Practice Makes Perfect
Solve Equations Using the General Strategy
In the following exercises, determine whether the given values are solutions to the equation.
1. \(6y+10=12y\)
a. \(y=\frac{5}{3}\)
b. \(y=−\frac{1}{2}\)
 Answer

a. yes
b. no
2. \(4x+9=8x\)
a. \(x=−\frac{7}{8}\)
b. \(x=\frac{9}{4}\)
3. \(8u−1=6u\)
a. \(u=−\frac{1}{2}\)
b. \(u=\frac{1}{2}\)
 Answer

a. no
b. yes
4. \(9v−2=3v\)
a. \(v=−\frac{1}{3}\)
b. \(v=\frac{1}{3}\)
In the following exercises, solve each linear equation.
5. \(15(y−9)=−60\)
 Answer

\(y=5\)
7. \(−(w−12)=30\)
 Answer

\(w=−18\)
9. \(51+5(4−q)=56\)
 Answer

\(q=3\)
11. \(3(10−2x)+54=0\)
 Answer

\(x=14\)
13. \(\frac{2}{3}(9c−3)=22\)
 Answer

\(c=4\)
14. \(\frac{3}{5}(10x−5)=27\)
15. \(\frac{1}{5}(15c+10)=c+7\)
 Answer

\(c=\frac{5}{2}\)
16. \(\frac{1}{4}(20d+12)=d+7\)
17. \(3(4n−1)−2=8n+3\)
 Answer

\(n=2\)
19. \(12+2(5−3y)=−9(y−1)−2\)
 Answer

\(y=−5\)
20. \(−15+4(2−5y)=−7(y−4)+4\)
21. \(5+6(3s−5)=−3+2(8s−1)\)
 Answer

\(s=10\)
22. \(−12+8(x−5)=−4+3(5x−2)\)
23. \(4(p−4)−(p+7)=5(p−3)\)
 Answer

\(p=−4\)
24. \(3(a−2)−(a+6)=4(a−1)\)
25. \(4[5−8(4c−3)]=12(1−13c)−8\)
 Answer

\(c=−4\)
26. \(5[9−2(6d−1)]=11(4−10d)−139\)
27. \(3[−9+8(4h−3)]=2(5−12h)−19\)
 Answer

\(h=\frac{3}{4}\)
28. \(3[−14+2(15k−6)]=8(3−5k)−24\)
29. \(5[2(m+4)+8(m−7)]=2[3(5+m)−(21−3m)]\)
 Answer

\(m=6\)
30. \(10[5(n+1)+4(n−1)]=11[7(5+n)−(25−3n)]\)
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
31. \(23z+19=3(5z−9)+8z+46\)
 Answer

identity; all real numbers
32. \(15y+32=2(10y−7)−5y+46\)
33. \(18(5j−1)+29=47\)
 Answer

conditional equation; \(j=\frac{2}{5}\)
35. \(22(3m−4)=8(2m+9)\)
 Answer

conditional equation; \(m=165\)
36. \(30(2n−1)=5(10n+8)\)
37. \(7v+42=11(3v+8)−2(13v−1)\)
 Answer

contradiction; no solution
38. \(18u−51=9(4u+5)−6(3u−10)\)
39. \(45(3y−2)=9(15y−6)\)
 Answer

contradiction; no solution
40. \(60(2x−1)=15(8x+5)\)
41. \(9(14d+9)+4d=13(10d+6)+3\)
 Answer

identity; all real numbers
42. \(11(8c+5)−8c=2(40c+25)+5\)
Solve Equations with Fraction or Decimal Coefficients
In the following exercises, solve each equation with fraction coefficients.
43. \(\frac{1}{4}x−\frac{1}{2}=−\frac{3}{4}\)
 Answer

\(x=−1\)
44. \(\frac{3}{4}x−\frac{1}{2}=\frac{1}{4}\)
45. \(\frac{5}{6}y−\frac{2}{3}=−\frac{3}{2}\)
 Answer

\(y=−1\)
46. \(\frac{5}{6}y−\frac{1}{3}=−\frac{7}{6}\)
47. \(\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}\)
 Answer

\(a=\frac{3}{4}\)
48. \(\frac{5}{8}b+\frac{1}{2}=−\frac{3}{4}\)
49. \(2=\frac{1}{3}x−\frac{1}{2}x+\frac{2}{3}x\)
 Answer

\(x=4\)
50. \(2=\frac{3}{5}x−\frac{1}{3}x+\frac{2}{5}x\)
51. \(\frac{1}{3}w+\frac{5}{4}=w−\frac{1}{4}\)
 Answer

\(w=\frac{9}{4}\)
52. \(\frac{1}{2}a−\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}\)
53. \(\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b−\frac{3}{5}\)
 Answer

\(b=12\)
54. \(\frac{1}{3}x+\frac{2}{5}=\frac{1}{5}x−\frac{2}{5}\)
55. \(\frac{1}{4}(p−7)=\frac{1}{3}(p+5)\)
 Answer

\(p=−41\)
56. \(\frac{1}{5}(q+3)=\frac{1}{2}(q−3)\)
57. \(\frac{1}{2}(x+4)=\frac{3}{4}\)
 Answer

\(x=−\frac{5}{2}\)
58. \(\frac{1}{3}(x+5)=\frac{5}{6}\)
59. \(\dfrac{4n+8}{4}=\dfrac{n}{3}\)
 Answer

\(n=−3\)
60. \(\dfrac{3p+6}{3}=\dfrac{p}{2}\)
61. \(\dfrac{3x+4}{2}+1=\dfrac{5x+10}{8}\)
 Answer

\(x=−2\)
62. \(\dfrac{10y−2}{3}+3=\dfrac{10y+1}{9}\)
63. \(\dfrac{7u−1}{4}−1=\dfrac{4u+8}{5}\)
 Answer

\(u=3\)
64. \(\dfrac{3v−6}{2}+5=\dfrac{11v−4}{5}\)
In the following exercises, solve each equation with decimal coefficients.
65. \(0.4x+0.6=0.5x−1.2\)
 Answer

\(x=18\)
66. \(0.7x+0.4=0.6x+2.4\)
67. \(0.9x−1.25=0.75x+1.75\)
 Answer

\(x=20\)
68. \(1.2x−0.91=0.8x+2.29\)
69. \(0.05n+0.10(n+8)=2.15\)
 Answer

\(n=9\)
70. \(0.05n+0.10(n+7)=3.55\)
71. \(0.10d+0.25(d+5)=4.05\)
 Answer

\(d=8\)
72. \(0.10d+0.25(d+7)=5.25\)
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no  I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.