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Math 64
4.8 "Solving Equations Containing Fractions"
Objectives:
*
Solve equations containing fractions.
*
Solve equations by multiplying by the LCD.
*
Review adding and subtracting fractions.
Preliminaries:
In Chapter 3, we solved linear equations in one variable. In this section, we practice this skill by solving linear equations
containing fractions. To help us solve these equations, let’s review the properties of equality.
De…nition:
"First-Degree Equation in x"
A …rst degree equation in x (or linear equation in x) is any equation that can be written in the form
ax + b = 0
;
where a, b; and c are constants and a 6= 0.
Solutions of a First-Degree Equation:
A …rst-degree equation has exactly one solution
:
Addition Property of Equality:
Let a; b; and c represent numbers. Then a = b and a + c = b + c are equivalent equations.
Also, a = b and a
c=b
c are equivalent equations.
Multiplication Property of Equality:
Let a; b; and c represent numbers and let c 6= 0. Then a = b and a c = b c are equivalent equations.
a
b
Also, a = b and
=
are equivalent equations.
c
c
Solving Equations Containing Fractions
Example 1: (Solving equations containing fractions)
Solve each equation.
5
2
=
a) y
3
12
c)
1
y=2
5
b) 9x
d)
Page: 1
2
7
8x =
11
14
5
b = 25
7
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
e)
4.8
7
2
x=
10
5
f) 5m =
3
4
Solving Equations by Multiplying by the LCD
If an equation contains fractions, it is often helpful to …rst multiply both sides of the equation by the LCD of the
fractions. This has the e¤ect of eliminating the fractions in the equation.
Solving an Equation in x :
1: If fractions are present, multiply both sides of the equation by the LCD of the fractions.
2: If parentheses are present, use the distributive property.
3: Combine any like terms on each side of the equation.
4: Use the addition property of equality to rewrite the equation so that variable terms
are on one side of the equation and constant terms are on the other side.
5: Divide both sides of the equation by the numerical coe¢ cient of x to solve.
6: Check the answer in the original equation.
Example 2 (Solving equations by multiplying by the LCD)
Solve the following equations.
y 3
a)
+ =2
8 4
c)
m
m 3
=
+
2
5
2
b)
x
5
d)
7
1
t=
6
4
Page: 2
x=
1
5
2
3
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
4.8
Review of Adding and Subtracting Fractions
Make sure you understand the di¤erence between solving an equation containing fractions and adding or subtracting
two fractions. To solve an equation containing fractions, we use the multiplication property of equality and multiply both
sides by the LCD of the fractions, thus eliminating the fractions. The multiplication property of equality applies only to
equations. To add or subtract unlike fractions, we write each fraction as an equivalent fraction using the LCD of the fractions
as the denominator.
Example 3 (Review of adding and subtracting fractions)
9
x
Add:
+
10 3
Example 4: (Solving equations by multiplying by the LCD)
x
1
2
Solve:
+ =
12 4
3
Page: 3
Notes by Bibiana Lopez