2-4 Solving Equations with the Variable on Each Side
Solve each equation. Check your solution.
1. 13x + 2 = 4x + 38
SOLUTION: Check:
2. SOLUTION: Check:
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2-4 Solving Equations with the Variable on Each Side
2. SOLUTION: Check:
3. 6(n + 4) = −18
SOLUTION: Check:
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4. 7 = −11 + 3(b + 5)
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2-4 Solving Equations with the Variable on Each Side
3. 6(n + 4) = −18
SOLUTION: Check:
4. 7 = −11 + 3(b + 5)
SOLUTION: Check:
5. 5 + 2(n + 1) = 2n
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2-4 Solving Equations with the Variable on Each Side
5. 5 + 2(n + 1) = 2n
SOLUTION: Since
Check:
, this equation has no solutions. To check, substitute any number for n.
6. 7 − 3r = r − 4(2 + r)
SOLUTION: Since
, this equation has no solutions. To check, substitute any number for r.
Check:
7. 14v + 6 = 2(5 + 7v) − 4
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2-4 Solving Equations with the Variable on Each Side
7. 14v + 6 = 2(5 + 7v) − 4
SOLUTION: Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this
equation. To check the solution, substitute any number for v.
Check:
8. 5h − 7 = 5(h − 2) + 3
SOLUTION: Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this
equation. To check the solution, substitute any number for h.
Check:
9. MULTIPLE CHOICE Find the value of x so that the figures have the same perimeter.
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A 4
B 5
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2-4 Solving Equations with the Variable on Each Side
9. MULTIPLE CHOICE Find the value of x so that the figures have the same perimeter.
A 4
B 5
C 6
D 7
SOLUTION: The following is an expression for the perimeter of the triangle. Let a = 3x + 4, b = 5x + 1, and c = 2x + 5.
The following is an expression for the perimeter of the rectangle.Let
and w = 2x.
Set the expressions equal to each other and solve for x.
Check to find the perimeters.
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2-4 Solving
Equations with the Variable on Each Side
Check to find the perimeters.
When x = 4, both figures have a perimeter of 50 units. Choice A is correct.
Solve each equation. Check your solution.
10. 7c + 12 = −4c + 78
SOLUTION: Check:
11. 2m − 13 = −8m + 27
SOLUTION: Check:
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2-4 Solving Equations with the Variable on Each Side
11. 2m − 13 = −8m + 27
SOLUTION: Check:
12. 9x − 4 = 2x + 3
SOLUTION: Check:
13. 6 + 3t = 8t − 14
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2-4 Solving Equations with the Variable on Each Side
13. 6 + 3t = 8t − 14
SOLUTION: Check:
14. SOLUTION: Check:
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15. SOLUTION: Page 9
2-4 Solving Equations with the Variable on Each Side
15. SOLUTION: Check:
16. 8 = 4(r + 4)
SOLUTION: Check:
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17. 6(n + 5) = 66
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2-4 Solving Equations with the Variable on Each Side
16. 8 = 4(r + 4)
SOLUTION: Check:
17. 6(n + 5) = 66
SOLUTION: Check:
18. 5(g + 8) − 7 = 103
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2-4 Solving Equations with the Variable on Each Side
18. 5(g + 8) − 7 = 103
SOLUTION: Check:
19. SOLUTION: Check:
20. 3(3m − 2) = 2(3m + 3)
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2-4 Solving Equations with the Variable on Each Side
20. 3(3m − 2) = 2(3m + 3)
SOLUTION: Check:
21. 6(3a + 1) − 30 = 3(2a − 4)
SOLUTION: Check:
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GEOMETRY
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2-4 Solving Equations with the Variable on Each Side
21. 6(3a + 1) − 30 = 3(2a − 4)
SOLUTION: Check:
22. GEOMETRY Find the value of x so the rectangles have the same area.
SOLUTION: The following is an expression for the area of the left rectangle.
The following is an expression for the area of the right rectangle.
Set the expressions equal to each other and solve for x.
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2-4 Solving Equations with the Variable on Each Side
22. GEOMETRY Find the value of x so the rectangles have the same area.
SOLUTION: The following is an expression for the area of the left rectangle.
The following is an expression for the area of the right rectangle.
Set the expressions equal to each other and solve for x.
Find the areas to check the solution.
When x = 8, both figures have an area of 96 square units.
23. NUMBER THEORY Four times the lesser of two consecutive even integers is 12 less than twice the greater
number. Find the integers.
SOLUTION: Let x be the first even integer and x + 2 be the next consecutive even integer.
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2-4 Solving
Equations with the Variable on Each Side
When x = 8, both figures have an area of 96 square units.
23. NUMBER THEORY Four times the lesser of two consecutive even integers is 12 less than twice the greater
number. Find the integers.
SOLUTION: Let x be the first even integer and x + 2 be the next consecutive even integer.
Substitute −4 into each expression to find the even numbers.
x = −4
x + 2 = −4 + 2 or −2
So, the consecutive even integers are −4 and −2.
24. CCSS SENSE-MAKING Two times the least of three consecutive odd integers exceeds three times the greatest
by 15. What are the integers?
SOLUTION: Let x = the least odd integer. Then x + 2 = the next greater odd integer, and x + 4 = the greatest of the three
integers.
Substitute −27 into each expression to find the odd numbers.
x = −27
x + 2 = −27 + 2 or −25
x + 4 = −27 + 4 or −23
So, the consecutive odd integers are −27, −25, and −23.
Solve each equation. Check your solution.
25. 2x = 2(x − 3)
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Page 16
x = −27
x + 2 = −27 + 2 or −25
x + 4 = −27 + 4 or −23
2-4 Solving
Equations with the Variable on Each Side
So, the consecutive odd integers are −27, −25, and −23.
Solve each equation. Check your solution.
25. 2x = 2(x − 3)
SOLUTION: Since
Check:
, this equation has no solutions. To check, substitute any number for x.
26. SOLUTION: Since
, this equation has no solutions. To check, substitute any number for h.
Check:
27. −5(3 − q) + 4 = 5q − 11
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Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this
equation. To check, substitute any number for q.
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2-4 Solving Equations with the Variable on Each Side
27. −5(3 − q) + 4 = 5q − 11
SOLUTION: Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this
equation. To check, substitute any number for q.
Check:
28. SOLUTION: Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this
equation. To check, substitute any number for r.
Check:
29. SOLUTION: eSolutions Manual - Powered by Cognero
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2-4 Solving
Equations with the Variable on Each Side
29. SOLUTION: Check:
30. SOLUTION: Check:
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2-4 Solving Equations with the Variable on Each Side
30. SOLUTION: Check:
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2-4 Solving Equations with the Variable on Each Side
31. SOLUTION: Check:
32. SOLUTION: eSolutions Manual - Powered by Cognero
Check:
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2-4 Solving Equations with the Variable on Each Side
32. SOLUTION: Check:
33. 6.78j − 5.2 = 4.33j + 2.15
SOLUTION: Check:
34. 14.2t – 25.2 = 3.8t + 26.8
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2-4 Solving Equations with the Variable on Each Side
34. 14.2t – 25.2 = 3.8t + 26.8
SOLUTION: Check:
35. 3.2k − 4.3 = 12.6k + 14.5
SOLUTION: Check:
36. 5[2p − 4(p + 5)] = 25
SOLUTION: Check:
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2-4 Solving Equations with the Variable on Each Side
36. 5[2p − 4(p + 5)] = 25
SOLUTION: Check:
37. NUMBER THEORY Three times the lesser of two consecutive even integers is 6 less than six times the greater
number. Find the integers.
SOLUTION: Let x be the first consecutive even integer and x + 2 be the next consecutive even integer.
Substitute −2 into each expression to find the even numbers.
x = −2
x + 2 = −2 + 2 or 0
So, the consecutive even integers are −2 and 0.
38. MONEY Chris has saved twice the number of quarters that Nora saved plus 6. The number of quarters Chris saved
is also five times the difference of the number of quarters and 3 that Nora has saved. Write and solve an equation to
find the number of quarters they each have saved.
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SOLUTION: Let q be the number of quarters Nora saved.
Substitute −2 into each expression to find the even numbers.
x = −2
x + 2 = −2 + 2 or 0
2-4 Solving
Equations with the Variable on Each Side
So, the consecutive even integers are −2 and 0.
38. MONEY Chris has saved twice the number of quarters that Nora saved plus 6. The number of quarters Chris saved
is also five times the difference of the number of quarters and 3 that Nora has saved. Write and solve an equation to
find the number of quarters they each have saved.
SOLUTION: Let q be the number of quarters Nora saved.
Substitute 7 for q to find the number of quarters Chris saved.
So, Nora saved 7 quarters and Chris saved 20 quarters.
39. DVD A company that replicates DVDs spends $1500 per day in building overhead plus $0.80 per DVD in supplies
and labor. If the DVDs sell for $1.59 per disk, how many DVDs must the company sell each day before it makes a
profit?
SOLUTION: Let d be the number of DVDs.
So, the company must sell more than 1,899 DVDs per day to make a profit.
40. MOBILE PHONES The table shows the number of mobile phone subscribers for two states for a recent year.
How long will it take for the numbers of subscribers to be the same? SOLUTION: eSolutions
Manual - Powered by Cognero
Let t be the number of years.
Let x be the number of years.
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2-4 Solving
Equations with the Variable on Each Side
So, the company must sell more than 1,899 DVDs per day to make a profit.
40. MOBILE PHONES The table shows the number of mobile phone subscribers for two states for a recent year.
How long will it take for the numbers of subscribers to be the same? SOLUTION: Let t be the number of years.
Let x be the number of years.
3765 + 325x
3765 – 3765 + 325x
325x
325x – 292x
33x
=
=
=
=
=
3842 + 292x
3842 – 3765 + 292x
77 + 292x
77 + 292x – 292x
77
So it will take about 2.3 years or 2 years 4 months to be the same
41. MULTIPLE REPRESENTATIONS In this problem, you will explore 2x + 4 = −x − 2.
a. GRAPHICAL Make a table of values with five points for y = 2x + 4 and y = −x − 2. Graph the points from the
tables.
b. ALGEBRAIC Solve 2x + 4 = −x − 2.
c. VERBAL Explain how the solution you found in part b is related to the intersection point of the graphs in part a.
SOLUTION: a.
b.
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c. Draw the lines y = 2x + 4 and y = x – 2. Then find the intersection of the lines.
Page 26
325x – 292x
33x
=
=
77 + 292x – 292x
77
So it willEquations
take about with
2.3 years
or 2 yearson
4 months
to be the same
2-4 Solving
the Variable
Each Side
41. MULTIPLE REPRESENTATIONS In this problem, you will explore 2x + 4 = −x − 2.
a. GRAPHICAL Make a table of values with five points for y = 2x + 4 and y = −x − 2. Graph the points from the
tables.
b. ALGEBRAIC Solve 2x + 4 = −x − 2.
c. VERBAL Explain how the solution you found in part b is related to the intersection point of the graphs in part a.
SOLUTION: a.
b.
c. Draw the lines y = 2x + 4 and y = x – 2. Then find the intersection of the lines.
The lines intersect when x = –2. This is the x-coordinate for the point of intersection on the graph.
42. REASONING Solve 5x + 2 = ax – 1 for x. Assume that a ≠ 0. Describe each step.
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2-4 Solving Equations with the Variable on Each Side
The lines intersect when x = –2. This is the x-coordinate for the point of intersection on the graph.
42. REASONING Solve 5x + 2 = ax – 1 for x. Assume that a ≠ 0. Describe each step.
SOLUTION: 43. CHALLENGE Write an equation with the variable on each side of the equals sign, at least one fractional
coefficient, and a solution of −6. Discuss the steps you used.
SOLUTION: First, chose a fractional coefficient.
Then, chose a coefficient for the variable on the other side of the equation. 2x Plugging in –6 on both sides.
To balance, 1 must be added to the left and 2 must be subtracted.
Check:
44. OPEN ENDED Create an equation with at least two grouping symbols for which there is no solution.
SOLUTION: Pick two expression where when distributed, the first terms are the same, and the constant terms are different. Page 28
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2-4 Solving Equations with the Variable on Each Side
44. OPEN ENDED Create an equation with at least two grouping symbols for which there is no solution.
SOLUTION: Pick two expression where when distributed, the first terms are the same, and the constant terms are different. .
45. CCSS CRITIQUE Determine whether each solution is correct. If the solution is not correct, describe the error and
give the correct solution.
a.
b.
c.
SOLUTION: a. This is incorrect. The 2 must be distributed over both g and 5.
The correct answer is g = 6.
b. This is correct. Check the answer. c. This is incorrect. To eliminate –6z on the left side of the equals sign, 6z must be added to each side of the
equation.
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.
2-4 Solving Equations with the Variable on Each Side
45. CCSS CRITIQUE Determine whether each solution is correct. If the solution is not correct, describe the error and
give the correct solution.
a.
b.
c.
SOLUTION: a. This is incorrect. The 2 must be distributed over both g and 5.
The correct answer is g = 6.
b. This is correct. Check the answer. c. This is incorrect. To eliminate –6z on the left side of the equals sign, 6z must be added to each side of the
equation.
The correct answer is z = 1.
Find
the value
46. CHALLENGE
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by Cognero
a. k(3x − 2) = 4 − 6x
b. 15y − 10 + k = 2(k y − 1) − y
of k for which each equation is an identity.
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2-4 Solving Equations with the Variable on Each Side
The correct answer is z = 1.
46. CHALLENGE Find the value of k for which each equation is an identity.
a. k(3x − 2) = 4 − 6x
b. 15y − 10 + k = 2(k y − 1) − y
SOLUTION: a. An equation that is true for every value of the variable is an identity. Find the value k such that k(3x − 2) is always 4 − 6x. The equation is an identity when k = –2.
b. Find the value k such that 15y − 10 + k is always 2(k y − 1) − y. The equation is an identity when k = 8.
47. WRITING IN MATH Compare and contrast solving equations with variables on both sides of the equation to
solving one-step or multi-step equations with a variable on one side of the equation.
SOLUTION: If the equation has variables on both sides of the equation, you must first add or subtract one of the terms from both
sides of the equation so that the variable is left on only one side of the equation. Then, solving the equations uses the
same steps.
For example, solve the equation 8 + 5x = 7x – 2 with variables on both side.
Consider the multi-step problem 8 + 5x = – 2.
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2-4 Solving
Equations with the Variable on Each Side
The equation is an identity when k = 8.
47. WRITING IN MATH Compare and contrast solving equations with variables on both sides of the equation to
solving one-step or multi-step equations with a variable on one side of the equation.
SOLUTION: If the equation has variables on both sides of the equation, you must first add or subtract one of the terms from both
sides of the equation so that the variable is left on only one side of the equation. Then, solving the equations uses the
same steps.
For example, solve the equation 8 + 5x = 7x – 2 with variables on both side.
Consider the multi-step problem 8 + 5x = – 2.
After step 4, the steps for solving an equation with variable on both size are the same as a multi-step equation.
48. A hang glider 25 meters above the ground starts to descend at a constant rate of 2 meters per second. Which
equation shows the height h after t seconds of descent?
A h = 25t + 2t
B h = −25t + 2
C h = 2t + 25
D h = −2t + 25
SOLUTION: The hang glider is starting at 25 feet. He is falling at 2 feet per second. Falling suggests a negative. So, the equation
that could be used to determine the hang glider’s height after t seconds is h = −2t +25. Choice D is correct.
49. GEOMETRY Two rectangular walls each with a length of 12 feet and a width of 23 feet need to be painted. It
costs $0.08 per square foot for paint. How much will it cost to paint the walls?
F $22.08
G $23.04
H $34.50
J $44.16
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SOLUTION: First find the area of both walls. Page 32
D h = −2t + 25
SOLUTION: The hangEquations
glider is starting
at 25
feet. Heon
is Each
fallingSide
at 2 feet per second. Falling suggests a negative. So, the equation
2-4 Solving
with the
Variable
that could be used to determine the hang glider’s height after t seconds is h = −2t +25. Choice D is correct.
49. GEOMETRY Two rectangular walls each with a length of 12 feet and a width of 23 feet need to be painted. It
costs $0.08 per square foot for paint. How much will it cost to paint the walls?
F $22.08
G $23.04
H $34.50
J $44.16
SOLUTION: First find the area of both walls. Since there are two walls, the total area that needs covered is 2 × 276 square feet, which is 552 square feet. To find
the total cost, multiply the total area by the cost per square foot, which is 552 × 0.08, or 44.16. So it costs $44.16 to
paint the walls. Choice J is correct.
50. SHORT RESPONSE Maddie works at Game Exchange. They are having a sale as shown.
Her employee discount is 15%. If sales tax is 7.25%, how much does she spend for a total of 4 video games?
SOLUTION: First, find Maddie’s discount.
0.15 × 20 = 3
Now, subtract her discount from the original cost.
20 − 3 = 17
Maddie will pay $17 per video game. For every two she buys, she gets one free. Since she is getting 4, she will buy
two, get 1 free, and then buy another. So, she will only pay for 3 video games.
3 × 17 = 51
Her total cost for the games is $51.00. Now find the sales tax.
51 × 0.0725 = 3.6975
Rounded to the nearest cent, the sales tax is $3.70. To find her total cost, add the tax to the cost of the games. So,
Maddie paid 3.70 + 51, or $54.70.
51. Solve
A
B
C D −10
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Page 33
3 × 17 = 51
Her total cost for the games is $51.00. Now find the sales tax.
51 × 0.0725 = 3.6975
RoundedEquations
to the nearest
sales tax
$3.70.
To find her total cost, add the tax to the cost of the games. So,
2-4 Solving
withcent,
the the
Variable
onisEach
Side
Maddie paid 3.70 + 51, or $54.70.
51. Solve
A
B
C D −10
SOLUTION: Choice A is correct.
Solve each equation. Check your solution.
52. 5n + 6 = −4
SOLUTION: Check:
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53. −1 = 7 + 3c
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2-4 Solving Equations with the Variable on Each Side
Choice A is correct.
Solve each equation. Check your solution.
52. 5n + 6 = −4
SOLUTION: Check:
53. −1 = 7 + 3c
SOLUTION: Check:
54. SOLUTION: eSolutions Manual - Powered by Cognero
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2-4 Solving Equations with the Variable on Each Side
54. SOLUTION: Check:
55. SOLUTION: eSolutions Manual - Powered by Cognero
Check:
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2-4 Solving Equations with the Variable on Each Side
55. SOLUTION: Check:
56. SOLUTION: Check:
57. SOLUTION: eSolutions Manual - Powered by Cognero
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2-4 Solving Equations with the Variable on Each Side
57. SOLUTION: Check:
58. WORLD RECORDS In 1998, Winchell’s House of Donuts in Pasadena, California, made the world’s largest
donut. It weighed 5000 pounds and had a circumference of 298.3 feet. What was the donut’s diameter to the nearest
tenth? (Hint: C = πd)
SOLUTION: We are given the circumference, so we can use this value and the formula for circumference to find the diameter.
The diameter of the donut is about 95.0 feet.
59. ZOO At a zoo, the cost of admission is posted on the sign. Find the cost of admission for two adults and two
children.
SOLUTION: Zoo admission for two adults and two children will cost $34.
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Find the value of n. Then name the property used in each step.
60. 25n = 25
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2-4 Solving
Equations with the Variable on Each Side
The diameter of the donut is about 95.0 feet.
59. ZOO At a zoo, the cost of admission is posted on the sign. Find the cost of admission for two adults and two
children.
SOLUTION: Zoo admission for two adults and two children will cost $34.
Find the value of n. Then name the property used in each step.
60. 25n = 25
SOLUTION: Since the product of 25 and 1 is 25, the property used is the Multiplicative Identity.
61. n • 1 = 2
SOLUTION: Since the product of 2 and 1 is 2, the property used is the Multiplicative Identity.
62. 12 • n = 12 • 6
SOLUTION: When 6 is substituted back into the equation, each side is equal to 72. Since 72 = 72, the property used is the
Reflexive Property.
63. n + 0 =
SOLUTION: eSolutions Manual - Powered by Cognero
Since the sum of and 0 is , the property used is the Additive Identity.
Page 39
When 6 Equations
is substituted
back
into
the equation,
eachSide
side is equal to 72. Since 72 = 72, the property used is the
2-4 Solving
with
the
Variable
on Each
Reflexive Property.
63. n + 0 =
SOLUTION: Since the sum of and 0 is , the property used is the Additive Identity.
64. SOLUTION: Since the product of 4 and
is 1, the property used is the Multiplicative Inverse.
65. (10 − 8)(7) = 2(n)
SOLUTION: Since the first quantity of (10 − 8) on the left is equal to the first quantity 2 on the right and the second quantity 7 on
the left is equal to the second quantity n or 7 on the right, the property used is the Transitive Property.
Translate each sentence into an equation.
66. Twice a number t decreased by eight equals seventy.
SOLUTION: Rewrite the verbal sentence so it is easier to translate. Twice a number t decreased by eight equals seventy is the
same as 2 times t minus 8 equals 70.
2
2
times
•
t
t
minus
–
8
8
equals
=
70
70
The equation is 2t − 8 = 70.
67. Five times the sum of m and k is the same as seven times k.
SOLUTION: 5
times
5
the sum
of m and
k
•
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The equation is 5(m + k) = 7k.
is the
same as
7
times
k
=
7
•
k
Page 40
2
2
times
•
t
t
minus
–
8
8
equals
=
70
70
2-4 Solving
Equations
the Variable on Each Side
The equation
is 2t −with
8 = 70.
67. Five times the sum of m and k is the same as seven times k.
SOLUTION: 5
times
5
the sum
of m and
k
is the
same as
7
times
k
=
7
•
k
•
The equation is 5(m + k) = 7k.
68. Half of p is the same as p minus 3.
SOLUTION: Half
of
p
is the
same as
p
minus
3
•
p
=
p
–
3
The equation is
p = p − 3.
Evaluate each expression.
69. −9 − (−14)
SOLUTION: 70. −10 + (20)
SOLUTION: −10 + (20) = 10
71. −15 − 9
SOLUTION: 72. 5(14)
SOLUTION: 5(14) = 70
73. −55 ÷ (−5)
SOLUTION: −55 ÷ −5 = 11
74. −25(−5)
SOLUTION: −25(−5) = 125
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73. −55 ÷ (−5)
SOLUTION: 2-4 Solving
Equations with the Variable on Each Side
−55 ÷ −5 = 11
74. −25(−5)
SOLUTION: −25(−5) = 125
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