Download Solving First-Degree Equations Containing Fractions

Math 952
4.5 "Solving Equations with Fractions"
Objectives:
*
Understand that fractions and mixed numbers are possible solutions to equations.
*
Learn how to solve equations with fractional coe¢ cients.
Preliminaries:
In Section 1.5, we discussed solving …rst-degree equations with whole number constants and coe¢ cients. In Section 2.7,
we did it with integers. Today, we’ll solve equations having fractions as coe¢ cients.
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
where a, b; and c are constants and a 6= 0.
;
Solutions of a First-Degree Equation:
A …rst-degree equation has exactly
:
Solving First-Degree Equations Containing Fractions
Example 1 (Using the addition and multiplication principle)
Solve each equation.
a) x + 10 =
22
b) 5y
2y =
30
To Understand How to Solve Equations:
1: Apply the distributive property to remove parentheses whenever necessary.
2: Combine like terms on each side of the equation.
3: If a constant is added to a variable, add its opposite to both sides of the equation.
4: If a variable has constant coe¢ cient other than 1, divide both sides by that coe¢ cient.
5: Generally, use the Addition Principle …rst so that terms with variables are on one side and constant
terms are on the other side. Then combine terms and use the Multiplication Principle.
6: Remember that the object is to isolate the variable on one side of the equation with a coe¢ cient of 1.
Example 2 (Using both principles)
Solve the following equations.
a) 4x + 22 = 18
b) 2 (x
Page: 1
5) + 3x = 20
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
4.5
Equations with Fractions
Special Note on Fractional Coe¢ cients:
An expression such as 51 x can be thought of as a product.
Similarly,
NOTE:
To solve equations where the variable has a fractional coe¢ cient (as above) we multiply both sides by the
reciprocal of the coe¢ cient.
Example 3: (solving equations with fractional coe¢ cients)
Solve the equations.
3x
a)
+ 2 = 17
4
c)
3x 2x
+
=
5
5
1
10
b)
d)
Page: 2
23 =
2
n
5
1
y
2
2
n=
3
3
4
Bibiana Lopez
Prealgebra by Franklin Wright
4.5
NOTE: More general equations with fractional coe¢ cients can be solved by …rst multiplying every term in the equation by
the LCM of all denominators.
Example 4: (solving equations with fractional coe¢ cients)
Solve each equation.
3
1
3
a) x
x=
4
3
8
b)
Page: 3
1
1
4
= x+
2
3
15
Bibiana Lopez