Can I use any common denominator, or must it be the least common denominator?

You can multiply by any common multiple of the denominators to clear the fractions. However, using the LCD keeps the coefficients as small as possible, reducing the chance of arithmetic errors.

Can I use cross-multiplication to solve these?

Cross-multiplication is only valid for proportions, which are equations consisting of exactly one fraction on each side, such as $\frac{a}{b} = \frac{c}{d}$. If there are multiple terms on either side, you must use the LCD method.

What if the variable is in the denominator?

If the variable is in the denominator, it is a rational equation, not a standard linear equation. You still multiply by the LCD, but you must check your final answers to ensure they do not make any original denominator equal to zero (extraneous solutions).

How to solve a linear equation with fractions

To solve a linear equation with fractions, multiply both sides of the equation by the least common denominator (LCD) of all the fractions present. This method applies to any linear equation where the variable or constants are divided by integer values.

The setup

Identify all fractional terms in the equation. Determine the least common denominator (LCD) for all denominators on both the left and right sides of the equals sign. Treat integers as fractions with a denominator of 1.

The steps

Find the LCD of all fractions in the equation. 2. Multiply every single term on both sides of the equation by this LCD. 3. Simplify each term; all denominators should cancel out, leaving an equation with only integers. 4. Solve the resulting standard linear equation by isolating the variable using addition, subtraction, multiplication, and division.

Checking the result

Substitute your final answer back into the original equation containing the fractions. Simplify both sides using standard fraction arithmetic to verify that the left side exactly equals the right side.

Common errors

The most frequent mistake is forgetting to multiply the integer terms by the LCD. Another common error is multiplying the numerator and denominator by the LCD, rather than just the numerator. Finally, beware of sign errors when a negative fraction precedes a set of parentheses.

Worked example

Solve for $x$ : $\frac{x}{3} - \frac{1}{4} = \frac{x}{2} + 1$

The denominators are 3, 4, 2, and 1. The LCD is 12. Multiply every term by 12: $12\left(\frac{x}{3}\right) - 12\left(\frac{1}{4}\right) = 12\left(\frac{x}{2}\right) + 12(1)$ Simplify each term to clear fractions: $4x - 3 = 6x + 12$ Subtract $4x$ from both sides: $-3 = 2x + 12$ Subtract 12 from both sides: $-15 = 2x$ Divide by 2: $x = -\frac{15}{2}$

FAQ

Run your own problem

Answer form

References: OpenStax Elementary Algebra, Section 2.3: Solve Equations with Fractions or Decimals · Khan Academy, Solving linear equations with fractions