Solving Equations Having Like Terms and Parentheses
LESSON
3.2 Review Vocabulary like terms, p. 78
Solving Equations Having Like Terms and Parentheses Now
BEFORE
WHY?
You used the distributive You’ll solve equations using property. the distributive property.
So you can budget for fishing rods, as in Ex. 20.
School Spirit Your school’s basketball team is playing in the championship game. For the game, the cheerleaders want to buy a banner that costs $47. They also want to buy small items to give to students in the stands. Pompoms cost $5.20 each. Noisemakers cost $.80 each. The cheerleaders have a total budget of $375 for the game. If they buy equal numbers of pompoms and noisemakers, how many can they afford to buy?
Example 1
Writing and Solving an Equation
Find how many pompoms and noisemakers the cheerleaders can afford to buy, as described above. Solution Let n represent the number of pompoms and the number of noisemakers. Then 5.20n represents the cost of n pompoms, and 0.80n represents the cost of n noisemakers. Write a verbal model. Cost of n pompoms
Cost of n noisemakers
5.20n 0.80n 47 375 6.00n 47 375 6n 47 47 375 47 6n 328 6n 328 _ _ 6 6 2 n 54 _ 3
Cost of banner
Total budget
Substitute. Combine like terms. Subtract 47 from each side. Simplify. Divide each side by 6. Simplify.
Answer The answer must be a whole number. Round down so the budget is not exceeded. The cheerleaders can afford to buy 54 pompoms and 54 noisemakers.
Video Tutor Go to thinkcentral.com
Lesson 3.2
Solving Equations Having Like Terms and Parentheses
125
Distributive Property You can use the distributive property to solve equations involving parentheses.
Example 2
Review Help For help with using the distributive property, see p. 73.
Solving Equations Using the Distributive Property
Solve the equation.
a. 21 7(3 x)
b. 3(8 4x) 12
Solution
a.
21 7(3 ⴚ x)
Write original equation.
21 21 ⴚ 7x
Distributive property
21 ⴚ 21 21 7x ⴚ 21
Study Strategy In part (a) of Example 2, the substitution property of equality allows you to substitute 21 7x for 7(3 x). This property states that if two expressions are equivalent, then one can be substituted for the other in an equation.
Subtract 21 from each side.
42 7x
Simplify.
42 7x ⴚ7 ⴚ7
Divide each side by ⴚ7.
6x
Simplify.
Answer The solution is 6.
b.
ⴚ3(8 ⴚ 4x) 12
Write original equation.
ⴚ24 ⴙ 12x 12
Distributive property
24 12x ⴙ 24 12 ⴙ 24
Add 24 to each side.
12x 36
Simplify.
12x 36 12 12
Divide each side by 12.
x3
Simplify.
Answer The solution is 3.
Example 3
Combining Like Terms After Distributing
Solve 5x ⴚ 2(x ⴚ 1) ⴝ 8. 5x ⴚ 2(x ⴚ 1) 8
Write original equation.
5x ⴚ 2x ⴙ 2 8
Distributive property
3x 2 8
Combine like terms.
3x 2 ⴚ 2 8 ⴚ 2
Subtract 2 from each side.
3x 6
Simplify.
3x 6 3 3
Divide each side by 3.
x2
Simplify.
Checkpoint Solve the equation. Check your solution. 1. 3n 40 2n 15
126
Chapter 3
Multi-Step Equations and Inequalities
2. 2(s 1) 6
3. 13 2y 3( y 4)
3.2
Exercises More Practice, p. 841
Go to thinkcentral.com Practice Exercises
Guided Practice Vocabulary Check
1. What property do you use when you rewrite the equation
6(x 1) 12 as 6x 6 12?
2. Identify the like terms you would combine to solve the equation
3x 5 2x 8 12.
Skill Check
Guided Problem Solving
Solve the equation. Check your solution. 3. 4 x 7 10
4. 3x 2x 25
5. 21 4x 9 x
6. 3(x 1) 6
7. 16 8(x 1)
8. 5 2(x 2) 19
9. Geometry The perimeter of the rectangle
x2
shown is 28 units. The length is 10 units. What is the width of the rectangle?
10
1
Write an equation for the perimeter of the rectangle in terms of x.
2
Solve the equation to find the value of x.
3
Find the width of the rectangle using the value of x.
4
Check your answer.
Practice and Problem Solving Homework Help Example 1 2 3
Exercises 11–13, 20, 21 14–19 22–31
10. Error Analysis Describe and
correct the error in solving the equation 2(5 n) 2.
10 2n 2 10 2n 10 2 10 2n 12 n 6
Lesson Resources Go to thinkcentral.com • More Examples • @HomeTutor
2(5 n) 2
Solve the equation. Check your solution. 11. 13t 7 10t 2
12. 22 4y 14 0
13. 2d 24 3d 84
14. 4(x 5) 16
15. 3(7 2y) 9
16. 2(z 11) 6
17. 5(3n 5) 20
18. 30 6( f 5)
19. 12 3(m 17)
20. Fishing A family of five people has $200 to spend on fishing rods and
fishing licenses. They spend a total of $20 on licenses. Assuming they buy 5 identical rods, what is the maximum amount they can spend on each rod?
Lesson 3.2
Solving Equations Having Like Terms and Parentheses
127
21. Karaoke You want to organize a group of friends to go to a karaoke
studio this Friday night. You must pay $30 to reserve a private karaoke room plus $5 for each person in the group. You also want to have snacks for the group at a cost of $2 per person. How many people can be in the group if you have $70 to spend? Solve the equation. Check your solution. 22. 5(2w 1) 25
23. 4(5 p) 8
24. 40 (2x 5) 61
25. 2 4(3k 8) 11k
26. 42 18t 4(t 5)
27. 3(2z 8) 10z 16
28. 5g (8 g) 12
29. 5 0.25(4 20r) 8r
30. 2m 0.5(m 4) 9
31. 12 2h 0.2(20 6h)
32. Photograph The perimeter of a rectangular photograph is 22 inches.
The length of the photograph is 1 inch more than the width. What are the dimensions of the photograph? Geometry Find the value of x for the given triangle, rectangle, or square. 33. Perimeter 40 units
34. Perimeter 22 units x
7
5
x1
x2
35. Perimeter 104 units
36. Perimeter 32 units x
x 11
2x 10
x 11
37. Cell Phones Your cell phone provider charges a monthly fee of
$19.50 for 200 minutes. You are also charged $.25 per minute for each minute over 200 minutes. Last month, your bill was $29.50.
a. Let m represent the total number of minutes you used last month. Use the verbal model below to write an equation. Total Charge for each Monthly phone bill fee additional minute p
Number of minutes over 200
b. Solve the equation you wrote in part (a). c. How many additional minutes did you use last month? 38. Duathlons A duathlon is an event that consists of running and
biking. While training for a duathlon, you run and bike a total of 23 kilometers in 1.25 hours. You run at an average speed of 10 kilometers per hour and bike at an average speed of 24 kilometers per hour. Write and solve an equation to find the time you spend running and the time you spend biking.
128
Chapter 3
Multi-Step Equations and Inequalities
39. Challenge The figure shown is composed
24
of a triangle and a rectangle. The figure has a total area of 1258 square units. Find the value of x.
25 3x 1
Mixed Review
Plot the point in a coordinate plane. Describe the location of the point. (Lesson 1.8) 40. J(3, 8)
41. K(8, 3)
42. L(4, 4)
43. M(1, 1)
44. N(0, 2)
45. P(5, 1)
46. Q(9, 0)
47. R(5, 8)
Simplify the expression. (Lesson 2.3) 48. a 2 (3 a)
49. 3b 8 2(b 4)
50. 2x 5 7(x 1)
51. 2y 4 3(y 1)
52. (2x 3) 4(x 2)
53. 3(2x 7) 8(4 x)
54. Family Party A family wants to hold a dinner party at a restaurant.
The restaurant charges $150 to rent space for the party. The food cost for each person at the party is $18. How many people can come to the party if the family has $600 to spend? (Lesson 3.1)
Standardized Test Practice
55. Multiple Choice What is the solution of the equation
3(2x 1) 21?
A. 4
B. 3
C. 3
D. 4
56. Short Response The length of a rectangle is 5 feet less than twice its
width. The perimeter of the rectangle is 38 feet. Let w represent the width. Write an equation for the perimeter of the rectangle in terms of w. Then solve the equation to find the length and width of the rectangle.
Patent Puzzle Solve each equation. In each group, there are two To Come equations that have the same solution. Write the value of this solution in the corresponding letter’s blank to find the year blue jeans were patented.
A.
C.
10x 7 17 2(7x 6) 40 (x 11) 10
B.
? A
7x (12) 61
6(2x 1) 90
7(x 2) 63
Lesson 3.2
? D
5x 4x 6
8x 15 47
7x 4x 24
? C
? B D.
2(6x 7) 50 5x 3x 56 11x 9 42
Solving Equations Having Like Terms and Parentheses
129
3.2 Review Vocabulary like terms, p. 78
Solving Equations Having Like Terms and Parentheses Now
BEFORE
WHY?
You used the distributive You’ll solve equations using property. the distributive property.
So you can budget for fishing rods, as in Ex. 20.
School Spirit Your school’s basketball team is playing in the championship game. For the game, the cheerleaders want to buy a banner that costs $47. They also want to buy small items to give to students in the stands. Pompoms cost $5.20 each. Noisemakers cost $.80 each. The cheerleaders have a total budget of $375 for the game. If they buy equal numbers of pompoms and noisemakers, how many can they afford to buy?
Example 1
Writing and Solving an Equation
Find how many pompoms and noisemakers the cheerleaders can afford to buy, as described above. Solution Let n represent the number of pompoms and the number of noisemakers. Then 5.20n represents the cost of n pompoms, and 0.80n represents the cost of n noisemakers. Write a verbal model. Cost of n pompoms
Cost of n noisemakers
5.20n 0.80n 47 375 6.00n 47 375 6n 47 47 375 47 6n 328 6n 328 _ _ 6 6 2 n 54 _ 3
Cost of banner
Total budget
Substitute. Combine like terms. Subtract 47 from each side. Simplify. Divide each side by 6. Simplify.
Answer The answer must be a whole number. Round down so the budget is not exceeded. The cheerleaders can afford to buy 54 pompoms and 54 noisemakers.
Video Tutor Go to thinkcentral.com
Lesson 3.2
Solving Equations Having Like Terms and Parentheses
125
Distributive Property You can use the distributive property to solve equations involving parentheses.
Example 2
Review Help For help with using the distributive property, see p. 73.
Solving Equations Using the Distributive Property
Solve the equation.
a. 21 7(3 x)
b. 3(8 4x) 12
Solution
a.
21 7(3 ⴚ x)
Write original equation.
21 21 ⴚ 7x
Distributive property
21 ⴚ 21 21 7x ⴚ 21
Study Strategy In part (a) of Example 2, the substitution property of equality allows you to substitute 21 7x for 7(3 x). This property states that if two expressions are equivalent, then one can be substituted for the other in an equation.
Subtract 21 from each side.
42 7x
Simplify.
42 7x ⴚ7 ⴚ7
Divide each side by ⴚ7.
6x
Simplify.
Answer The solution is 6.
b.
ⴚ3(8 ⴚ 4x) 12
Write original equation.
ⴚ24 ⴙ 12x 12
Distributive property
24 12x ⴙ 24 12 ⴙ 24
Add 24 to each side.
12x 36
Simplify.
12x 36 12 12
Divide each side by 12.
x3
Simplify.
Answer The solution is 3.
Example 3
Combining Like Terms After Distributing
Solve 5x ⴚ 2(x ⴚ 1) ⴝ 8. 5x ⴚ 2(x ⴚ 1) 8
Write original equation.
5x ⴚ 2x ⴙ 2 8
Distributive property
3x 2 8
Combine like terms.
3x 2 ⴚ 2 8 ⴚ 2
Subtract 2 from each side.
3x 6
Simplify.
3x 6 3 3
Divide each side by 3.
x2
Simplify.
Checkpoint Solve the equation. Check your solution. 1. 3n 40 2n 15
126
Chapter 3
Multi-Step Equations and Inequalities
2. 2(s 1) 6
3. 13 2y 3( y 4)
3.2
Exercises More Practice, p. 841
Go to thinkcentral.com Practice Exercises
Guided Practice Vocabulary Check
1. What property do you use when you rewrite the equation
6(x 1) 12 as 6x 6 12?
2. Identify the like terms you would combine to solve the equation
3x 5 2x 8 12.
Skill Check
Guided Problem Solving
Solve the equation. Check your solution. 3. 4 x 7 10
4. 3x 2x 25
5. 21 4x 9 x
6. 3(x 1) 6
7. 16 8(x 1)
8. 5 2(x 2) 19
9. Geometry The perimeter of the rectangle
x2
shown is 28 units. The length is 10 units. What is the width of the rectangle?
10
1
Write an equation for the perimeter of the rectangle in terms of x.
2
Solve the equation to find the value of x.
3
Find the width of the rectangle using the value of x.
4
Check your answer.
Practice and Problem Solving Homework Help Example 1 2 3
Exercises 11–13, 20, 21 14–19 22–31
10. Error Analysis Describe and
correct the error in solving the equation 2(5 n) 2.
10 2n 2 10 2n 10 2 10 2n 12 n 6
Lesson Resources Go to thinkcentral.com • More Examples • @HomeTutor
2(5 n) 2
Solve the equation. Check your solution. 11. 13t 7 10t 2
12. 22 4y 14 0
13. 2d 24 3d 84
14. 4(x 5) 16
15. 3(7 2y) 9
16. 2(z 11) 6
17. 5(3n 5) 20
18. 30 6( f 5)
19. 12 3(m 17)
20. Fishing A family of five people has $200 to spend on fishing rods and
fishing licenses. They spend a total of $20 on licenses. Assuming they buy 5 identical rods, what is the maximum amount they can spend on each rod?
Lesson 3.2
Solving Equations Having Like Terms and Parentheses
127
21. Karaoke You want to organize a group of friends to go to a karaoke
studio this Friday night. You must pay $30 to reserve a private karaoke room plus $5 for each person in the group. You also want to have snacks for the group at a cost of $2 per person. How many people can be in the group if you have $70 to spend? Solve the equation. Check your solution. 22. 5(2w 1) 25
23. 4(5 p) 8
24. 40 (2x 5) 61
25. 2 4(3k 8) 11k
26. 42 18t 4(t 5)
27. 3(2z 8) 10z 16
28. 5g (8 g) 12
29. 5 0.25(4 20r) 8r
30. 2m 0.5(m 4) 9
31. 12 2h 0.2(20 6h)
32. Photograph The perimeter of a rectangular photograph is 22 inches.
The length of the photograph is 1 inch more than the width. What are the dimensions of the photograph? Geometry Find the value of x for the given triangle, rectangle, or square. 33. Perimeter 40 units
34. Perimeter 22 units x
7
5
x1
x2
35. Perimeter 104 units
36. Perimeter 32 units x
x 11
2x 10
x 11
37. Cell Phones Your cell phone provider charges a monthly fee of
$19.50 for 200 minutes. You are also charged $.25 per minute for each minute over 200 minutes. Last month, your bill was $29.50.
a. Let m represent the total number of minutes you used last month. Use the verbal model below to write an equation. Total Charge for each Monthly phone bill fee additional minute p
Number of minutes over 200
b. Solve the equation you wrote in part (a). c. How many additional minutes did you use last month? 38. Duathlons A duathlon is an event that consists of running and
biking. While training for a duathlon, you run and bike a total of 23 kilometers in 1.25 hours. You run at an average speed of 10 kilometers per hour and bike at an average speed of 24 kilometers per hour. Write and solve an equation to find the time you spend running and the time you spend biking.
128
Chapter 3
Multi-Step Equations and Inequalities
39. Challenge The figure shown is composed
24
of a triangle and a rectangle. The figure has a total area of 1258 square units. Find the value of x.
25 3x 1
Mixed Review
Plot the point in a coordinate plane. Describe the location of the point. (Lesson 1.8) 40. J(3, 8)
41. K(8, 3)
42. L(4, 4)
43. M(1, 1)
44. N(0, 2)
45. P(5, 1)
46. Q(9, 0)
47. R(5, 8)
Simplify the expression. (Lesson 2.3) 48. a 2 (3 a)
49. 3b 8 2(b 4)
50. 2x 5 7(x 1)
51. 2y 4 3(y 1)
52. (2x 3) 4(x 2)
53. 3(2x 7) 8(4 x)
54. Family Party A family wants to hold a dinner party at a restaurant.
The restaurant charges $150 to rent space for the party. The food cost for each person at the party is $18. How many people can come to the party if the family has $600 to spend? (Lesson 3.1)
Standardized Test Practice
55. Multiple Choice What is the solution of the equation
3(2x 1) 21?
A. 4
B. 3
C. 3
D. 4
56. Short Response The length of a rectangle is 5 feet less than twice its
width. The perimeter of the rectangle is 38 feet. Let w represent the width. Write an equation for the perimeter of the rectangle in terms of w. Then solve the equation to find the length and width of the rectangle.
Patent Puzzle Solve each equation. In each group, there are two To Come equations that have the same solution. Write the value of this solution in the corresponding letter’s blank to find the year blue jeans were patented.
A.
C.
10x 7 17 2(7x 6) 40 (x 11) 10
B.
? A
7x (12) 61
6(2x 1) 90
7(x 2) 63
Lesson 3.2
? D
5x 4x 6
8x 15 47
7x 4x 24
? C
? B D.
2(6x 7) 50 5x 3x 56 11x 9 42
Solving Equations Having Like Terms and Parentheses
129
