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1-3 Solving Equations
Write an algebraic expression to represent each verbal expression.
2. the difference between the product of 4 and a number and the square of the number
SOLUTION: Let x be the number.
2
4 times of x is 4x. The square of x is x .
The keyword ‘difference’ means subtraction.
2
So, the algebraic expression is 4x – x .
Write a verbal sentence to represent each equation.
4. SOLUTION: The difference between the square of a number and 9 is 27.
6. SOLUTION: Eight more than the quotient of a number and four is –16.
Name the property illustrated by each statement.
and
, then
.
8. If SOLUTION: Transitive Property; the Transitive Property of Equality states that for any real numbers a, b, and c, if a = b and b =
c, then a = c.
Solve each equation. Check your solution.
10. SOLUTION: Substitute x = –6 in the equation.
The solution of the equation is
x = –6.
12. SOLUTION: eSolutions
Manual x- Powered
= –7 inbytheCognero
Substitute
equation.
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The solution of the equation is
x = –6.
1-3 Solving
Equations
12. SOLUTION: Substitute x = –7 in the equation.
So, the solution is x = –7.
14. SOLUTION: Substitute y = –4 in the equation.
So, the solution is y = –4.
16. SOLUTION: Substitute c = 8 in the equation.
So, the solution is c = 8.
18. SOLUTION: eSolutions Manual - Powered by Cognero
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1-3 Solving Equations
So, the solution is c = 8.
18. SOLUTION: Substitute m = –5 in the equation.
So, the solution is m = –5.
20. Solve each equation or formula for the specified variable.
, for n
SOLUTION: Write an algebraic expression to represent each verbal expression.
22. the difference between the product of four and a number and 6
SOLUTION: Let the number be n.
The product of four and n is 4n.
The keyword ‘difference’ indicates subtraction.
The algebraic expression is 4n – 6.
24. fifteen less than the cube of a number
SOLUTION: Let the number be x.
3
Cube of x is x .
3
3
15 less than x is x – 15.
Write a verbal sentence to represent each equation.
26. SOLUTION: Four less than 8 times a number is 16.
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28. Page 3
SOLUTION: Let the number be x.
3
Cube of Equations
x is x .
1-3 Solving
3
3
15 less than x is x – 15.
Write a verbal sentence to represent each equation.
26. SOLUTION: Four less than 8 times a number is 16.
28. SOLUTION: Three less than four times the square of a number is 13.
Name the property illustrated by each statement.
30. If x + 9 = 2, then x + 9 – 9 = 2 – 9
SOLUTION: Subtraction Property of Equality; the Subtraction Property of Equality states that for any real numbers a, b, and c, if
a = b, then a - c = b - c.
32. If g = 3h and 3h = 16, then g = 16
SOLUTION: Transitive Property; the Transitive Property of Equality states that for any real numbers a, b, and c, if a = b and b =
c, then a = c.
34. MONEY Aiko and Kendra arrive at the state fair with $32.50. What is the total number of rides they can go on if
they each pay the entrance fee?
SOLUTION: Let n be the total number of rides.
The entrance fee for two persons = 2($7.50) = $15.00
So, Aiko and Kendra can go on a total of 7 rides.
Solve each equation. Check your solution.
36. SOLUTION: eSolutions Manual - Powered by Cognero
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1-3 Solving Equations
So, Aiko and Kendra can go on a total of 7 rides.
Solve each equation. Check your solution.
36. SOLUTION: Substitute x = –7 in the equation.
The solution is x = –7.
38. SOLUTION: Substitute g = –5 in the original equation.
The solution is g = –5.
40. SOLUTION: Substitute y = 8 in the equation.
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1-3 Solving Equations
The solution is g = –5.
40. SOLUTION: Substitute y = 8 in the equation.
The solution is y = 8.
42. SOLUTION: Substitute d = 4 in the equation.
The solution is d = 4.
44. MEDICINE For Nina’s illness her doctor gives her a prescription for 28 pills. The doctor says that she should take
4 pills the first day and then 2 pills each day until her prescription runs out. For how many days does she take 2 pills?
SOLUTION: Let x be the number of days Nina takes 2 pills.
Total number of pills = 28.
So:
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1-3 Solving Equations
The solution is d = 4.
44. MEDICINE For Nina’s illness her doctor gives her a prescription for 28 pills. The doctor says that she should take
4 pills the first day and then 2 pills each day until her prescription runs out. For how many days does she take 2 pills?
SOLUTION: Let x be the number of days Nina takes 2 pills.
Total number of pills = 28.
So:
Nina takes 2 pills a day for 12 days.
Solve each equation or formula for the specified variable.
, for a
46. SOLUTION: 48. , for y
SOLUTION: 50. , for z
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1-3 Solving Equations
50. , for z
SOLUTION: 52. AWARDS BANQUET A banquet room can seat a maximum of 69 people. The coach, principal, and vice principal
have invited the award-winning girls’ tennis team to the banquet. If the tennis team consists of 22 girls, how many
guests can each student bring?
SOLUTION: Let n be the number of guests each student can bring.
The maximum number of people can be seated in the room is 69. The tennis team, coach, principal and vice principal
gives 25 to attend the banquet.
Solve for n.
Therefore, each student can bring 2 guests.
Solve each equation. Check your solution.
54. SOLUTION: Check:
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Therefore,
each student can bring 2 guests.
1-3 Solving
Equations
Solve each equation. Check your solution.
54. SOLUTION: Check:
The solution is x = 3.
56. SOLUTION: Check:
The solution is p = 3.
58. SOLUTION: eSolutions Manual - Powered by Cognero
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1-3 Solving
Equations
The solution is p = 3.
58. SOLUTION: Check:
The solution is
.
60. BRIDGES The Sunshine Skyway Bridge spans Tampa Bay, Florida. Suppose one crew began building south from
St. Petersburg, and another crew began building north from Bradenton. The two crews met 10,560 feet south of St.
Petersburg approximately 5 years after construction began.
a. Suppose the St. Petersburg crew built an average of 176 feet per month. Together the two crews built 21,120 feet
of bridge. Determine the average number of feet built per month by the Bradenton crew.
b. About how many miles of bridge did each crew build?
c. Is this answer reasonable? Explain.
SOLUTION: a. Let x be represent the average number of feet built per month by the Bradenton crew.
The number of feet the St. Petersburg crew built in 5 years is
.
Therefore:
The Bradenton crew built an average of 176 feet per month.
b. Each crew built a distance of 10,560 feet.
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5,280 feet = 1 mile
Therefore, each crew built a distance of 2 miles.
c. Yes; it seems reasonable that two crews working 4 miles apart would be able to complete the same amount of
1-3 Solving
Equations
The solution
is
.
60. BRIDGES The Sunshine Skyway Bridge spans Tampa Bay, Florida. Suppose one crew began building south from
St. Petersburg, and another crew began building north from Bradenton. The two crews met 10,560 feet south of St.
Petersburg approximately 5 years after construction began.
a. Suppose the St. Petersburg crew built an average of 176 feet per month. Together the two crews built 21,120 feet
of bridge. Determine the average number of feet built per month by the Bradenton crew.
b. About how many miles of bridge did each crew build?
c. Is this answer reasonable? Explain.
SOLUTION: a. Let x be represent the average number of feet built per month by the Bradenton crew.
The number of feet the St. Petersburg crew built in 5 years is
.
Therefore:
The Bradenton crew built an average of 176 feet per month.
b. Each crew built a distance of 10,560 feet.
5,280 feet = 1 mile
Therefore, each crew built a distance of 2 miles.
c. Yes; it seems reasonable that two crews working 4 miles apart would be able to complete the same amount of
miles in the same amount of time.
62. ERROR ANALYSIS Steven and Jade are solving
for b 2. Is either of them correct? Explain your
reasoning.
SOLUTION: Sample answer: Jade; in the last step, when Steven subtracted b 1 from each side, he mistakenly put the –b 1 in the
numerator instead of after the entire fraction. To solve for b 2, b 1must be subtracted from each side.
64. REASONING Use what you have learned in this lesson to explain why the following number trick works.
• Take any number.
• Multiply it by ten.
• Subtract 30 from the result.
• Divide the new result by 5.
• Add 6 to the result.
• Your new number is twice your original.
SOLUTION: This number trick is a series of steps that can be represented by an equation with x representing the number chosen.
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SOLUTION: Sample answer: Jade; in the last step, when Steven subtracted b 1 from each side, he mistakenly put the –b 1 in the
1-3 Solving Equations
numerator instead of after the entire fraction. To solve for b 2, b 1must be subtracted from each side.
64. REASONING Use what you have learned in this lesson to explain why the following number trick works.
• Take any number.
• Multiply it by ten.
• Subtract 30 from the result.
• Divide the new result by 5.
• Add 6 to the result.
• Your new number is twice your original.
SOLUTION: This number trick is a series of steps that can be represented by an equation with x representing the number chosen.
66. WRITING IN MATH Compare and contrast the Substitution Property of Equality and the Transitive Property of
Equality.
SOLUTION: Sample answer: The Transitive Property utilizes the Substitution Property. While the Substitution Property is done
with two values, that is, one being substituted for another, the Transitive Property deals with three values,
determining that since two values are equal to a third value, then they must be equal.
68. SAT/ACT What is
F
G
H
subtracted from its reciprocal?
J
K
SOLUTION: The reciprocal of
is .
So, the correct choice is G.
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70. SHORT RESPONSE A local theater sold 1200 tickets during the opening weekend of a movie. On the following
weekend, 840 tickets were sold. What was the percent decrease of tickets sold?
SOLUTION: Sample answer: The Transitive Property utilizes the Substitution Property. While the Substitution Property is done
with twoEquations
values, that is, one being substituted for another, the Transitive Property deals with three values,
1-3 Solving
determining that since two values are equal to a third value, then they must be equal.
68. SAT/ACT What is
F
G
H
subtracted from its reciprocal?
J
K
SOLUTION: The reciprocal of
is .
So, the correct choice is G.
70. SHORT RESPONSE A local theater sold 1200 tickets during the opening weekend of a movie. On the following
weekend, 840 tickets were sold. What was the percent decrease of tickets sold?
SOLUTION: Difference = 1200 – 840 = 360
72. BAKING Tamera is making two types of bread. The first type of bread needs
needs
cups of flour, and the second cups of flour. Tamera wants to make 2 loaves of the first recipe and 3 loaves of the second recipe. How many cups of flour does she need?
SOLUTION: eSolutions Manual - Powered by Cognero
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1-3 Solving Equations
72. BAKING Tamera is making two types of bread. The first type of bread needs
needs
cups of flour, and the second cups of flour. Tamera wants to make 2 loaves of the first recipe and 3 loaves of the second recipe. How many cups of flour does she need?
SOLUTION: 74. Evaluate
, if a = 5, b = 7, and c = 2.
SOLUTION: Identify the additive inverse for each number or expression.
76. 3.5
SOLUTION: Since 3.5 – 3.5 = 0, the additive inverse of 3.5 is –3.5.
78. SOLUTION: The additive inverse of 6 is –6 and the additive inverse of –7y is 7y.
The additive inverse of 6 –7y is –6 + 7y.
80. SOLUTION: Since (–1.25) + 1.25 = 0, the additive inverse of –1.25 is 1.25.
82. SOLUTION: The additive inverse of 4 is –4 and the additive inverse of 9x is –9x.
So, the additive inverse of 4 – 9x is –4 + 9x.
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