# Exercises: Solve Equations Using the Subtraction and Addition Properties of Equality (3.1)

## Exercises: Verify a Solution of an Equation

Instructions: For questions 1-4, determine whether the given value is a solution to the equation.

1. Is $y=\frac{5}{3}$ a solution of $6y+10=12y$?

Solution

Yes

2. Is $x=\frac{9}{4}$ a solution of $4x+9=8x$?

3. Is $u=-\frac{1}{2}$ a solution of $8u-1=6u$?

Solution

No

4. Is $v=-\frac{1}{3}$ a solution of $9v-2=3v$?

## Exercises: Solve Equations using the Subtraction and Addition Properties of Equality

Instructions: For questions 5-28, solve each equation using the Subtraction and Addition Properties of Equality.

5. $x+24=35$

Solution

$x=11$

6. $x+17=22$

7. $y+45=-66$

Solution

$y=-111$

8. $y+39=-83$

9. $b+\frac{1}{4}=\frac{3}{4}$

Solution

$b=\frac{1}{2}$

10. $a+\frac{2}{5}=\frac{4}{5}$

11. $p+2.4=-9.3$

Solution

$p=-11.7$

12. $m+7.9=11.6$

13. $a-45=76$

Solution

$a=121$

14. $a-30=57$

15. $m-18=-200$

Solution

$m=-182$

16. $m-12=-12$

17. $x-\frac{1}{3}=2$

Solution

$x=\frac{7}{3}$

18. $x-\frac{1}{5}=4$

19. $y-3.8=10$

Solution

$y=13.8$

20. $y-7.2=5$

21. $x-165=-420$

Solution

$x=-255$

22. $z-101=-314$

23. $z+0.52=-8.5$

Solution

$z=-9.02$

24. $x+0.93=-4.1$

25. $q+\frac{3}{4}=\frac{1}{2}$

Solution

$q=-\frac{1}{4}$

26. $p+\frac{1}{3}=\frac{5}{6}$

27. $p-\frac{2}{5}=\frac{2}{3}$

Solution

$p=\frac{16}{15}$

28. $y-\frac{3}{4}=\frac{3}{5}$

## Exercises: Solve Equations that Require Simplification

Instructions: For questions 29-50, solve each equation.

29. $c+31-10=46$

Solution

$c=25$

30. $m+16-28=5$

31. $9x+5-8x+14=20$

Solution

$x=1$

32. $6x+8-5x+16=32$

33. $-6x-11+7x-5=-16$

Solution

$x=0$

34. $-8n-17+9n-4=-41$

35. $5(y-6)-4y=-6$

Solution

$y=24$

36. $9(y-2)-8y=-16$

37. $8(u+1.5)-7u=4.9$

Solution

$u=-7.1$

38. $5(w+2.2)-4w=9.3$

39. $6a-5(a-2)+9=-11$

Solution

$a=-30$

40. $8c-7(c-3)+4=-16$

41. $6(y-2)-5y=4(y+3)-4(y-1)$

Solution

$y=28$

42. $9(x-1)-8x=-3(x+5)+3(x-5)$

43. $3(5n-1)-14n+9=10(n-4)-6n-4(n+1)$

Solution

$n=-50$

44. $2(8m+3)-15m-4=9(m+6)-2(m-1)-7m$

45. $-(j+2)+2j-1=5$

Solution

$j=8$

46. $-(k+7)+2k+8=7$

47. $-\left(\frac{1}{4}a-\frac{3}{4}\right)+\frac{5}{4}a=-2$

Solution

$a=-\frac{11}{4}$

48. $-\left(\frac{2}{3}d-\frac{1}{3}\right)+\frac{5}{3}d=-4$

49. $8(4x+5)-5(6x)-x=53-6(x+1)+3(2x+2)$

Solution

$x=13$

50. $6(9y-1)-10(5y)-3y=22-4(2y-12)+8(y-6)$

## Exercises: Translate to an Equation and Solve

Instructions: For questions 51-62, translate to an equation and then solve it.

51. Nine more than $x$ is equal to $52$.

Solution

$\begin{array}{rcl}x+9&=&52\\x&=&43\end{array}$

52. The sum of $x$ and $-15$ is $23$.

53. Ten less than $m$ is $-14$.

Solution

$\begin{array}{rcl}m-10&=&-14\\m&=&-4\end{array}$

54. Three less than $y$ is $-19$.

55. The sum of $y$ and $-30$ is $40$.

Solution

$\begin{array}{rcl}y+(-30)&=&40\\y&=&70\end{array}$

56. Twelve more than $p$ is equal to $67$.

57. The difference of $9x$ and $8x$ is $107$.

Solution

$\begin{array}{rcl}9x-8x&=&107\\x&=&107\end{array}$

58. The difference of $5c$ and $4c$ is $602$.

59. The difference of $n$ and $\frac{1}{6}$ is $\frac{1}{2}$.

Solution

$\begin{array}{rcl}n-\frac{1}{6}&=&\frac{1}{2}\\n&=&\frac{2}{3}\end{array}$

60. The difference of $f$ and $\frac{1}{3}$ is $\frac{1}{12}$.

61. The sum of $-4n$ and $5n$ is $-82$.

Solution

$\begin{array}{rcl}-4n+5n&=&-82\\n&=&-82\end{array}$

62. The sum of $-9m$ and $10m$ is $-95$.

## Exercises: Translate and Solve Applications

Instructions: For questions 63-72, translate into an equation and solve.

63. Distance. Avril rode her bike a total of $18$ miles, from home to the library and then to the beach. The distance from Avril’s house to the library is $7$ miles. What is the distance from the library to the beach?

Solution

$11$ miles

64. Reading. Jeff read a total of $54$ pages in his History and Sociology textbooks. He read $41$ pages in his History textbook. How many pages did he read in his Sociology textbook?

65. Age. Eva’s daughter is $15$ years younger than her son. Eva’s son is $22$ years old. How old is her daughter?

Solution

$7$ years old

66. Age. Pablo’s father is $3$ years older than his mother. Pablo’s mother is $42$ years old. How old is his father?

67. Groceries. For a family birthday dinner, Celeste bought a turkey that weighed $5$ pounds less than the one she bought for Thanksgiving. The birthday turkey weighed $16$ pounds. How much did the Thanksgiving turkey weigh?

Solution

$21$ pounds

68. Weight. Allie weighs $8$ pounds less than her twin sister Lorrie. Allie weighs $124$ pounds. How much does Lorrie weigh?

69. Health. Connor’s temperature was $0.7$ degrees higher this morning than it had been last night. His temperature this morning was $101.2$ degrees. What was his temperature last night?

Solution

$100.5$ degrees

70. Health. The nurse reported that Tricia’s daughter had gained $4.2$ pounds since her last checkup and now weighs $31.6$ pounds. How much did Tricia’s daughter weigh at her last checkup?

71. Salary. Ron’s paycheck this week was $17.43$ less than his paycheck last week. His paycheck this week was $103.76$. How much was Ron’s paycheck last week?

Solution

$121.19$

72. Textbooks. Melissa’s math book cost $22.85$ less than her art book cost. Her math book cost $93.75$. How much did her art book cost?

## Exercises: Everyday Math

Instructions: For questions 73-74, answer the given everyday math word problems.

73. Construction. Miguel wants to drill a hole for a $\frac{5}{8}$ inch screw. The hole should be $\frac{1}{12}$ inch smaller than the screw. Let $d$ equal the size of the hole he should drill. Solve the equation $d-\frac{1}{12}=\frac{5}{8}$ to see what size the hole should be.

Solution

$d=\frac{17}{24}\text{ inch}$

74. Baking. Kelsey needs $\frac{2}{3}$ cup of sugar for the cookie recipe she wants to make. She only has $\frac{3}{8}$ cup of sugar and will borrow the rest from her neighbor. Let $s$ equal the amount of sugar she will borrow. Solve the equation $\frac{3}{8}+s=\frac{2}{3}$ to find the amount of sugar she should ask to borrow.

## Exercises: Writing Exercises

Instructions: For questions 75-76, answer the given writing exercises.

75. Is $-8$ a solution to the equation $3x=16-5x$? How do you know?

Solution

No. Justifications will vary.

76. What is the first step in your solution to the equation $10x+2=4x+26$?