Problem Set: Multi-Step Linear Equations

THIS IS OPTIONAL ADDITIONAL PRACTICE

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 [latex]y=\Large\frac{1}{3}[/latex] a solution of [latex]4y+2=10y?[/latex]

Is [latex]x=\Large\frac{3}{4}[/latex] a solution of [latex]5x+3=9x?[/latex]

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

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

In the following exercises, solve each equation.

[latex]x+7=12[/latex]

[latex]y+5=-6[/latex]

[latex]b+\Large\frac{1}{4}\normalsize =\Large\frac{3}{4}[/latex]

[latex]a+\Large\frac{2}{5}\normalsize =\Large\frac{4}{5}[/latex]

[latex]p+2.4=-9.3[/latex]

[latex]m+7.9=11.6[/latex]

[latex]a - 3=7[/latex]

[latex]m - 8=-20[/latex]

[latex]x-\Large\frac{1}{3}\normalsize=2[/latex]

[latex]x-\Large\frac{1}{5}\normalsize =4[/latex]

[latex]y - 3.8=10[/latex]

[latex]y - 7.2=5[/latex]

[latex]x - 15=-42[/latex]

[latex]z+5.2=-8.5[/latex]

[latex]q+\Large\frac{3}{4}\normalsize =\Large\frac{1}{2}[/latex]

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

[latex]p-\Large\frac{2}{5}\normalsize =\Large\frac{2}{3}[/latex]

[latex]y-\Large\frac{3}{4}\normalsize =\Large\frac{3}{5}[/latex]

[latex]y=\Large\frac{27}{20}[/latex]

Solve Equations that Need to be Simplified
In the following exercises, solve each equation.

[latex]c+3 - 10=18[/latex]

[latex]m+6 - 8=15[/latex]

17

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

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

8

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

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

−20

[latex]3\left(y - 5\right)-2y=-7[/latex]

[latex]4\left(y - 2\right)-3y=-6[/latex]

2

[latex]8\left(u+1.5\right)-7u=4.9[/latex]

[latex]5\left(w+2.2\right)-4w=9.3[/latex]

1.7

[latex]-5\left(y - 2\right)+6y=-7+4[/latex]

[latex]-8\left(x - 1\right)+9x=-3+9[/latex]

−2

[latex]3\left(5n - 1\right)-14n+9=1 - 2[/latex]

[latex]2\left(8m+3\right)-15m - 4=3 - 5[/latex]

−4

[latex]-\left(j+2\right)+2j - 1=5[/latex]

[latex]-\left(k+7\right)+2k+8=7[/latex]

6

[latex]6a - 5\left(a - 2\right)+9=-11[/latex]

[latex]8c - 7\left(c - 3\right)+4=-16[/latex]

−41

[latex]8\left(4x+5\right)-5\left(6x\right)-x=53[/latex]

[latex]6\left(9y - 1\right)-10\left(5y\right)-3y=22[/latex]

28

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

Five more than [latex]x[/latex] is equal to [latex]21[/latex].

The sum of [latex]x[/latex] and [latex]-5[/latex] is [latex]33[/latex].

x + (−5) = 33; x = 38

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

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

y − 3 = −19; y = −16

The sum of [latex]y[/latex] and [latex]-3[/latex] is [latex]40[/latex].

Eight more than [latex]p[/latex] is equal to [latex]52[/latex].

p + 8 = 52; p = 44

The difference of [latex]9x[/latex] and [latex]8x[/latex] is [latex]17[/latex].

The difference of [latex]5c[/latex] and [latex]4c[/latex] is [latex]60[/latex].

5c − 4c = 60; 60

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

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

[latex]f-\Large\frac{1}{3}\normalsize =\Large\frac{1}{12}\normalsize ;\Large\frac{5}{12}[/latex]

The sum of [latex]-4n[/latex] and [latex]5n[/latex] is [latex]-32[/latex].

The sum of [latex]-9m[/latex] and [latex]10m[/latex] is [latex]-25[/latex].

−9m + 10m = −25; m = −25

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Pilar drove from home to school and then to her aunt’s house, a total of [latex]18[/latex] miles. The distance from Pilar’s house to school is [latex]7[/latex] miles. What is the distance from school to her aunt’s house?

Jeff read a total of [latex]54[/latex] pages in his English and Psychology textbooks. He read [latex]41[/latex] pages in his English textbook. How many pages did he read in his Psychology textbook?

Let p equal the number of pages read in the Psychology book 41 + p = 54. Jeff read pages in his Psychology book.

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?

Eva’s daughter is [latex]5[/latex] years younger than her son. Eva’s son is [latex]12[/latex] years old. How old is her daughter?

Let d equal the daughter’s age. d = 12 − 5. Eva’s daughter’s age is 7 years old.

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

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 dinner turkey weighed [latex]16[/latex] pounds. How much did the Thanksgiving turkey weigh?

21 pounds

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?

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?

100.5 degrees

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?

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?

$121.19

everyday math

Baking

Kelsey needs [latex]\Large\frac{2}{3}[/latex] cup of sugar for the cookie recipe she wants to make. She only has [latex]\Large\frac{1}{4}[/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]\Large\frac{1}{4}\normalsize +s=\Large\frac{2}{3}[/latex] to find the amount of sugar she should ask to borrow.

Construction

Miguel wants to drill a hole for a [latex]\Large\frac{5}{\text{8}}\normalsize\text{-inch}[/latex] screw. The screw should be [latex]\Large\frac{1}{12}[/latex] inch larger than the hole. Let [latex]d[/latex] equal the size of the hole he should drill. Solve the equation [latex]d+\Large\frac{1}{12}\normalsize =\Large\frac{5}{8}[/latex] to see what size the hole should be.

[latex]d=\Large\frac{13}{24}[/latex]

 

writing exercises

Is [latex]-18[/latex] a solution to the equation [latex]3x=16 - 5x?[/latex] How do you know?

Write a word sentence that translates the equation [latex]y - 18=41[/latex] and then make up an application that uses this equation in its solution.

Answers will vary.

 

Solve Equations Using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

[latex]8x=32[/latex]

[latex]7p=63[/latex]

9

[latex]-5c=55[/latex]

[latex]-9x=-27[/latex]

3

[latex]-90=6y[/latex]

[latex]-72=12y[/latex]

−7

[latex]-16p=-64[/latex]

[latex]-8m=-56[/latex]

7

[latex]0.25z=3.25[/latex]

[latex]0.75a=11.25[/latex]

15

[latex]-3x=0[/latex]

[latex]4x=0[/latex]

0

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

[latex]\Large\frac{x}{4}\normalsize =15[/latex]

[latex]\Large\frac{z}{2}\normalsize =14[/latex]

28

[latex]-20=\Large\frac{q}{-5}[/latex]

[latex]\Large\frac{c}{-3}\normalsize =-12[/latex]

36

[latex]\Large\frac{y}{9}\normalsize =-6[/latex]

[latex]\Large\frac{q}{6}\normalsize =-8[/latex]

−48

[latex]\Large\frac{m}{-12}\normalsize =5[/latex]

[latex]-4=\Large\frac{p}{-20}[/latex]

80

[latex]\Large\frac{2}{3}\normalsize y=18[/latex]

[latex]\Large\frac{3}{5}\normalsize r=15[/latex]

25

[latex]-\Large\frac{5}{8}\normalsize w=40[/latex]

[latex]24=-\Large\frac{3}{4}\normalsize x[/latex]

−32

[latex]-\Large\frac{2}{5}\normalsize =\Large\frac{1}{10}\normalsize a[/latex]

[latex]-\Large\frac{1}{3}\normalsize q=-\Large\frac{5}{6}[/latex]

5/2

Solve Equations That Need to be Simplified
In the following exercises, solve the equation.

[latex]8a+3a - 6a=-17+27[/latex]

[latex]6y - 3y+12y=-43+28[/latex]

y = −1

[latex]-9x - 9x+2x=50 - 2[/latex]

[latex]-5m+7m - 8m=-6+36[/latex]

m = −5

[latex]100 - 16=4p - 10p-p[/latex]

[latex]-18 - 7=5t - 9t - 6t[/latex]

[latex]t=\Large\frac{5}{2}[/latex]

[latex]\Large\frac{7}{8}\normalsize n-\Large\frac{3}{4}\normalsize n=9+2[/latex]

[latex]\Large\frac{5}{12}\normalsize q+\Large\frac{1}{2}\normalsize q=25 - 3[/latex]

q = 24

[latex]0.25d+0.10d=6 - 0.75[/latex]

[latex]0.05p - 0.01p=2+0.24[/latex]

p = 56

Everyday math

Balloons Ramona bought [latex]18[/latex] balloons for a party. She wants to make [latex]3[/latex] equal bunches. Find the number of balloons in each bunch, [latex]b[/latex], by solving the equation [latex]3b=18[/latex].

Teaching Connie’s kindergarten class has [latex]24[/latex] children. She wants them to get into [latex]4[/latex] equal groups. Find the number of children in each group, [latex]g[/latex], by solving the equation [latex]4g=24[/latex].

6 children

Ticket price Daria paid [latex]{$36.25}[/latex] for [latex]5[/latex] children’s tickets at the ice skating rink. Find the price of each ticket, [latex]p[/latex], by solving the equation [latex]5p=36.25[/latex].

Unit price Nishant paid [latex]{$12.96}[/latex] for a pack of [latex]12[/latex] juice bottles. Find the price of each bottle, [latex]b[/latex], by solving the equation [latex]12b=12.96[/latex].

$1.08

Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets [latex]\text{18 mpg}[/latex]. Find the miles per gallons, [latex]m[/latex], of the hybrid car, by solving the equation [latex]\Large\frac{1}{2}\normalsize m=18[/latex].

Fabric The drill team used [latex]14[/latex] yards of fabric to make flags for one-third of the members. Find how much fabric, [latex]f[/latex], they would need to make flags for the whole team by solving the equation [latex]\Large\frac{1}{3}\normalsize f=14[/latex].

42 yards

 

writing exercises

Frida started to solve the equation [latex]-3x=36[/latex] by adding [latex]3[/latex] to both sides. Explain why Frida’s method will result in the correct solution.

Emiliano thinks [latex]x=40[/latex] is the solution to the equation [latex]\Large\frac{1}{2}\normalsize x=80[/latex]. 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]6x - 2=40[/latex]

[latex]7x - 8=34[/latex]

6

[latex]11w+6=93[/latex]

[latex]14y+7=91[/latex]

6

[latex]3a+8=-46[/latex]

[latex]4m+9=-23[/latex]

−8

[latex]-50=7n - 1[/latex]

[latex]-47=6b+1[/latex]

−8

[latex]25=-9y+7[/latex]

[latex]29=-8x - 3[/latex]

−4

[latex]-12p - 3=15[/latex]

[latex]-14\text{q}-15=13[/latex]

−2

Solve an Equation with Variables on Both Sides
In the following exercises, solve the equation for the variable.

[latex]8z=7z - 7[/latex]

[latex]9k=8k - 11[/latex]

−11

[latex]4x+36=10x[/latex]

[latex]6x+27=9x[/latex]

9

[latex]c=-3c - 20[/latex]

[latex]b=-4b - 15[/latex]

−3

[latex]5q=44 - 6q[/latex]

[latex]7z=39 - 6z[/latex]

3

[latex]3y+\Large\frac{1}{2}\normalsize =2y[/latex]

[latex]8x+\Large\frac{3}{4}\normalsize =7x[/latex]

−3/4

[latex]-12a - 8=-16a[/latex]

[latex]-15r - 8=-11r[/latex]

2

Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the equations for the variable.

[latex]6x - 15=5x+3[/latex]

[latex]4x - 17=3x+2[/latex]

19

[latex]26+8d=9d+11[/latex]

[latex]21+6f=7f+14[/latex]

7

[latex]3p - 1=5p - 33[/latex]

[latex]8q - 5=5q - 20[/latex]

−5

[latex]4a+5=-a - 40[/latex]

[latex]9c+7=-2c - 37[/latex]

−4

[latex]8y - 30=-2y+30[/latex]

[latex]12x - 17=-3x+13[/latex]

2

[latex]2\text{z}-4=23-\text{z}[/latex]

[latex]3y - 4=12-y[/latex]

4

[latex]\Large\frac{5}{4}\normalsize c - 3=\Large\frac{1}{4}\normalsize c - 16[/latex]

[latex]\Large\frac{4}{3}\normalsize m - 7=\Large\frac{1}{3}\normalsize m - 13[/latex]

6

[latex]8-\Large\frac{2}{5}\normalsize q=\Large\frac{3}{5}\normalsize q+6[/latex]

[latex]11-\Large\frac{1}{4}\normalsize a=\Large\frac{3}{4}\normalsize a+4[/latex]

7

[latex]\Large\frac{4}{3}\normalsize n+9=\Large\frac{1}{3}\normalsize n - 9[/latex]

[latex]\Large\frac{5}{4}\normalsize a+15=\Large\frac{3}{4}\normalsize a - 5[/latex]

−40

[latex]\Large\frac{1}{4}\normalsize y+7=\Large\frac{3}{4}\normalsize y - 3[/latex]

[latex]\Large\frac{3}{5}\normalsize p+2=\Large\frac{4}{5}\normalsize p - 1[/latex]

3

[latex]14n+8.25=9n+19.60[/latex]

[latex]13z+6.45=8z+23.75[/latex]

3.46

[latex]2.4w - 100=0.8w+28[/latex]

[latex]2.7w - 80=1.2w+10[/latex]

60

[latex]5.6r+13.1=3.5r+57.2[/latex]

[latex]6.6x - 18.9=3.4x+54.7[/latex]

23

Solve an Equation Using the General Strategy
In the following exercises, solve the linear equation using the general strategy.

[latex]5\left(x+3\right)=75[/latex]

[latex]4\left(y+7\right)=64[/latex]

9

[latex]8=4\left(x - 3\right)[/latex]

[latex]9=3\left(x - 3\right)[/latex]

6

[latex]20\left(y - 8\right)=-60[/latex]

[latex]14\left(y - 6\right)=-42[/latex]

3

[latex]-4\left(2n+1\right)=16[/latex]

[latex]-7\left(3n+4\right)=14[/latex]

−2

[latex]3\left(10+5r\right)=0[/latex]

[latex]8\left(3+3\text{p}\right)=0[/latex]

−1

[latex]\Large\frac{2}{3}\normalsize\left(9c - 3\right)=22[/latex]

[latex]\Large\frac{3}{5}\normalsize\left(10x - 5\right)=27[/latex]

5

[latex]5\left(1.2u - 4.8\right)=-12[/latex]

[latex]4\left(2.5v - 0.6\right)=7.6[/latex]

0.52

[latex]0.2\left(30n+50\right)=28[/latex]

[latex]0.5\left(16m+34\right)=-15[/latex]

0.25

[latex]-\left(w - 6\right)=24[/latex]

[latex]-\left(t - 8\right)=17[/latex]

−9

[latex]9\left(3a+5\right)+9=54[/latex]

[latex]8\left(6b - 7\right)+23=63[/latex]

2

[latex]10+3\left(z+4\right)=19[/latex]

[latex]13+2\left(m - 4\right)=17[/latex]

6

[latex]7+5\left(4-q\right)=12[/latex]

[latex]-9+6\left(5-k\right)=12[/latex]

3/2

[latex]15-\left(3r+8\right)=28[/latex]

[latex]18-\left(9r+7\right)=-16[/latex]

3

[latex]11 - 4\left(y - 8\right)=43[/latex]

[latex]18 - 2\left(y - 3\right)=32[/latex]

−4

[latex]9\left(p - 1\right)=6\left(2p - 1\right)[/latex]

[latex]3\left(4n - 1\right)-2=8n+3[/latex]

2

[latex]9\left(2m - 3\right)-8=4m+7[/latex]

[latex]5\left(x - 4\right)-4x=14[/latex]

34

[latex]8\left(x - 4\right)-7x=14[/latex]

[latex]5+6\left(3s - 5\right)=-3+2\left(8s - 1\right)[/latex]

10

[latex]-12+8\left(x - 5\right)=-4+3\left(5x - 2\right)[/latex]

[latex]4\left(x - 1\right)-8=6\left(3x - 2\right)-7[/latex]

2

[latex]7\left(2x - 5\right)=8\left(4x - 1\right)-9[/latex]

everyday math

Making a fence

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

30 feet

Concert tickets

At a school concert, the total value of tickets sold was [latex]{$1,506.}[/latex] Student tickets sold for [latex]{$6}[/latex] and adult tickets sold for [latex]{$9.}[/latex] The number of adult tickets sold was [latex]5[/latex] less than [latex]3[/latex] times the number of student tickets. Find the number of student tickets sold, [latex]s[/latex], by solving the equation [latex]6s+9\left(3s - 5\right)=1506[/latex].

Coins Rhonda has [latex]{$1.90}[/latex] in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, [latex]n[/latex], by solving the equation [latex]0.05n+0.10\left(2n - 1\right)=1.90[/latex].

8 nickels

Fencing

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

 

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 [latex]10x+14=-2x+38[/latex], explaining all the steps of your solution.

What is the first step you take when solving the equation [latex]3 - 7\left(y - 4\right)=38?[/latex] Explain why this is your first step.

Answers will vary.

Solve the equation [latex]\Large\frac{1}{4}\normalsize\left(8x+20\right)=3x - 4[/latex] 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]\Large\frac{1}{4}\normalsize x-\Large\frac{1}{2}\normalsize =-\Large\frac{3}{4}[/latex]

x = −1

[latex]\Large\frac{3}{4}\normalsize x-\Large\frac{1}{2}\normalsize =\Large\frac{1}{4}[/latex]

[latex]\Large\frac{5}{6}\normalsize y-\Large\frac{2}{3}\normalsize =-\Large\frac{3}{2}[/latex]

y = −1

[latex]\Large\frac{5}{6}\normalsize y-\Large\frac{1}{3}\normalsize =-\Large\frac{7}{6}[/latex]

[latex]\Large\frac{1}{2}\normalsize a+\Large\frac{3}{8}\normalsize =\Large\frac{3}{4}[/latex]

[latex]a=\Large\frac{3}{4}[/latex]

[latex]\Large\frac{5}{8}\normalsize b+\Large\frac{1}{2}\normalsize =-\Large\frac{3}{4}[/latex]

[latex]2=\Large\frac{1}{3}\normalsize x-\Large\frac{1}{2}\normalsize x+\Large\frac{2}{3}\normalsize x[/latex]

x = 4

[latex]2=\Large\frac{3}{5}\normalsize x-\Large\frac{1}{3}\normalsize x+\Large\frac{2}{5}\normalsize x[/latex]

[latex]\Large\frac{1}{4}\normalsize m-\Large\frac{4}{5}\normalsize m+\Large\frac{1}{2}\normalsize m=-1[/latex]

m = 20

[latex]\Large\frac{5}{6}\normalsize n-\Large\frac{1}{4}\normalsize n-\Large\frac{1}{2}\normalsize n=-2[/latex]

[latex]x+\Large\frac{1}{2}\normalsize =\Large\frac{2}{3}\normalsize x-\Large\frac{1}{2}[/latex]

x = −3

[latex]x+\Large\frac{3}{4}\normalsize =\Large\frac{1}{2}\normalsize x-\Large\frac{5}{4}[/latex]

[latex]\Large\frac{1}{3}\normalsize w+\Large\frac{5}{4}\normalsize =w-\Large\frac{1}{4}[/latex]

[latex]w=\Large\frac{9}{4}[/latex]

[latex]\Large\frac{3}{2}\normalsize z+\Large\frac{1}{3}\normalsize =z-\Large\frac{2}{3}[/latex]

[latex]\Large\frac{1}{2}\normalsize x-\Large\frac{1}{4}\normalsize =\Large\frac{1}{12}\normalsize x+\Large\frac{1}{6}[/latex]

x = 1

[latex]\Large\frac{1}{2}\normalsize a-\Large\frac{1}{4}\normalsize =\Large\frac{1}{6}\normalsize a+\Large\frac{1}{12}[/latex]

[latex]\Large\frac{1}{3}\normalsize b+\Large\frac{1}{5}\normalsize =\Large\frac{2}{5}\normalsize b-\Large\frac{3}{5}[/latex]

b = 12

[latex]\Large\frac{1}{3}\normalsize x+\Large\frac{2}{5}\normalsize =\Large\frac{1}{5}\normalsize x-\Large\frac{2}{5}[/latex]

[latex]1=\Large\frac{1}{6}\normalsize\left(12x - 6\right)[/latex]

x = 1

[latex]1=\Large\frac{1}{5}\normalsize\left(15x - 10\right)[/latex]

[latex]\Large\frac{1}{4}\normalsize\left(p - 7\right)=\Large\frac{1}{3}\normalsize\left(p+5\right)[/latex]

p = −41

[latex]\Large\frac{1}{5}\normalsize\left(q+3\right)=\Large\frac{1}{2}\normalsize\left(q - 3\right)[/latex]

[latex]\Large\frac{1}{2}\normalsize\left(x+4\right)=\Large\frac{3}{4}[/latex]

[latex]x=-\Large\frac{5}{2}[/latex]

[latex]\Large\frac{1}{3}\normalsize\left(x+5\right)=\Large\frac{5}{6}[/latex]

Solve Equations with Decimal Coefficients

In the following exercises, solve the equation by clearing the decimals.

[latex]0.6y+3=9[/latex]

y = 10

[latex]0.4y - 4=2[/latex]

[latex]3.6j - 2=5.2[/latex]

j = 2

[latex]2.1k+3=7.2[/latex]

[latex]0.4x+0.6=0.5x - 1.2[/latex]

x = 18

[latex]0.7x+0.4=0.6x+2.4[/latex]

[latex]0.23x+1.47=0.37x - 1.05[/latex]

x = 18

[latex]0.48x+1.56=0.58x - 0.64[/latex]

[latex]0.9x - 1.25=0.75x+1.75[/latex]

x = 20

[latex]1.2x - 0.91=0.8x+2.29[/latex]

[latex]0.05n+0.10\left(n+8\right)=2.15[/latex]

n = 9

[latex]0.05n+0.10\left(n+7\right)=3.55[/latex]

[latex]0.10d+0.25\left(d+5\right)=4.05[/latex]

d = 8

[latex]0.10d+0.25\left(d+7\right)=5.25[/latex]

[latex]0.05\left(q - 5\right)+0.25q=3.05[/latex]

q = 11

[latex]0.05\left(q - 8\right)+0.25q=4.10[/latex]

Everyday math

Coins Taylor has [latex]{$2.00}[/latex] in dimes and pennies. The number of pennies is [latex]2[/latex] more than the number of dimes. Solve the equation [latex]0.10d+0.01\left(d+2\right)=2[/latex] for [latex]d[/latex], the number of dimes.

d = 18

Stamps Travis bought [latex]{$9.45}[/latex] worth of [latex]\text{49-cent}[/latex] stamps and [latex]\text{21-cent}[/latex] stamps. The number of [latex]\text{21-cent}[/latex] stamps was [latex]5[/latex] less than the number of [latex]\text{49-cent}[/latex] stamps. Solve the equation [latex]0.49s+0.21\left(s - 5\right)=9.45[/latex] for [latex]s[/latex], to find the number of [latex]\text{49-cent}[/latex] stamps Travis bought.

 

writing exercises

Explain how to find the least common denominator of [latex]\Large\frac{3}{8}\normalsize ,\Large\frac{1}{6}\normalsize ,\text{and}\Large\frac{2}{3}[/latex].

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 [latex]0.35x+2.1=3.85[/latex], what is the LCD? How do you know?

 

Chapter Review Exercises

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]x+16=31,x=15[/latex]

yes

[latex]w - 8=5,w=3[/latex]

[latex]-9n=45,n=54[/latex]

no

[latex]4a=72,a=18[/latex]

In the following exercises, solve the equation using the Subtraction Property of Equality.

[latex]x+7=19[/latex]

12

[latex]y+2=-6[/latex]

[latex]a+\Large\frac{1}{3}\normalsize =\Large\frac{5}{3}[/latex]

[latex]a=\Large\frac{4}{3}[/latex]

[latex]n+3.6=5.1[/latex]

In the following exercises, solve the equation using the Addition Property of Equality.

[latex]u - 7=10[/latex]

u = 17

[latex]x - 9=-4[/latex]

[latex]c-\Large\frac{3}{11}\normalsize =\Large\frac{9}{11}[/latex]

[latex]c=\Large\frac{12}{11}[/latex]

[latex]p - 4.8=14[/latex]

In the following exercises, solve the equation.

[latex]n - 12=32[/latex]

n = 44

[latex]y+16=-9[/latex]

[latex]f+\Large\frac{2}{3}\normalsize =4[/latex]

[latex]f=\Large\frac{10}{3}[/latex]

[latex]d - 3.9=8.2[/latex]

[latex]y+8 - 15=-3[/latex]

y = 4

[latex]7x+10 - 6x+3=5[/latex]

[latex]6\left(n - 1\right)-5n=-14[/latex]

n = −8

[latex]8\left(3p+5\right)-23\left(p - 1\right)=35[/latex]

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

The sum of [latex]-6[/latex] and [latex]m[/latex] is [latex]25[/latex].

−6 + m = 25; m = 31

Four less than [latex]n[/latex] is [latex]13[/latex].

In the following exercises, translate into an algebraic equation and solve.

Rochelle’s daughter is [latex]11[/latex] years old. Her son is [latex]3[/latex] years younger. How old is her son?

s = 11 − 3; 8 years old

Tan weighs [latex]146[/latex] pounds. Minh weighs [latex]15[/latex] pounds more than Tan. How much does Minh weigh?

Peter paid [latex]{$9.75}[/latex] to go to the movies, which was [latex]{$46.25}[/latex] 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 [latex]{$152.84}[/latex] this week, which was [latex]{$21.65}[/latex] 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]8x=72[/latex]

x = 9

[latex]13a=-65[/latex]

[latex]0.25p=5.25[/latex]

p = 21

[latex]-y=4[/latex]

In the following exercises, solve each equation using the Multiplication Property of Equality.

[latex]\Large\frac{n}{6}\normalsize =18[/latex]

n = 108

[latex]\Large\frac{y}{-10}\normalsize =30[/latex]

[latex]36=\Large\frac{3}{4}\normalsize x[/latex]

x = 48

[latex]\Large\frac{5}{8}\normalsize u=\Large\frac{15}{16}[/latex]

In the following exercises, solve each equation.

[latex]-18m=-72[/latex]

m = 4

[latex]\Large\frac{c}{9}\normalsize =36[/latex]

[latex]0.45x=6.75[/latex]

x = 15

[latex]\Large\frac{11}{12}\normalsize =\Large\frac{2}{3}\normalsize y[/latex]

[latex]5r - 3r+9r=35 - 2[/latex]

r = 3

[latex]24x+8x - 11x=-7 - 14[/latex]

 

Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

[latex]8p+7=47[/latex]

p = 5

[latex]10w - 5=65[/latex]

[latex]3x+19=-47[/latex]

x = −22

[latex]32=-4 - 9n[/latex]

In the following exercises, solve the equations with variables on both sides.

[latex]7y=6y - 13[/latex]

y = −13

[latex]5a+21=2a[/latex]

[latex]k=-6k - 35[/latex]

k = −5

[latex]4x-\Large\frac{3}{8}\normalsize =3x[/latex]

In the following exercises, solve the equations with constants and variables on both sides.

[latex]12x - 9=3x+45[/latex]

x = 6

[latex]5n - 20=-7n - 80[/latex]

[latex]4u+16=-19-u[/latex]

u = −7

[latex]\Large\frac{5}{8}\normalsize c - 4=\Large\frac{3}{8}\normalsize c+4[/latex]

In the following exercises, solve each linear equation using the general strategy.

[latex]6\left(x+6\right)=24[/latex]

x = −2

[latex]9\left(2p - 5\right)=72[/latex]

[latex]-\left(s+4\right)=18[/latex]

s = −22

[latex]8+3\left(n - 9\right)=17[/latex]

[latex]23 - 3\left(y - 7\right)=8[/latex]

y = 12

[latex]\Large\frac{1}{3}\normalsize\left(6m+21\right)=m - 7[/latex]

[latex]8\left(r - 2\right)=6\left(r+10\right)[/latex]

r = 38

[latex]5+7\left(2 - 5x\right)=2\left(9x+1\right)-\left(13x - 57\right)[/latex]

[latex]4\left(3.5y+0.25\right)=365[/latex]

y = 26

[latex]0.25\left(q - 8\right)=0.1\left(q+7\right)[/latex]

 

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

[latex]\Large\frac{2}{5}\normalsize n-\Large\frac{1}{10}\normalsize =\Large\frac{7}{10}[/latex]

n = 2

[latex]\Large\frac{1}{3}\normalsize x+\Large\frac{1}{5}\normalsize x=8[/latex]

[latex]\Large\frac{3}{4}\normalsize a-\Large\frac{1}{3}\normalsize =\Large\frac{1}{2}\normalsize a+\Large\frac{5}{6}[/latex]

[latex]a=\Large\frac{14}{3}[/latex]

[latex]\Large\frac{1}{2}\normalsize\left(k+3\right)=\Large\frac{1}{3}\normalsize\left(k+16\right)[/latex]

In the following exercises, solve each equation by clearing the decimals.

[latex]0.8x - 0.3=0.7x+0.2[/latex]

x = 5

[latex]0.36u+2.55=0.41u+6.8[/latex]

[latex]0.6p - 1.9=0.78p+1.7[/latex]

p = −20

[latex]0.10d+0.05\left(d - 4\right)=2.05[/latex]

 

Chapter Practice Test

Determine whether each number is a solution to the equation.

[latex]3x+5=23[/latex].

ⓐ [latex]6[/latex]
ⓑ [latex]\Large\frac{23}{5}[/latex]

ⓐ yes
ⓑ no

In the following exercises, solve each equation.

[latex]n - 18=31[/latex]

[latex]9c=144[/latex]

c = 16

[latex]4y - 8=16[/latex]

[latex]-8x - 15+9x - 1=-21[/latex]

x = −5

[latex]-15a=120[/latex]

[latex]\Large\frac{2}{3}\normalsize x=6[/latex]

x = 9

[latex]x+3.8=8.2[/latex]

[latex]10y=-5y+60[/latex]

y = 4

[latex]8n+2=6n+12[/latex]

[latex]9m - 2 - 4m+m=42 - 8[/latex]

m = 6

[latex]-5\left(2x+1\right)=45[/latex]

[latex]-\left(d+9\right)=23[/latex]

d = −32

[latex]\Large\frac{1}{3}\normalsize\left(6m+21\right)=m - 7[/latex]

[latex]2\left(6x+5\right)-8=-22[/latex]

x = −2

[latex]8\left(3a+5\right)-7\left(4a - 3\right)=20 - 3a[/latex]

[latex]\Large\frac{1}{4}\normalsize p+\Large\frac{1}{3}\normalsize =\Large\frac{1}{2}[/latex]

[latex]p=\Large\frac{2}{3}[/latex]

[latex]0.1d+0.25\left(d+8\right)=4.1[/latex]

Translate and solve: The difference of twice [latex]x[/latex] and [latex]4[/latex] is [latex]16[/latex].

2x − 4 = 16; x = 10

Samuel paid [latex]{$25.82}[/latex] for gas this week, which was [latex]{$3.47}[/latex] 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.

[latex]x - 0.8=2.3[/latex]

ⓐ [latex]x=2[/latex] ⓑ [latex]x=-1.5[/latex] ⓒ [latex]x=3.1[/latex]

ⓐ no
ⓑ no
ⓒ yes

[latex]y+0.6=-3.4[/latex]

ⓐ [latex]y=-4[/latex] ⓑ [latex]y=-2.8[/latex] ⓒ [latex]y=2.6[/latex]

[latex]\Large\frac{h}{1.5}\normalsize =-4.3[/latex]

ⓐ [latex]h=6.45[/latex] ⓑ [latex]h=-6.45[/latex] ⓒ [latex]h=-2.1[/latex]

ⓐ no
ⓑ yes
ⓒ no

[latex]0.75k=-3.6[/latex]

ⓐ [latex]k=-0.48[/latex] ⓑ [latex]k=-4.8[/latex] ⓒ [latex]k=-2.7[/latex]

Solve Equations with Decimals

In the following exercises, solve the equation.

[latex]y+2.9=5.7[/latex]

y = 2.8

[latex]m+4.6=6.5[/latex]

[latex]f+3.45=2.6[/latex]

f = −0.85

[latex]h+4.37=3.5[/latex]

[latex]a+6.2=-1.7[/latex]

a = −7.9

[latex]b+5.8=-2.3[/latex]

[latex]c+1.15=-3.5[/latex]

c = −4.65

[latex]d+2.35=-4.8[/latex]

[latex]n - 2.6=1.8[/latex]

n = 4.4

[latex]p - 3.6=1.7[/latex]

[latex]x - 0.4=-3.9[/latex]

x = −3.5

[latex]y - 0.6=-4.5[/latex]

[latex]j - 1.82=-6.5[/latex]

j = −4.68

[latex]k - 3.19=-4.6[/latex]

[latex]m - 0.25=-1.67[/latex]

m = −1.42

[latex]q - 0.47=-1.53[/latex]

[latex]0.5x=3.5[/latex]

x = 7

[latex]0.4p=9.2[/latex]

[latex]-1.7c=8.5[/latex]

c = −5

[latex]-2.9x=5.8[/latex]

[latex]-1.4p=-4.2[/latex]

p = 3

[latex]-2.8m=-8.4[/latex]

[latex]-120=1.5q[/latex]

q = −80

[latex]-75=1.5y[/latex]

[latex]0.24x=4.8[/latex]

x = 20

[latex]0.18n=5.4[/latex]

[latex]-3.4z=-9.18[/latex]

z = 2.7

[latex]-2.7u=-9.72[/latex]

[latex]\Large\frac{a}{0.4}\normalsize =-20[/latex]

a = −8

[latex]\Large\frac{b}{0.3}\normalsize =-9[/latex]

[latex]\Large\frac{x}{0.7}\normalsize =-0.4[/latex]

x = −0.28

[latex]\Large\frac{y}{0.8}\normalsize =-0.7[/latex]

[latex]\Large\frac{p}{-5}\normalsize =-1.65[/latex]

p = 8.25

[latex]\Large\frac{q}{-4}\normalsize =-5.92[/latex]

[latex]\Large\frac{r}{-1.2}\normalsize =-6[/latex]

r = 7.2

[latex]\Large\frac{s}{-1.5}\normalsize =-3[/latex]

Mixed Practice

In the following exercises, solve the equation. Then check your solution.

[latex]x - 5=-11[/latex]

x = −6

[latex]-\Large\frac{2}{5}\normalsize =x+\Large\frac{3}{4}[/latex]

[latex]p+8=-2[/latex]

p = −10

[latex]p+\Large\frac{2}{3}\normalsize =\Large\frac{1}{12}[/latex]

[latex]-4.2m=-33.6[/latex]

m = 8

[latex]q+9.5=-14[/latex]

[latex]q+\Large\frac{5}{6}\normalsize =\Large\frac{1}{12}[/latex]

[latex]q=-\Large\frac{3}{4}[/latex]

[latex]\Large\frac{8.6}{15}\normalsize =-d[/latex]

[latex]\Large\frac{7}{8}\normalsize m=\Large\frac{1}{10}[/latex]

[latex]m=\Large\frac{4}{35}[/latex]

[latex]\Large\frac{j}{-6.2}\normalsize =-3[/latex]

[latex]-\Large\frac{2}{3}\normalsize =y+\Large\frac{3}{8}[/latex]

[latex]y=-\Large\frac{25}{24}[/latex]

[latex]s - 1.75=-3.2[/latex]

[latex]\Large\frac{11}{20}\normalsize =-f[/latex]

[latex]f=-\Large\frac{11}{20}[/latex]

[latex]-3.6b=2.52[/latex]

[latex]-4.2a=3.36[/latex]

a = −0.8

[latex]-9.1n=-63.7[/latex]

[latex]r - 1.25=-2.7[/latex]

r = −1.45

[latex]\Large\frac{1}{4}\normalsize n=\Large\frac{7}{10}[/latex]

[latex]\Large\frac{h}{-3}\normalsize =-8[/latex]

h = 24

[latex]y - 7.82=-16[/latex]

Translate to an Equation and Solve

In the following exercises, translate and solve.

The difference of [latex]n[/latex] and [latex]1.9[/latex] is [latex]3.4[/latex].

[latex]n - 1.9=3.4;5.3[/latex]

The difference [latex]n[/latex] and [latex]1.5[/latex] is [latex]0.8[/latex].

The product of [latex]-6.2[/latex] and [latex]x[/latex] is [latex]-4.96[/latex].

−6.2x = −4.96; 0.8

The product of [latex]-4.6[/latex] and [latex]x[/latex] is [latex]-3.22[/latex].

The quotient of [latex]y[/latex] and [latex]-1.7[/latex] is [latex]-5[/latex].

[latex]\Large\frac{y}{-1.7}\normalsize =-5;8.5[/latex]

The quotient of [latex]z[/latex] and [latex]-3.6[/latex] is [latex]3[/latex].

The sum of [latex]n[/latex] and [latex]-7.3[/latex] is [latex]2.4[/latex].

n + (−7.3) = 2.4; 9.7

The sum of [latex]n[/latex] and [latex]-5.1[/latex] is [latex]3.8[/latex].

Everyday math

Shawn bought a pair of shoes on sale for [latex]$78[/latex] . Solve the equation [latex]0.75p=78[/latex] to find the original price of the shoes, [latex]p[/latex].

$104

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

 

writing exercises

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

Answers will vary.

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