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# Problem Set: Multi-Step Linear Equations

## Solve Equations Using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given value is a solution to the equation. Is $y=\Large\frac{1}{3}$ a solution of $4y+2=10y?$ Is $x=\Large\frac{3}{4}$ a solution of $5x+3=9x?$ Is $u=-\Large\frac{1}{2}$ a solution of $8u - 1=6u?$ Is $v=-\Large\frac{1}{3}$ a solution of $9v - 2=3v?$ In the following exercises, solve each equation. [latex-display]x+7=12[/latex-display] [latex-display]y+5=-6[/latex-display] [latex-display]b+\Large\frac{1}{4}\normalsize =\Large\frac{3}{4}[/latex-display] [latex-display]a+\Large\frac{2}{5}\normalsize =\Large\frac{4}{5}[/latex-display] [latex-display]p+2.4=-9.3[/latex-display] [latex-display]m+7.9=11.6[/latex-display] [latex-display]a - 3=7[/latex-display] [latex-display]m - 8=-20[/latex-display] [latex-display]x-\Large\frac{1}{3}\normalsize=2[/latex-display] [latex-display]x-\Large\frac{1}{5}\normalsize =4[/latex-display] [latex-display]y - 3.8=10[/latex-display] [latex-display]y - 7.2=5[/latex-display] [latex-display]x - 15=-42[/latex-display] [latex-display]z+5.2=-8.5[/latex-display] [latex-display]q+\Large\frac{3}{4}\normalsize =\Large\frac{1}{2}[/latex-display] [latex-display]q=-\Large\frac{1}{4}[/latex-display] [latex-display]p-\Large\frac{2}{5}\normalsize =\Large\frac{2}{3}[/latex-display] [latex-display]y-\Large\frac{3}{4}\normalsize =\Large\frac{3}{5}[/latex-display] [latex-display]y=\Large\frac{27}{20}[/latex-display] Solve Equations that Need to be Simplified In the following exercises, solve each equation. [latex-display]c+3 - 10=18[/latex-display] [latex-display]m+6 - 8=15[/latex-display] 17 [latex-display]9x+5 - 8x+14=20[/latex-display] [latex-display]6x+8 - 5x+16=32[/latex-display] 8 [latex-display]-6x - 11+7x - 5=-16[/latex-display] [latex-display]-8n - 17+9n - 4=-41[/latex-display] −20 [latex-display]3\left(y - 5\right)-2y=-7[/latex-display] [latex-display]4\left(y - 2\right)-3y=-6[/latex-display] 2 [latex-display]8\left(u+1.5\right)-7u=4.9[/latex-display] [latex-display]5\left(w+2.2\right)-4w=9.3[/latex-display] 1.7 [latex-display]-5\left(y - 2\right)+6y=-7+4[/latex-display] [latex-display]-8\left(x - 1\right)+9x=-3+9[/latex-display] −2 [latex-display]3\left(5n - 1\right)-14n+9=1 - 2[/latex-display] [latex-display]2\left(8m+3\right)-15m - 4=3 - 5[/latex-display] −4 [latex-display]-\left(j+2\right)+2j - 1=5[/latex-display] [latex-display]-\left(k+7\right)+2k+8=7[/latex-display] 6 [latex-display]6a - 5\left(a - 2\right)+9=-11[/latex-display] [latex-display]8c - 7\left(c - 3\right)+4=-16[/latex-display] −41 [latex-display]8\left(4x+5\right)-5\left(6x\right)-x=53[/latex-display] [latex-display]6\left(9y - 1\right)-10\left(5y\right)-3y=22[/latex-display] 28

### Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve. Five more than $x$ is equal to $21$. The sum of $x$ and $-5$ is $33$. x + (−5) = 33; x = 38 Ten less than $m$ is $-14$. Three less than $y$ is $-19$. y − 3 = −19; y = −16 The sum of $y$ and $-3$ is $40$. Eight more than $p$ is equal to $52$. p + 8 = 52; p = 44 The difference of $9x$ and $8x$ is $17$. The difference of $5c$ and $4c$ is $60$. 5c − 4c = 60; 60 The difference of $n$ and $\Large\frac{1}{6}$ is $\Large\frac{1}{2}$. The difference of $f$ and $\Large\frac{1}{3}$ is $\Large\frac{1}{12}$. [latex-display]f-\Large\frac{1}{3}\normalsize =\Large\frac{1}{12}\normalsize ;\Large\frac{5}{12}[/latex-display] The sum of $-4n$ and $5n$ is $-32$. The sum of $-9m$ and $10m$ is $-25$. −9m + 10m = −25; m = −25

### writing exercises

Frida started to solve the equation $-3x=36$ by adding $3$ to both sides. Explain why Frida’s method will result in the correct solution. Emiliano thinks $x=40$ is the solution to the equation $\Large\frac{1}{2}\normalsize x=80$. Explain why he is wrong. Answer will vary.

## Solve Equations with Variables and Constants on Both Sides

### Solve an Equation with Constants on Both Sides

In the following exercises, solve the equation for the variable. [latex-display]6x - 2=40[/latex-display] [latex-display]7x - 8=34[/latex-display] 6 [latex-display]11w+6=93[/latex-display] [latex-display]14y+7=91[/latex-display] 6 [latex-display]3a+8=-46[/latex-display] [latex-display]4m+9=-23[/latex-display] −8 [latex-display]-50=7n - 1[/latex-display] [latex-display]-47=6b+1[/latex-display] −8 [latex-display]25=-9y+7[/latex-display] [latex-display]29=-8x - 3[/latex-display] −4 [latex-display]-12p - 3=15[/latex-display] [latex-display]-14\text{q}-15=13[/latex-display] −2 Solve an Equation with Variables on Both Sides In the following exercises, solve the equation for the variable. [latex-display]8z=7z - 7[/latex-display] [latex-display]9k=8k - 11[/latex-display] −11 [latex-display]4x+36=10x[/latex-display] [latex-display]6x+27=9x[/latex-display] 9 [latex-display]c=-3c - 20[/latex-display] [latex-display]b=-4b - 15[/latex-display] −3 [latex-display]5q=44 - 6q[/latex-display] [latex-display]7z=39 - 6z[/latex-display] 3 [latex-display]3y+\Large\frac{1}{2}\normalsize =2y[/latex-display] [latex-display]8x+\Large\frac{3}{4}\normalsize =7x[/latex-display] −3/4 [latex-display]-12a - 8=-16a[/latex-display] [latex-display]-15r - 8=-11r[/latex-display] 2 Solve an Equation with Variables and Constants on Both Sides In the following exercises, solve the equations for the variable. [latex-display]6x - 15=5x+3[/latex-display] [latex-display]4x - 17=3x+2[/latex-display] 19 [latex-display]26+8d=9d+11[/latex-display] [latex-display]21+6f=7f+14[/latex-display] 7 [latex-display]3p - 1=5p - 33[/latex-display] [latex-display]8q - 5=5q - 20[/latex-display] −5 [latex-display]4a+5=-a - 40[/latex-display] [latex-display]9c+7=-2c - 37[/latex-display] −4 [latex-display]8y - 30=-2y+30[/latex-display] [latex-display]12x - 17=-3x+13[/latex-display] 2 [latex-display]2\text{z}-4=23-\text{z}[/latex-display] [latex-display]3y - 4=12-y[/latex-display] 4 [latex-display]\Large\frac{5}{4}\normalsize c - 3=\Large\frac{1}{4}\normalsize c - 16[/latex-display] [latex-display]\Large\frac{4}{3}\normalsize m - 7=\Large\frac{1}{3}\normalsize m - 13[/latex-display] 6 [latex-display]8-\Large\frac{2}{5}\normalsize q=\Large\frac{3}{5}\normalsize q+6[/latex-display] [latex-display]11-\Large\frac{1}{4}\normalsize a=\Large\frac{3}{4}\normalsize a+4[/latex-display] 7 [latex-display]\Large\frac{4}{3}\normalsize n+9=\Large\frac{1}{3}\normalsize n - 9[/latex-display] [latex-display]\Large\frac{5}{4}\normalsize a+15=\Large\frac{3}{4}\normalsize a - 5[/latex-display] −40 [latex-display]\Large\frac{1}{4}\normalsize y+7=\Large\frac{3}{4}\normalsize y - 3[/latex-display] [latex-display]\Large\frac{3}{5}\normalsize p+2=\Large\frac{4}{5}\normalsize p - 1[/latex-display] 3 [latex-display]14n+8.25=9n+19.60[/latex-display] [latex-display]13z+6.45=8z+23.75[/latex-display] 3.46 [latex-display]2.4w - 100=0.8w+28[/latex-display] [latex-display]2.7w - 80=1.2w+10[/latex-display] 60 [latex-display]5.6r+13.1=3.5r+57.2[/latex-display] [latex-display]6.6x - 18.9=3.4x+54.7[/latex-display] 23 Solve an Equation Using the General Strategy In the following exercises, solve the linear equation using the general strategy. [latex-display]5\left(x+3\right)=75[/latex-display] [latex-display]4\left(y+7\right)=64[/latex-display] 9 [latex-display]8=4\left(x - 3\right)[/latex-display] [latex-display]9=3\left(x - 3\right)[/latex-display] 6 [latex-display]20\left(y - 8\right)=-60[/latex-display] [latex-display]14\left(y - 6\right)=-42[/latex-display] 3 [latex-display]-4\left(2n+1\right)=16[/latex-display] [latex-display]-7\left(3n+4\right)=14[/latex-display] −2 [latex-display]3\left(10+5r\right)=0[/latex-display] [latex-display]8\left(3+3\text{p}\right)=0[/latex-display] −1 [latex-display]\Large\frac{2}{3}\normalsize\left(9c - 3\right)=22[/latex-display] [latex-display]\Large\frac{3}{5}\normalsize\left(10x - 5\right)=27[/latex-display] 5 [latex-display]5\left(1.2u - 4.8\right)=-12[/latex-display] [latex-display]4\left(2.5v - 0.6\right)=7.6[/latex-display] 0.52 [latex-display]0.2\left(30n+50\right)=28[/latex-display] [latex-display]0.5\left(16m+34\right)=-15[/latex-display] 0.25 [latex-display]-\left(w - 6\right)=24[/latex-display] [latex-display]-\left(t - 8\right)=17[/latex-display] −9 [latex-display]9\left(3a+5\right)+9=54[/latex-display] [latex-display]8\left(6b - 7\right)+23=63[/latex-display] 2 [latex-display]10+3\left(z+4\right)=19[/latex-display] [latex-display]13+2\left(m - 4\right)=17[/latex-display] 6 [latex-display]7+5\left(4-q\right)=12[/latex-display] [latex-display]-9+6\left(5-k\right)=12[/latex-display] 3/2 [latex-display]15-\left(3r+8\right)=28[/latex-display] [latex-display]18-\left(9r+7\right)=-16[/latex-display] 3 [latex-display]11 - 4\left(y - 8\right)=43[/latex-display] [latex-display]18 - 2\left(y - 3\right)=32[/latex-display] −4 [latex-display]9\left(p - 1\right)=6\left(2p - 1\right)[/latex-display] [latex-display]3\left(4n - 1\right)-2=8n+3[/latex-display] 2 [latex-display]9\left(2m - 3\right)-8=4m+7[/latex-display] [latex-display]5\left(x - 4\right)-4x=14[/latex-display] 34 [latex-display]8\left(x - 4\right)-7x=14[/latex-display] [latex-display]5+6\left(3s - 5\right)=-3+2\left(8s - 1\right)[/latex-display] 10 [latex-display]-12+8\left(x - 5\right)=-4+3\left(5x - 2\right)[/latex-display] [latex-display]4\left(x - 1\right)-8=6\left(3x - 2\right)-7[/latex-display] 2 [latex-display]7\left(2x - 5\right)=8\left(4x - 1\right)-9[/latex-display]

### everyday math

#### Making a fence

Jovani has a fence around the rectangular garden in his backyard. The perimeter of the fence is $150$ feet. The length is $15$ feet more than the width. Find the width, $w$, by solving the equation $150=2\left(w+15\right)+2w$. 30 feet

#### Concert tickets

At a school concert, the total value of tickets sold was ${1,506.}$ Student tickets sold for ${6}$ and adult tickets sold for ${9.}$ The number of adult tickets sold was $5$ less than $3$ times the number of student tickets. Find the number of student tickets sold, $s$, by solving the equation $6s+9\left(3s - 5\right)=1506$. Coins Rhonda has ${1.90}$ in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, $n$, by solving the equation $0.05n+0.10\left(2n - 1\right)=1.90$. 8 nickels

#### Fencing

Micah has $74$ feet of fencing to make a rectangular dog pen in his yard. He wants the length to be $25$ feet more than the width. Find the length, $L$, by solving the equation $2L+2\left(L - 25\right)=74$.

### writing exercises

When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient as the variable side? Answers will vary. Solve the equation $10x+14=-2x+38$, explaining all the steps of your solution. What is the first step you take when solving the equation $3 - 7\left(y - 4\right)=38?$ Explain why this is your first step. Answers will vary. Solve the equation $\Large\frac{1}{4}\normalsize\left(8x+20\right)=3x - 4$ explaining all the steps of your solution as in the examples in this section. Using your own words, list the steps in the General Strategy for Solving Linear Equations. Answers will vary. Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.

## Solve Equations with Fraction or Decimal Coefficients

### Solve equations with fraction coefficients

In the following exercises, solve the equation by clearing the fractions. [latex-display]\Large\frac{1}{4}\normalsize x-\Large\frac{1}{2}\normalsize =-\Large\frac{3}{4}[/latex-display] x = −1 [latex-display]\Large\frac{3}{4}\normalsize x-\Large\frac{1}{2}\normalsize =\Large\frac{1}{4}[/latex-display] [latex-display]\Large\frac{5}{6}\normalsize y-\Large\frac{2}{3}\normalsize =-\Large\frac{3}{2}[/latex-display] y = −1 [latex-display]\Large\frac{5}{6}\normalsize y-\Large\frac{1}{3}\normalsize =-\Large\frac{7}{6}[/latex-display] [latex-display]\Large\frac{1}{2}\normalsize a+\Large\frac{3}{8}\normalsize =\Large\frac{3}{4}[/latex-display] [latex-display]a=\Large\frac{3}{4}[/latex-display] [latex-display]\Large\frac{5}{8}\normalsize b+\Large\frac{1}{2}\normalsize =-\Large\frac{3}{4}[/latex-display] [latex-display]2=\Large\frac{1}{3}\normalsize x-\Large\frac{1}{2}\normalsize x+\Large\frac{2}{3}\normalsize x[/latex-display] x = 4 [latex-display]2=\Large\frac{3}{5}\normalsize x-\Large\frac{1}{3}\normalsize x+\Large\frac{2}{5}\normalsize x[/latex-display] [latex-display]\Large\frac{1}{4}\normalsize m-\Large\frac{4}{5}\normalsize m+\Large\frac{1}{2}\normalsize m=-1[/latex-display] m = 20 [latex-display]\Large\frac{5}{6}\normalsize n-\Large\frac{1}{4}\normalsize n-\Large\frac{1}{2}\normalsize n=-2[/latex-display] [latex-display]x+\Large\frac{1}{2}\normalsize =\Large\frac{2}{3}\normalsize x-\Large\frac{1}{2}[/latex-display] x = −3 [latex-display]x+\Large\frac{3}{4}\normalsize =\Large\frac{1}{2}\normalsize x-\Large\frac{5}{4}[/latex-display] [latex-display]\Large\frac{1}{3}\normalsize w+\Large\frac{5}{4}\normalsize =w-\Large\frac{1}{4}[/latex-display] [latex-display]w=\Large\frac{9}{4}[/latex-display] [latex-display]\Large\frac{3}{2}\normalsize z+\Large\frac{1}{3}\normalsize =z-\Large\frac{2}{3}[/latex-display] [latex-display]\Large\frac{1}{2}\normalsize x-\Large\frac{1}{4}\normalsize =\Large\frac{1}{12}\normalsize x+\Large\frac{1}{6}[/latex-display] x = 1 [latex-display]\Large\frac{1}{2}\normalsize a-\Large\frac{1}{4}\normalsize =\Large\frac{1}{6}\normalsize a+\Large\frac{1}{12}[/latex-display] [latex-display]\Large\frac{1}{3}\normalsize b+\Large\frac{1}{5}\normalsize =\Large\frac{2}{5}\normalsize b-\Large\frac{3}{5}[/latex-display] b = 12 [latex-display]\Large\frac{1}{3}\normalsize x+\Large\frac{2}{5}\normalsize =\Large\frac{1}{5}\normalsize x-\Large\frac{2}{5}[/latex-display] [latex-display]1=\Large\frac{1}{6}\normalsize\left(12x - 6\right)[/latex-display] x = 1 [latex-display]1=\Large\frac{1}{5}\normalsize\left(15x - 10\right)[/latex-display] [latex-display]\Large\frac{1}{4}\normalsize\left(p - 7\right)=\Large\frac{1}{3}\normalsize\left(p+5\right)[/latex-display] p = −41 [latex-display]\Large\frac{1}{5}\normalsize\left(q+3\right)=\Large\frac{1}{2}\normalsize\left(q - 3\right)[/latex-display] [latex-display]\Large\frac{1}{2}\normalsize\left(x+4\right)=\Large\frac{3}{4}[/latex-display] [latex-display]x=-\Large\frac{5}{2}[/latex-display] [latex-display]\Large\frac{1}{3}\normalsize\left(x+5\right)=\Large\frac{5}{6}[/latex-display]

### Solve Equations with Decimal Coefficients

In the following exercises, solve the equation by clearing the decimals. [latex-display]0.6y+3=9[/latex-display] y = 10 [latex-display]0.4y - 4=2[/latex-display] [latex-display]3.6j - 2=5.2[/latex-display] j = 2 [latex-display]2.1k+3=7.2[/latex-display] [latex-display]0.4x+0.6=0.5x - 1.2[/latex-display] x = 18 [latex-display]0.7x+0.4=0.6x+2.4[/latex-display] [latex-display]0.23x+1.47=0.37x - 1.05[/latex-display] x = 18 [latex-display]0.48x+1.56=0.58x - 0.64[/latex-display] [latex-display]0.9x - 1.25=0.75x+1.75[/latex-display] x = 20 [latex-display]1.2x - 0.91=0.8x+2.29[/latex-display] [latex-display]0.05n+0.10\left(n+8\right)=2.15[/latex-display] n = 9 [latex-display]0.05n+0.10\left(n+7\right)=3.55[/latex-display] [latex-display]0.10d+0.25\left(d+5\right)=4.05[/latex-display] d = 8 [latex-display]0.10d+0.25\left(d+7\right)=5.25[/latex-display] [latex-display]0.05\left(q - 5\right)+0.25q=3.05[/latex-display] q = 11 [latex-display]0.05\left(q - 8\right)+0.25q=4.10[/latex-display]

### Everyday math

Coins Taylor has ${2.00}$ in dimes and pennies. The number of pennies is $2$ more than the number of dimes. Solve the equation $0.10d+0.01\left(d+2\right)=2$ for $d$, the number of dimes. d = 18 Stamps Travis bought ${9.45}$ worth of $\text{49-cent}$ stamps and $\text{21-cent}$ stamps. The number of $\text{21-cent}$ stamps was $5$ less than the number of $\text{49-cent}$ stamps. Solve the equation $0.49s+0.21\left(s - 5\right)=9.45$ for $s$, to find the number of $\text{49-cent}$ stamps Travis bought.

### writing exercises

Explain how to find the least common denominator of $\Large\frac{3}{8}\normalsize ,\Large\frac{1}{6}\normalsize ,\text{and}\Large\frac{2}{3}$. Answers will vary. If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve? If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD? Answers will vary. In the equation $0.35x+2.1=3.85$, what is the LCD? How do you know?

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

In the following exercises, determine whether the given number is a solution to the equation. [latex-display]x+16=31,x=15[/latex-display] yes [latex-display]w - 8=5,w=3[/latex-display] [latex-display]-9n=45,n=54[/latex-display] no [latex-display]4a=72,a=18[/latex-display] In the following exercises, solve the equation using the Subtraction Property of Equality. [latex-display]x+7=19[/latex-display] 12 [latex-display]y+2=-6[/latex-display] [latex-display]a+\Large\frac{1}{3}\normalsize =\Large\frac{5}{3}[/latex-display] [latex-display]a=\Large\frac{4}{3}[/latex-display] [latex-display]n+3.6=5.1[/latex-display] In the following exercises, solve the equation using the Addition Property of Equality. [latex-display]u - 7=10[/latex-display] u = 17 [latex-display]x - 9=-4[/latex-display] [latex-display]c-\Large\frac{3}{11}\normalsize =\Large\frac{9}{11}[/latex-display] [latex-display]c=\Large\frac{12}{11}[/latex-display] [latex-display]p - 4.8=14[/latex-display] In the following exercises, solve the equation. [latex-display]n - 12=32[/latex-display] n = 44 [latex-display]y+16=-9[/latex-display] [latex-display]f+\Large\frac{2}{3}\normalsize =4[/latex-display] [latex-display]f=\Large\frac{10}{3}[/latex-display] [latex-display]d - 3.9=8.2[/latex-display] [latex-display]y+8 - 15=-3[/latex-display] y = 4 [latex-display]7x+10 - 6x+3=5[/latex-display] [latex-display]6\left(n - 1\right)-5n=-14[/latex-display] n = −8 [latex-display]8\left(3p+5\right)-23\left(p - 1\right)=35[/latex-display] In the following exercises, translate each English sentence into an algebraic equation and then solve it. The sum of $-6$ and $m$ is $25$. −6 + m = 25; m = 31 Four less than $n$ is $13$. In the following exercises, translate into an algebraic equation and solve. Rochelle’s daughter is $11$ years old. Her son is $3$ years younger. How old is her son? s = 11 − 3; 8 years old Tan weighs $146$ pounds. Minh weighs $15$ pounds more than Tan. How much does Minh weigh? Peter paid ${9.75}$ to go to the movies, which was ${46.25}$ less than he paid to go to a concert. How much did he pay for the concert? c − 46.25 = 9.75; $56.00 Elissa earned ${152.84}$ this week, which was ${21.65}$ more than she earned last week. How much did she earn last week? ### Solve Equations using the Division and Multiplication Properties of Equality In the following exercises, solve each equation using the Division Property of Equality. [latex-display]8x=72[/latex-display] x = 9 [latex-display]13a=-65[/latex-display] [latex-display]0.25p=5.25[/latex-display] p = 21 [latex-display]-y=4[/latex-display] In the following exercises, solve each equation using the Multiplication Property of Equality. [latex-display]\Large\frac{n}{6}\normalsize =18[/latex-display] n = 108 [latex-display]\Large\frac{y}{-10}\normalsize =30[/latex-display] [latex-display]36=\Large\frac{3}{4}\normalsize x[/latex-display] x = 48 [latex-display]\Large\frac{5}{8}\normalsize u=\Large\frac{15}{16}[/latex-display] In the following exercises, solve each equation. [latex-display]-18m=-72[/latex-display] m = 4 [latex-display]\Large\frac{c}{9}\normalsize =36[/latex-display] [latex-display]0.45x=6.75[/latex-display] x = 15 [latex-display]\Large\frac{11}{12}\normalsize =\Large\frac{2}{3}\normalsize y[/latex-display] [latex-display]5r - 3r+9r=35 - 2[/latex-display] r = 3 [latex-display]24x+8x - 11x=-7 - 14[/latex-display] ### Solve Equations with Variables and Constants on Both Sides In the following exercises, solve the equations with constants on both sides. [latex-display]8p+7=47[/latex-display] p = 5 [latex-display]10w - 5=65[/latex-display] [latex-display]3x+19=-47[/latex-display] x = −22 [latex-display]32=-4 - 9n[/latex-display] In the following exercises, solve the equations with variables on both sides. [latex-display]7y=6y - 13[/latex-display] y = −13 [latex-display]5a+21=2a[/latex-display] [latex-display]k=-6k - 35[/latex-display] k = −5 [latex-display]4x-\Large\frac{3}{8}\normalsize =3x[/latex-display] In the following exercises, solve the equations with constants and variables on both sides. [latex-display]12x - 9=3x+45[/latex-display] x = 6 [latex-display]5n - 20=-7n - 80[/latex-display] [latex-display]4u+16=-19-u[/latex-display] u = −7 [latex-display]\Large\frac{5}{8}\normalsize c - 4=\Large\frac{3}{8}\normalsize c+4[/latex-display] In the following exercises, solve each linear equation using the general strategy. [latex-display]6\left(x+6\right)=24[/latex-display] x = −2 [latex-display]9\left(2p - 5\right)=72[/latex-display] [latex-display]-\left(s+4\right)=18[/latex-display] s = −22 [latex-display]8+3\left(n - 9\right)=17[/latex-display] [latex-display]23 - 3\left(y - 7\right)=8[/latex-display] y = 12 [latex-display]\Large\frac{1}{3}\normalsize\left(6m+21\right)=m - 7[/latex-display] [latex-display]8\left(r - 2\right)=6\left(r+10\right)[/latex-display] r = 38 [latex-display]5+7\left(2 - 5x\right)=2\left(9x+1\right)-\left(13x - 57\right)[/latex-display] [latex-display]4\left(3.5y+0.25\right)=365[/latex-display] y = 26 [latex-display]0.25\left(q - 8\right)=0.1\left(q+7\right)[/latex-display] ### Solve Equations with Fraction or Decimal Coefficients In the following exercises, solve each equation by clearing the fractions. [latex-display]\Large\frac{2}{5}\normalsize n-\Large\frac{1}{10}\normalsize =\Large\frac{7}{10}[/latex-display] n = 2 [latex-display]\Large\frac{1}{3}\normalsize x+\Large\frac{1}{5}\normalsize x=8[/latex-display] [latex-display]\Large\frac{3}{4}\normalsize a-\Large\frac{1}{3}\normalsize =\Large\frac{1}{2}\normalsize a+\Large\frac{5}{6}[/latex-display] [latex-display]a=\Large\frac{14}{3}[/latex-display] [latex-display]\Large\frac{1}{2}\normalsize\left(k+3\right)=\Large\frac{1}{3}\normalsize\left(k+16\right)[/latex-display] In the following exercises, solve each equation by clearing the decimals. [latex-display]0.8x - 0.3=0.7x+0.2[/latex-display] x = 5 [latex-display]0.36u+2.55=0.41u+6.8[/latex-display] [latex-display]0.6p - 1.9=0.78p+1.7[/latex-display] p = −20 [latex-display]0.10d+0.05\left(d - 4\right)=2.05[/latex-display] ### Chapter Practice Test Determine whether each number is a solution to the equation. $3x+5=23$. ⓐ $6$ ⓑ $\Large\frac{23}{5}$ ⓐ yes ⓑ no In the following exercises, solve each equation. [latex-display]n - 18=31[/latex-display] [latex-display]9c=144[/latex-display] c = 16 [latex-display]4y - 8=16[/latex-display] [latex-display]-8x - 15+9x - 1=-21[/latex-display] x = −5 [latex-display]-15a=120[/latex-display] [latex-display]\Large\frac{2}{3}\normalsize x=6[/latex-display] x = 9 [latex-display]x+3.8=8.2[/latex-display] [latex-display]10y=-5y+60[/latex-display] y = 4 [latex-display]8n+2=6n+12[/latex-display] [latex-display]9m - 2 - 4m+m=42 - 8[/latex-display] m = 6 [latex-display]-5\left(2x+1\right)=45[/latex-display] [latex-display]-\left(d+9\right)=23[/latex-display] d = −32 [latex-display]\Large\frac{1}{3}\normalsize\left(6m+21\right)=m - 7[/latex-display] [latex-display]2\left(6x+5\right)-8=-22[/latex-display] x = −2 [latex-display]8\left(3a+5\right)-7\left(4a - 3\right)=20 - 3a[/latex-display] [latex-display]\Large\frac{1}{4}\normalsize p+\Large\frac{1}{3}\normalsize =\Large\frac{1}{2}[/latex-display] [latex-display]p=\Large\frac{2}{3}[/latex-display] [latex-display]0.1d+0.25\left(d+8\right)=4.1[/latex-display] Translate and solve: The difference of twice $x$ and $4$ is $16$. 2x − 4 = 16; x = 10 Samuel paid ${25.82}$ for gas this week, which was ${3.47}$ less than he paid last week. How much did he pay last week? Determine Whether a Decimal is a Solution of an Equation In the following exercises, determine whether each number is a solution of the given equation. $x - 0.8=2.3$ $x=2$ $x=-1.5$ $x=3.1$ no no yes $y+0.6=-3.4$ $y=-4$ $y=-2.8$ $y=2.6$ $\Large\frac{h}{1.5}\normalsize =-4.3$ $h=6.45$ $h=-6.45$ $h=-2.1$ no yes no $0.75k=-3.6$ $k=-0.48$ $k=-4.8$ $k=-2.7$ #### Solve Equations with Decimals In the following exercises, solve the equation. $y+2.9=5.7$ y = 2.8 $m+4.6=6.5$ $f+3.45=2.6$ f = −0.85 $h+4.37=3.5$ $a+6.2=-1.7$ a = −7.9 $b+5.8=-2.3$ $c+1.15=-3.5$ c = −4.65 $d+2.35=-4.8$ $n - 2.6=1.8$ n = 4.4 $p - 3.6=1.7$ $x - 0.4=-3.9$ x = −3.5 $y - 0.6=-4.5$ $j - 1.82=-6.5$ j = −4.68 $k - 3.19=-4.6$ $m - 0.25=-1.67$ m = −1.42 $q - 0.47=-1.53$ $0.5x=3.5$ x = 7 $0.4p=9.2$ $-1.7c=8.5$ c = −5 $-2.9x=5.8$ $-1.4p=-4.2$ p = 3 $-2.8m=-8.4$ $-120=1.5q$ q = −80 $-75=1.5y$ $0.24x=4.8$ x = 20 $0.18n=5.4$ $-3.4z=-9.18$ z = 2.7 $-2.7u=-9.72$ $\Large\frac{a}{0.4}\normalsize =-20$ a = −8 $\Large\frac{b}{0.3}\normalsize =-9$ $\Large\frac{x}{0.7}\normalsize =-0.4$ x = −0.28 $\Large\frac{y}{0.8}\normalsize =-0.7$ $\Large\frac{p}{-5}\normalsize =-1.65$ p = 8.25 $\Large\frac{q}{-4}\normalsize =-5.92$ $\Large\frac{r}{-1.2}\normalsize =-6$ r = 7.2 $\Large\frac{s}{-1.5}\normalsize =-3$ #### Mixed Practice In the following exercises, solve the equation. Then check your solution. $x - 5=-11$ x = −6 $-\Large\frac{2}{5}\normalsize =x+\Large\frac{3}{4}$ $p+8=-2$ p = −10 $p+\Large\frac{2}{3}\normalsize =\Large\frac{1}{12}$ $-4.2m=-33.6$ m = 8 $q+9.5=-14$ $q+\Large\frac{5}{6}\normalsize =\Large\frac{1}{12}$ $q=-\Large\frac{3}{4}$ $\Large\frac{8.6}{15}\normalsize =-d$ $\Large\frac{7}{8}\normalsize m=\Large\frac{1}{10}$ $m=\Large\frac{4}{35}$ $\Large\frac{j}{-6.2}\normalsize =-3$ $-\Large\frac{2}{3}\normalsize =y+\Large\frac{3}{8}$ $y=-\Large\frac{25}{24}$ $s - 1.75=-3.2$ $\Large\frac{11}{20}\normalsize =-f$ $f=-\Large\frac{11}{20}$ $-3.6b=2.52$ $-4.2a=3.36$ a = −0.8 $-9.1n=-63.7$ $r - 1.25=-2.7$ r = −1.45 $\Large\frac{1}{4}\normalsize n=\Large\frac{7}{10}$ $\Large\frac{h}{-3}\normalsize =-8$ h = 24 $y - 7.82=-16$ #### Translate to an Equation and Solve In the following exercises, translate and solve. The difference of $n$ and $1.9$ is $3.4$. $n - 1.9=3.4;5.3$ The difference $n$ and $1.5$ is $0.8$. The product of $-6.2$ and $x$ is $-4.96$. −6.2x = −4.96; 0.8 The product of $-4.6$ and $x$ is $-3.22$. The quotient of $y$ and $-1.7$ is $-5$. $\Large\frac{y}{-1.7}\normalsize =-5;8.5$ The quotient of $z$ and $-3.6$ is $3$. The sum of $n$ and $-7.3$ is $2.4$. n + (−7.3) = 2.4; 9.7 The sum of $n$ and $-5.1$ is $3.8$. ### Everyday math Shawn bought a pair of shoes on sale for $78$ . Solve the equation $0.75p=78$ to find the original price of the shoes, $p$.$104

Mary bought a new refrigerator. The total price including sales tax was ${1,350}$. Find the retail price, $r$, of the refrigerator before tax by solving the equation $1.08r=1,350$.

### writing exercises

Think about solving the equation $1.2y=60$, but do not actually solve it. Do you think the solution should be greater than $60$ or less than $60?$ Explain your reasoning. Then solve the equation to see if your thinking was correct.

Think about solving the equation $0.8x=200$, but do not actually solve it. Do you think the solution should be greater than $200$ or less than $200?$ Explain your reasoning. Then solve the equation to see if your thinking was correct.