Lesson 3: Equations with Rational Coefficients (Fractions and Decimals)

Property

An alternate method to solve equations with fractions is to eliminate 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.

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

Examples

• Solve $\frac{1}{5}y - \frac{1}{2} = \frac{3}{10}$. The LCD is 10. Multiplying by 10 gives $10(\frac{1}{5}y) - 10(\frac{1}{2}) = 10(\frac{3}{10})$, which simplifies to $2y - 5 = 3$. Solving gives $y = 4$.

Property

When an equation involves a fraction multiplied by a quantity in parentheses, you can either distribute the fraction first and then clear all resulting fractions with the LCD, or you can clear the initial fraction by multiplying by its denominator first.

Examples

• Solve $-5 = \frac{1}{4}(8x+4)$. Distribute first: $-5 = \frac{1}{4}(8x) + \frac{1}{4}(4)$, which simplifies to $-5 = 2x+1$. Then solve for $x$: $-6 = 2x$, so $x=-3$.

• Solve $\frac{1}{5}(q+3) = \frac{1}{2}(q-3)$. The LCD is 10. Multiply by 10: $10 \cdot \frac{1}{5}(q+3) = 10 \cdot \frac{1}{2}(q-3)$ becomes $2(q+3) = 5(q-3)$. Then $2q+6 = 5q-15$, so $21 = 3q$ and $q=7$.

Property

Some equations have decimals in them. Decimals can be expressed as fractions. For example, $0.3 = \frac{3}{10}$ and $0.17 = \frac{17}{100}$.
So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Examples

• Solve $0.05a + 0.03 = 0.2a - 1.2$. The place with the most decimals is hundredths, so multiply by 100. This gives $5a + 3 = 20a - 120$. Solving gives $123 = 15a$, so $a = 8.2$.

• Solve $0.14y + 0.07 = 0.3y - 1.53$. Multiply by 100: $14y + 7 = 30y - 153$. Solving gives $160 = 16y$, so $y=10$.