Lesson 4: Solve Equations with Fraction or Decimal Coefficients

New Concept

Learn to simplify equations with fractions or decimals by 'clearing' them. Multiplying both sides by the least common denominator (LCD) transforms a complex problem into a simpler one with integers, which you already know how to solve.

What’s next

Next, you'll see this strategy applied in worked examples. Then, you'll master it through a series of interactive practice cards and challenge problems.

Property

Solve equations with fraction coefficients by clearing the fractions.

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

• To solve $\frac{1}{6}x + \frac{1}{3} = \frac{1}{2}$, the LCD is 6. Multiplying by 6 gives $6(\frac{1}{6}x) + 6(\frac{1}{3}) = 6(\frac{1}{2})$, which simplifies to $x + 2 = 3$, so $x = 1$.

Property

Decimals are another way to represent fractions. To solve equations with decimals, use the same process as clearing fractions: multiply both sides of the equation by the least common denominator (LCD) of the equivalent fractions. For decimals, the LCD will be a power of 10 (like 10, 100, or 1000).

Examples

• To solve $0.6x - 2 = 4$, the LCD is 10. Multiplying by 10 gives $10(0.6x) - 10(2) = 10(4)$, which simplifies to $6x - 20 = 40$. This solves to $6x = 60$, so $x = 10$.

• To solve $0.05x + 0.03 = 0.1x - 1.22$, the LCD is 100. Multiplying by 100 gives $5x + 3 = 10x - 122$. This simplifies to $125 = 5x$, so $x = 25$.

Property

When an equation has a fraction outside parentheses, apply the Distributive Property first. This may clear the fractions immediately. If fractions remain after distributing, find the LCD of all remaining fractions and multiply both sides of the equation to clear them.

Examples

• In the equation $3 = \frac{1}{4}(8x + 4)$, distribute $\frac{1}{4}$ to get $3 = \frac{1}{4}(8x) + \frac{1}{4}(4)$, which simplifies directly to $3 = 2x + 1$. The solution is $x=1$.

• To solve $\frac{1}{3}(y - 2) = \frac{1}{6}(y + 1)$, first distribute to get $\frac{1}{3}y - \frac{2}{3} = \frac{1}{6}y + \frac{1}{6}$. The LCD is 6. Multiplying by 6 gives $2y - 4 = y + 1$, so $y = 5$.

Property

For equations with a decimal factor outside parentheses, perform these steps:

1. Distribute the decimal to the terms inside the parentheses.
2. Combine any like terms.
3. Clear all decimals by multiplying both sides of the equation by the LCD (a power of 10).

Examples

• To solve $0.20x + 0.05(x + 10) = 3$, distribute to get $0.20x + 0.05x + 0.5 = 3$. Combine terms to get $0.25x + 0.5 = 3$. Multiply by 100 to get $25x + 50 = 300$, so $x=10$.

• To solve $0.25x + 0.40(x - 5) = 7.75$, distribute to get $0.25x + 0.40x - 2 = 7.75$. Combine terms to get $0.65x - 2 = 7.75$. Multiply by 100 to get $65x - 200 = 775$, so $x = 15$.