1. Home
  2. Elementary algebra
  3. Solving linear equations and
  4. Solve equations with fractions

2.5 Solve equations with fractions or decimals

<< Chapter < Page
  Elementary algebra     Page 1 / 2
Chapter >> Page >
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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 1 4 x + 1 2 = 5 8 .

x = 1 2

Got questions? Get instant answers now!

Solve: 1 8 x + 1 2 = 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.

  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 . .
.
.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 7 = 1 2 x + 3 4 x − 2 3 x .

x = 12

Got questions? Get instant answers now!

Solve: −1 = 1 2 u + 1 4 u − 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 + 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 . .
.
.
.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: x + 1 3 = 1 6 x − 1 2 .

x = −1

Got questions? Get instant answers now!

Solve: c + 3 4 = 1 2 c − 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 = 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!
Got questions? Get instant answers now!

Questions & Answers

differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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