2.5 Solve equations with fractions or decimals

Elementary algebra Page 1 / 2

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.

Multiply: $8 \cdot \frac{3}{8} .$
If you missed this problem, review [link] .
Find the LCD of $\frac{5}{6}$ and $\frac{1}{4}$ .
If you missed this problem, review [link] .
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, $\frac{1}{8} x + \frac{1}{2} = \frac{1}{4}$ .


To isolate the $x$ term, subtract $\frac{1}{2}$ from both sides.
Simplify the left side.
Change the constants to equivalent fractions with the LCD.
Subtract.
Multiply both sides by the reciprocal of $\frac{1}{8}$ .
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: $\frac{1}{6} y - \frac{1}{3} = \frac{5}{6}$ .

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.

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.

Got questions? Get instant answers now!

Solve: $\frac{1}{4} x + \frac{1}{2} = \frac{5}{8}$ .

$x = \frac{1}{2}$

Got questions? Get instant answers now!

Solve: $\frac{1}{8} x + \frac{1}{2} = \frac{1}{4}$ .

$x = −2$

Got questions? Get instant answers now!

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.

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

Solve: $6 = \frac{1}{2} v + \frac{2}{5} v - \frac{3}{4} v$ .

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 $v = 40$ .

Got questions? Get instant answers now!

Solve: $7 = \frac{1}{2} x + \frac{3}{4} x - \frac{2}{3} x$ .

$x = 12$

Got questions? Get instant answers now!

Solve: $−1 = \frac{1}{2} u + \frac{1}{4} u - \frac{2}{3} u$ .

$u = −12$

Got questions? Get instant answers now!

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

Solve: $a + \frac{3}{4} = \frac{3}{8} a - \frac{1}{2}$ .

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 $a = −2$ .

Got questions? Get instant answers now!

Solve: $x + \frac{1}{3} = \frac{1}{6} x - \frac{1}{2}$ .

$x = −1$

Got questions? Get instant answers now!

Solve: $c + \frac{3}{4} = \frac{1}{2} c - \frac{1}{4}$ .

$c = −2$

Got questions? Get instant answers now!

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

Solve: $−5 = \frac{1}{4} (8 x + 4)$ .

Solution


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

Got questions? Get instant answers now!

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

	Interventionism Mixed Economy By Robert Murphy Start Test
	Microbiology Practice Test By Sandhills MLT Start Test
	Are you part of the R5 Family? By Anonymous User Start Quiz
©flickr: Hey	Anatomy Physiology Final By Joli Julianna Start Test
	17 Sociology 17 Government and Politics MCQ By OpenStax Start Quiz
	Financial Intelligence Quiz By Yasser Ibrahim Start Quiz
	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	Society and Culture By Mahee Boo Start Flashcards
©flickr:	Anatomy Physiology By Jemekia Weeden Start Quiz
	Power Enigeering types of bearing lubrication By Sam Luong Start Quiz