Exercises: Solve Equations using the Division and Multiplication Properties of Equality (3.2)
Exercises: Solve Equations Using the Division and Multiplication Properties of Equality
Instructions: For questions 1-36, solve each equation using the Division and Multiplication Properties of Equality and check the solution.
1. [latex]8x=56[/latex]
Solution
[latex]x=7[/latex]
2. [latex]7p=63[/latex]
3. [latex]-5c=55[/latex]
Solution
[latex]c=-11[/latex]
4. [latex]-9x=-27[/latex]
5. [latex]-809=15y[/latex]
Solution
[latex]y=-\frac{809}{15}[/latex]
6. [latex]-731=19y[/latex]
7. [latex]-37p=-541[/latex]
Solution
[latex]p=-\frac{541}{37}[/latex]
8. [latex]-19m=-586[/latex]
9. [latex]0.25z=3.25[/latex]
Solution
[latex]z=13[/latex]
10. [latex]0.75a=11.25[/latex]
11. [latex]-13x=0[/latex]
Solution
[latex]x=0[/latex]
12. [latex]24x=0[/latex]
13. [latex]\frac{x}{4}=35[/latex]
Solution
[latex]x=140[/latex]
14. [latex]\frac{z}{2}=54[/latex]
15. [latex]-20=\frac{q}{-5}[/latex]
Solution
[latex]q=100[/latex]
16. [latex]\frac{c}{-3}=-12[/latex]
17. [latex]\frac{y}{9}=-16[/latex]
Solution
[latex]y=-144[/latex]
18. [latex]\frac{q}{6}=-38[/latex]
19. [latex]\frac{m}{-12}=45[/latex]
Solution
[latex]m=-540[/latex]
20. [latex]-24=\frac{p}{-20}[/latex]
21. [latex]-y=6[/latex]
Solution
[latex]y=-6[/latex]
22. [latex]-u=15[/latex]
23. [latex]-v=-72[/latex]
Solution
[latex]v=72[/latex]
24. [latex]-x=-39[/latex]
25. [latex]\frac{2}{3}y=48[/latex]
Solution
[latex]y=72[/latex]
26. [latex]\frac{3}{5}r=75[/latex]
27. [latex]-\frac{5}{8}w=40[/latex]
Solution
[latex]w=-64[/latex]
28. [latex]24=-\frac{3}{4}x[/latex]
29. [latex]-\frac{2}{5}=\frac{1}{10}a[/latex]
Solution
[latex]a=-4[/latex]
30. [latex]-\frac{1}{3}q=-\frac{5}{6}[/latex]
31. [latex]-\frac{7}{10}x=-\frac{14}{3}[/latex]
Solution
[latex]x=\frac{20}{3}[/latex]
32. [latex]\frac{3}{8}y=-\frac{1}{4}[/latex]
33. [latex]\frac{7}{12}=-\frac{3}{4}p[/latex]
Solution
[latex]p=-\frac{7}{9}[/latex]
34. [latex]\frac{11}{18}=-\frac{5}{6}q[/latex]
35. [latex]-\frac{5}{18}=-\frac{10}{9}u[/latex]
Solution
[latex]u=\frac{1}{4}[/latex]
36. [latex]-\frac{7}{20}=-\frac{7}{4}v[/latex]
Exercises: Solve Equations that Require Simplification
Instructions: For questions 37-46, solve each equation requiring simplification.
37. [latex]100-16=4p-10p-p[/latex]
Solution
[latex]p=-12[/latex]
38. [latex]-18-7=5t-9t-6t[/latex]
39. [latex]\frac{7}{8}n-\frac{3}{4}n=9+2[/latex]
Solution
[latex]n=88[/latex]
40. [latex]\frac{5}{12}q+\frac{1}{2}q=25-3[/latex]
41. [latex]0.25d+0.10d=6-0.75[/latex]
Solution
[latex]d=15[/latex]
42. [latex]0.05p-0.01p=2+0.24[/latex]
43. [latex]-10(q-4)-57=93[/latex]
Solution
[latex]q=-11[/latex]
44. [latex]-12(d-5)-29=43[/latex]
45. [latex]-10(x+4)-19=85[/latex]
Solution
[latex]x=-\frac{72}{5}[/latex]
46. [latex]-15(z+9)-11=75[/latex]
Exercises: Mixed Practice
Instructions: For questions 47-65, solve each equation.
47. [latex]\frac{9}{10}x=90[/latex]
Solution
[latex]x=100[/latex]
48. [latex]\frac{5}{12}y=60[/latex]
49. [latex]y+46=55[/latex]
Solution
[latex]y=9[/latex]
50. [latex]x+33=41[/latex]
51. [latex]\frac{w}{-2}=99[/latex]
Solution
[latex]w=-198[/latex]
52. [latex]\frac{s}{-3}=-60[/latex]
53. [latex]27=6a[/latex]
Solution
[latex]a=\frac{9}{2}[/latex]
54. [latex]-a=7[/latex]
55. [latex]-x=2[/latex]
Solution
[latex]x=-2[/latex]
56. [latex]z-16=-59[/latex]
57. [latex]m-41=-14[/latex]
Solution
[latex]m=27[/latex]
58. [latex]0.04r=52.60[/latex]
59. [latex]63.90=0.03p[/latex]
Solution
[latex]p=2130[/latex]
60. [latex]-15x=-120[/latex]
61. [latex]84=-12z[/latex]
Solution
[latex]y=-7[/latex]
62. [latex]19.36=x-0.2x[/latex]
63. [latex]c-0.3c=35.70[/latex]
Solution
[latex]c=51[/latex]
64. [latex]-y=-9[/latex]
65. [latex]-x=-8[/latex]
Solution
[latex]x=8[/latex]
Exercises: Translate to an Equation and Solve
Instructions: For questions 66-85, translate to an equation and then solve.
66. [latex]187[/latex] is the product of [latex]-17[/latex] and [latex]m[/latex].
67. [latex]133[/latex] is the product of [latex]-19[/latex] and [latex]n[/latex].
Solution
[latex]\begin{array}{rcl}133&=&-19n\\n&=&-7\end{array}[/latex]
68. [latex]-184[/latex] is the product of [latex]23[/latex] and [latex]p[/latex].
69. [latex]-152[/latex] is the product of [latex]8[/latex] and [latex]q[/latex].
Solution
[latex]\begin{array}{rcl}-152&=&8q\\q&=&-19\end{array}[/latex]
70. [latex]u[/latex] divided by [latex]7[/latex] is equal to [latex]-49[/latex].
71. [latex]r[/latex] divided by 12 is equal to [latex]-48[/latex].
Solution
[latex]\begin{array}{rcl}\frac{r}{12}&=&-48\\r&=&-576\end{array}[/latex]
72. [latex]h[/latex] divided by [latex]-13[/latex] is equal to [latex]-65[/latex].
73. [latex]j[/latex] divided by [latex]-20[/latex] is equal to [latex]-80[/latex].
Solution
[latex]\begin{array}{rcl}\frac{j}{-20}&=&-80\\j&=&1,600\end{array}[/latex]
74. The quotient [latex]c[/latex] and [latex]-19[/latex] is [latex]38[/latex].
75. The quotient of [latex]b[/latex] and [latex]-6[/latex] is [latex]18[/latex].
Solution
[latex]\begin{array}{rcl}\frac{b}{-6}&=&18\\b&=&-108\end{array}[/latex]
76. The quotient of [latex]h[/latex] and [latex]26[/latex] is [latex]-52[/latex].
77. The quotient [latex]k[/latex] and [latex]22[/latex] is [latex]-66[/latex].
Solution
[latex]\frac{k}{22}=-66;k=-1,452[/latex]
78. Five-sixths of [latex]y[/latex] is [latex]15[/latex].
79. Three-tenths of [latex]x[/latex] is [latex]15[/latex].
Solution
[latex]\begin{array}{rcl}\frac{3}{10}x&=&15\\x&=&50\end{array}[/latex]
80. Four-thirds of [latex]w[/latex] is [latex]36[/latex].
81. Five-halves of [latex]v[/latex] is [latex]50[/latex].
Solution
[latex]\begin{array}{rcl}\frac{5}{2}v&=&50\\v&=&20\end{array}[/latex]
82. The sum of nine-tenths and [latex]g[/latex] is two-thirds.
83. The sum of two-fifths and [latex]f[/latex] is one-half.
Solution
[latex]\begin{array}{rcl}\frac{2}{5}+f&=&\frac{1}{2}\\f&=&\frac{1}{10}\end{array}[/latex]
84. The difference of [latex]p[/latex] and one-sixth is two-thirds.
85. The difference of [latex]q[/latex] and one-eighth is three-fourths.
Solution
[latex]\begin{array}{rcl}q-\frac{1}{8}&=&\frac{3}{4}\\q&=&\frac{7}{8}\end{array}[/latex]
Exercises: Translate and Solve Applications
Instructions: For questions 86-93, translate into an equation and solve.
86. Kindergarten. Connie’s kindergarten class has [latex]24[/latex] children. She wants them to get into [latex]4[/latex] equal groups. How many children will she put in each group?
87. Balloons. Ramona bought [latex]18[/latex] balloons for a party. She wants to make [latex]3[/latex] equal bunches. How many balloons did she use in each bunch?
Solution
[latex]6[/latex] balloons
88. Tickets. Mollie paid [latex]$36.25[/latex] for [latex]5[/latex] movie tickets. What was the price of each ticket?
89. Shopping. Serena paid [latex]$12.96[/latex] for a pack of [latex]12[/latex] pairs of sport socks. What was the price of pair of sport socks?
Solution
[latex]$1.08[/latex]
90. Sewing. Nancy used [latex]14[/latex] yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?
91. MPG. John’s SUV gets [latex]18[/latex] miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?
Solution
[latex]36[/latex] mpg
92. Height. Aiden is [latex]27[/latex] inches tall. He is [latex]\frac{3}{8}[/latex] as tall as his father. How tall is his father?
93. Real estate. Bea earned [latex]$11,700[/latex] commission for selling a house, calculated as [latex]\frac{6}{100}[/latex] of the selling price. What was the selling price of the house?
Solution
[latex]$195,000[/latex]
Exercises: Everyday Math
Instructions: For questions 94-95, solve the given everyday math word problems.
94. Commission. Every week Perry gets paid [latex]$150[/latex] plus [latex]12\%[/latex] of his total sales amount. Solve the equation [latex]840=150+0.12(a-1250)[/latex] for [latex]a[/latex], to find the total amount Perry must sell in order to be paid [latex]$840[/latex] one week.
95. Stamps. Travis bought [latex]$9.45[/latex] worth of [latex]49[/latex]-cent stamps and [latex]21[/latex]-cent stamps. The number of [latex]21[/latex]-cent stamps was [latex]5[/latex] less than the number of [latex]49[/latex]-cent stamps. Solve the equation [latex]0.49s+0.21(s-5)=9.45[/latex] for [latex]s[/latex], to find the number of [latex]49[/latex]-cent stamps Travis bought.
Solution
[latex]15[/latex] [latex]49[/latex]-cent stamps
Exercises: Writing Exercises
Instructions: For questions 96-97, answer the given writing exercises.
96. Frida started to solve the equation [latex]-3x=36[/latex] by adding [latex]3[/latex] to both sides. Explain why Frida’s method will not solve the equation.
97. Emiliano thinks [latex]x=40[/latex] is the solution to the equation [latex]\frac{1}{2}x=80[/latex]. Explain why he is wrong.
Solution
Answers will vary.
