siblings. Hint: Add all 3 together to equal $148. Solve. Sibling Account Balance Cindy s Pete 2(s + 3) Nila 4s 5 a. Write an equation to find the amount of money in Pete s account if the total of all of their accounts is $148. b. How much does each person have in their account? 3. Jason s Gym charges a one-time fee of $50 plus $30 per session for a personal trainer. A new fitness center charges a yearly fee of $250 plus $10 for each session with a trainer. For how many sessions is the cost of the two plans the same? 26
4.. An electrician charges $75 to make a house call. Plus an additional $40 per hour for each hour on site. How many hours did she work if the bill was 255 dollars? Write and solve an equation. 5. Linden Car Rental charges $40 a day plus $0.25 per mile. Morrison Rent-A-Car charges $25 a day plus $0.45 per mile. How many miles in one day would you have to drive to pay the same amount? Write and solve an equation. 6. Five more than twice a number is 9. Write and solve an equation. 27
803 Unit 1 Test Review Solve each equation. Check your answers. 1. 10x 3(2x + 7) = 5 2. 1 x = 16 3. 2 x = 20 4 5 4. x 3 + 4 = 13 5. 2z 31 = 9z + 24 6. 12 = 12 4a 7. 9(j 4) = 81 8. 9(k 5) + 7(k + 9) = 14k 9. 2y + 8 4y = 8 5y 12 10. An online movie streaming plan has no annual fee but charges $4.25 per movie watched. Another plan charges an annual fee of $36 plus $3.50 per movie watched. For how many movies is the cost of the plans the same? 28
Define a variable, write an equation, and solve to find the number. 11. Eighteen less than three times a number is twice the number. 12. Four times a number increased by 3 is 89. 13. Three less than one-half a number is 71. SOLVE 14. 11x + 4 = 48 15. 30 = 2( n + 3) 16. 1 n = 2 2 n + 21 3 3 17. 2c + 3 = 9 18. 6p 5 = 17 29
Solve each equation. Check your answers. 19. 3(y 2) + 15 = 3(y 3) + 6y 20. 5(c 2) = 20 5c + 10 21. Find the value of x so that the polygons have the same perimeter. 22. The table shows the number of hits made by three players in yesterday s softball game. If Mercedes and Kiaya had the same number of hits, how many hits did Evelyn have? Player Evelyn Points Mercedes 3x 1 x Kiaya 4x 2 23. The table shows the number of fish Callie and Jada each caught. If they caught the same number of fish, how many did each catch? Name Number of Fish Caught Callie 2(3t + 1) Jada 4(2t 1) 30
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