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By the end of this section, you will be able to:
  • Solve equations with fraction coefficients
  • Solve equations with decimal coefficients

Before you get started, take this readiness quiz.

  1. Multiply: 8 · 3 8 .
    If you missed this problem, review [link] .
  2. Find the LCD of 5 6 and 1 4 .
    If you missed this problem, review [link] .
  3. Multiply 4.78 by 100.
    If you missed this problem, review [link] .

Solve equations with fraction coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, 1 8 x + 1 2 = 1 4 .

.
To isolate the x term, subtract 1 2 from both sides. .
Simplify the left side. .
Change the constants to equivalent fractions with the LCD. .
Subtract. .
Multiply both sides by the reciprocal of 1 8 . .
Simplify. .

This method worked fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

How to solve equations with fraction coefficients

Solve: 1 6 y 1 3 = 5 6 .

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Find the least common denominator of all the fractions in the equation.” The text in the second cell reads: “What is the LCD of 1/6, 1/3, and 5/6?” The third cell contains the equation one-sixth y minus 1/3 equals 5/6, with LCD equals 6 written next to it. In the second row of the table, the first cell says: “Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.” In the second cell, the instructions say: “Multiply both sides of the equation by the LCD 6. Use the Distributive Property. Simplify—and notice, no more fractions!” The third cell contains the equation 6 times one-sixth y minus 1/3, with one-sixth y minus 1/3 in brackets, equals 6 times 5/6, with “6 times” written in red on both sides. Below this is the same equation with the 6 distributed on both sides: 6 times one-sixth y minus 6 times 1/3 equals 6 times 5/6. Below this is the equation y minus 2 equals 5. In the third row of the table, the first cell says: “Step 3. Solve using the General Strategy for Solving Linear Equations.” In the second cell, the instructions say: “Isolate the x term, add 2. Simplify.” The third cell contains the equation with 2 added to both sides: y minus 2 plus 2 equals 5 plus 2, with “plus 2” written in red on both sides. Below this is the equation y equals 7.
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Solve: 1 4 x + 1 2 = 5 8 .

x = 1 2

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Solve: 1 8 x + 1 2 = 1 4 .

x = −2

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Notice in [link] , once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

Strategy to solve equations with fraction coefficients.

  1. Find the least common denominator of all the fractions in the equation.
  2. Multiply both sides of the equation by that LCD. This clears the fractions.
  3. Solve using the General Strategy for Solving Linear Equations.

Solve: 6 = 1 2 v + 2 5 v 3 4 v .

Solution

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the LCD of all fractions in the equation. .
The LCD is 20.
Multiply both sides of the equation by 20. .
Distribute. .
Simplify—notice, no more fractions! .
Combine like terms. .
Divide by 3. .
Simplify. .
Check: .
Let v = 40 . .
.
.
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Solve: 7 = 1 2 x + 3 4 x 2 3 x .

x = 12

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Solve: −1 = 1 2 u + 1 4 u 2 3 u .

u = −12

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In the next example, we again have variables on both sides of the equation.

Solve: a + 3 4 = 3 8 a 1 2 .

Solution

.
Find the LCD of all fractions in the equation.
The LCD is 8.
Multiply both sides by the LCD. .
Distribute. .
Simplify—no more fractions. .
Subtract 3 a from both sides. .
Simplify. .
Subtract 6 from both sides. .
Simplify. .
Divide by 5. .
Simplify. .
Check: .
Let a = −2 . .
.
.
.
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Solve: x + 1 3 = 1 6 x 1 2 .

x = −1

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Solve: c + 3 4 = 1 2 c 1 4 .

c = −2

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In the next example, we start by using the Distributive Property. This step clears the fractions right away.

Solve: −5 = 1 4 ( 8 x + 4 ) .

Solution

.
Distribute. .
Simplify.
Now there are no fractions.
.
Subtract 1 from both sides. .
Simplify. .
Divide by 2. .
Simplify. .
Check: .
Let x = −3 . .
.
.
.
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Questions & Answers

A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Kendra Reply
Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
What is observation
adeyemi Reply
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Sunnyshay Reply
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Sophia Reply
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs $9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?
Parker Reply
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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