# 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 [latex]y=\frac{5}{3}[/latex] a solution of [latex]6y+10=12y[/latex]?**

**Solution**

Yes

**2. Is [latex]x=\frac{9}{4}[/latex] a solution of [latex]4x+9=8x[/latex]?**

**3. Is [latex]u=-\frac{1}{2}[/latex] a solution of [latex]8u-1=6u[/latex]?**

**Solution**

No

**4. Is [latex]v=-\frac{1}{3}[/latex] a solution of [latex]9v-2=3v[/latex]?**

**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. [latex]x+24=35[/latex]**

**Solution**

[latex]x=11[/latex]

**6. [latex]x+17=22[/latex]**

**7. [latex]y+45=-66[/latex]**

**Solution**

[latex]y=-111[/latex]

**8. [latex]y+39=-83[/latex]**

**9. [latex]b+\frac{1}{4}=\frac{3}{4}[/latex]**

**Solution**

[latex]b=\frac{1}{2}[/latex]

**10. [latex]a+\frac{2}{5}=\frac{4}{5}[/latex]**

**11. [latex]p+2.4=-9.3[/latex]**

**Solution**

[latex]p=-11.7[/latex]

**12. [latex]m+7.9=11.6[/latex]**

**13. [latex]a-45=76[/latex]**

**Solution**

[latex]a=121[/latex]

**14. [latex]a-30=57[/latex]**

**15. [latex]m-18=-200[/latex]**

**Solution**

[latex]m=-182[/latex]

**16. [latex]m-12=-12[/latex]**

**17. [latex]x-\frac{1}{3}=2[/latex]**

**Solution**

[latex]x=\frac{7}{3}[/latex]

**18. [latex]x-\frac{1}{5}=4[/latex]**

**19. [latex]y-3.8=10[/latex]**

**Solution**

[latex]y=13.8[/latex]

**20. [latex]y-7.2=5[/latex]**

**21. [latex]x-165=-420[/latex]**

**Solution**

[latex]x=-255[/latex]

**22. [latex]z-101=-314[/latex]**

**23. [latex]z+0.52=-8.5[/latex]**

**Solution**

[latex]z=-9.02[/latex]

**24. [latex]x+0.93=-4.1[/latex]**

**25. [latex]q+\frac{3}{4}=\frac{1}{2}[/latex]**

**Solution**

[latex]q=-\frac{1}{4}[/latex]

**26. [latex]p+\frac{1}{3}=\frac{5}{6}[/latex]**

**27. [latex]p-\frac{2}{5}=\frac{2}{3}[/latex]**

**Solution**

[latex]p=\frac{16}{15}[/latex]

**28. [latex]y-\frac{3}{4}=\frac{3}{5}[/latex]**

**Exercises: Solve Equations that Require Simplification**

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

**29. [latex]c+31-10=46[/latex]**

**Solution**

[latex]c=25[/latex]

**30. [latex]m+16-28=5[/latex]**

**31. [latex]9x+5-8x+14=20[/latex]**

**Solution**

[latex]x=1[/latex]

**32. [latex]6x+8-5x+16=32[/latex]**

**33. [latex]-6x-11+7x-5=-16[/latex]**

**Solution**

[latex]x=0[/latex]

**34. [latex]-8n-17+9n-4=-41[/latex]**

**35. [latex]5(y-6)-4y=-6[/latex]**

**Solution**

[latex]y=24[/latex]

**36. [latex]9(y-2)-8y=-16[/latex]**

**37. [latex]8(u+1.5)-7u=4.9[/latex]**

**Solution**

[latex]u=-7.1[/latex]

**38. [latex]5(w+2.2)-4w=9.3[/latex]**

**39. [latex]6a-5(a-2)+9=-11[/latex]**

**Solution**

[latex]a=-30[/latex]

**40. [latex]8c-7(c-3)+4=-16[/latex]**

**41. [latex]6(y-2)-5y=4(y+3)-4(y-1)[/latex]**

**Solution**

[latex]y=28[/latex]

**42. [latex]9(x-1)-8x=-3(x+5)+3(x-5)[/latex]**

**43. [latex]3(5n-1)-14n+9=10(n-4)-6n-4(n+1)[/latex]**

**Solution**

[latex]n=-50[/latex]

**44. [latex]2(8m+3)-15m-4=9(m+6)-2(m-1)-7m[/latex]**

**45. [latex]-(j+2)+2j-1=5[/latex]**

**Solution**

[latex]j=8[/latex]

**46. [latex]-(k+7)+2k+8=7[/latex]**

**47. [latex]-\left(\frac{1}{4}a-\frac{3}{4}\right)+\frac{5}{4}a=-2[/latex]**

**Solution**

[latex]a=-\frac{11}{4}[/latex]

**48. [latex]-\left(\frac{2}{3}d-\frac{1}{3}\right)+\frac{5}{3}d=-4[/latex]**

**49. [latex]8(4x+5)-5(6x)-x=53-6(x+1)+3(2x+2)[/latex]**

**Solution**

[latex]x=13[/latex]

**50. [latex]6(9y-1)-10(5y)-3y=22-4(2y-12)+8(y-6)[/latex]**

**Exercises: Translate to an Equation and Solve**

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

**51. Nine more than [latex]x[/latex] is equal to [latex]52[/latex].**

**Solution**

[latex]\begin{array}{rcl}x+9&=&52\\x&=&43\end{array}[/latex]

**52. The sum of [latex]x[/latex] and [latex]-15[/latex] is [latex]23[/latex].**

**53. Ten less than [latex]m[/latex] is [latex]-14[/latex].**

**Solution**

[latex]\begin{array}{rcl}m-10&=&-14\\m&=&-4\end{array}[/latex]

**54. Three less than [latex]y[/latex] is [latex]-19[/latex].**

**55. The sum of [latex]y[/latex] and [latex]-30[/latex] is [latex]40[/latex].**

**Solution**

[latex]\begin{array}{rcl}y+(-30)&=&40\\y&=&70\end{array}[/latex]

**56. Twelve more than [latex]p[/latex] is equal to [latex]67[/latex].**

**57. The difference of [latex]9x[/latex] and [latex]8x[/latex] is [latex]107[/latex].**

**Solution**

[latex]\begin{array}{rcl}9x-8x&=&107\\x&=&107\end{array}[/latex]

**58. The difference of [latex]5c[/latex] and [latex]4c[/latex] is [latex]602[/latex].**

**59. The difference of [latex]n[/latex] and [latex]\frac{1}{6}[/latex] is [latex]\frac{1}{2}[/latex].**

**Solution**

[latex]\begin{array}{rcl}n-\frac{1}{6}&=&\frac{1}{2}\\n&=&\frac{2}{3}\end{array}[/latex]

**60. The difference of [latex]f[/latex] and [latex]\frac{1}{3}[/latex] is [latex]\frac{1}{12}[/latex].**

**61. The sum of [latex]-4n[/latex] and [latex]5n[/latex] is [latex]-82[/latex].**

**Solution**

[latex]\begin{array}{rcl}-4n+5n&=&-82\\n&=&-82\end{array}[/latex]

**62. The sum of [latex]-9m[/latex] and [latex]10m[/latex] is [latex]-95[/latex].**

**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 [latex]18[/latex] miles, from home to the library and then to the beach. The distance from Avril’s house to the library is [latex]7[/latex] miles. What is the distance from the library to the beach?**

**Solution**

[latex]11[/latex] miles

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

**65. Age.** **Eva’s daughter is [latex]15[/latex] years younger than her son. Eva’s son is [latex]22[/latex] years old. How old is her daughter?**

**Solution**

[latex]7[/latex] years old

**66. Age.** **Pablo’s father is [latex]3[/latex] years older than his mother. Pablo’s mother is [latex]42[/latex] years old. How old is his father?**

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

**Solution**

[latex]21[/latex] pounds

**68. Weight.** **Allie weighs [latex]8[/latex] pounds less than her twin sister Lorrie. Allie weighs [latex]124[/latex] pounds. How much does Lorrie weigh?**

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

**Solution**

[latex]100.5[/latex] degrees

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

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

**Solution**

[latex]$121.19[/latex]

**72. Textbooks.** **Melissa’s math book cost [latex]$22.85[/latex] less than her art book cost. Her math book cost [latex]$93.75[/latex]. 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 [latex]\frac{5}{8}[/latex] inch screw. The hole should be [latex]\frac{1}{12}[/latex] inch smaller than the screw. Let [latex]d[/latex] equal the size of the hole he should drill. Solve the equation [latex]d-\frac{1}{12}=\frac{5}{8}[/latex] to see what size the hole should be.**

**Solution**

[latex]d=\frac{17}{24}\text{ inch}[/latex]

**74. Baking.** **Kelsey needs [latex]\frac{2}{3}[/latex] cup of sugar for the cookie recipe she wants to make. She only has [latex]\frac{3}{8}[/latex] cup of sugar and will borrow the rest from her neighbor. Let [latex]s[/latex] equal the amount of sugar she will borrow. Solve the equation [latex]\frac{3}{8}+s=\frac{2}{3}[/latex] 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 [latex]-8[/latex] a solution to the equation [latex]3x=16-5x[/latex]? How do you know?**

**Solution**

No. Justifications will vary.

**76. What is the first step in your solution to the equation [latex]10x+2=4x+26[/latex]?**