Linear Equations with Fractions

Fractions make a linear equation look harder than it is. The equation is solved with exactly the same moves as — the only extra work is a single step at the start that removes every denominator at once. Once that step is done, nothing about the rest is any different from the Linear Equations lesson.

That step is multiplying every term on both sides by the least common denominator (LCD) of all the fractions. This lesson is about doing that step cleanly: finding the right denominator, distributing it to every term without missing one, and handling the situations that trip people up — a whole number sitting next to the fractions, a binomial like in a numerator, and coefficients written as rather than .

The Fraction-Clearing Formula

, , and stand for whatever is in each numerator — a number, a single variable term, or a whole binomial. , , and are the constant denominators. Multiplying both entire sides by cancels every denominator in one move, because the LCD is divisible by each of them. What is left is a linear equation you finish with the standard method.

The reason this is legal is the multiplication property of equality: you may multiply both sides of a true equation by the same nonzero number and it stays true. The LCD is a fixed number (never zero, since the denominators are nonzero constants), so multiplying by it changes how the equation looks without changing which value of makes it true.

Step 1: Find the Least Common Denominator

Write down every denominator in the equation, on both sides of the equals sign. Their least common multiple is the LCD.

When the denominators share no common factor

Then the LCD is simply their product. For denominators , , and , the LCD is . There is nothing smaller that all three divide into.

When the denominators share a factor

Then the LCD is smaller than the product. For and , the product is , but both divide into , so the LCD is . Using still works — the equation clears — but the numbers you carry are twice as large, which means twice as many chances for an arithmetic slip. Finding the true least common multiple is worth the few seconds.

A quick way to build the LCD: factor each denominator into primes, then take the highest power of each prime that appears anywhere. For , , and : the primes are (highest power ) and (highest power ), so the LCD is .

Step 2: Multiply Every Term by the LCD

Every term on both sides is multiplied by the LCD — the fractions, and any plain numbers too. Write the multiplication out term by term the first several times; skipping straight to the cleared line is where a term gets dropped.

For each fraction, divide the LCD by that fraction’s denominator, then multiply the result by the numerator: , and . Do the cancellation before you distribute — cancel the against the to get , and only then does the get attached to the numerator. The equation is now completely free of fractions.

Step 3: Solve the Fraction-Free Equation

From here it is the ordinary four-step method from Linear Equations: simplify each side, gather the variable on one side, gather constants on the other, divide by the coefficient.

Nothing new happens in this step. Every technique — distributing, combining like terms, handling variables on both sides — is exactly as it was before fractions entered the picture.

Handling a Binomial Numerator

When a numerator is a binomial like or , keep it in parentheses while you multiply, then distribute afterward.

The most common error in the whole lesson is dropping those parentheses and multiplying only the first term of the numerator. The fraction bar acts as a grouping symbol: everything on top of it is being divided by the denominator, so everything on top must be multiplied by whatever replaces that denominator.

Watch the sign carefully when the term is subtracted and the numerator has a minus inside it:

Coefficients Written as Fractions

An equation might present a fractional coefficient two different ways: or . They mean the same thing, and the LCD clears both identically — and . Read as “the fraction times ,” and its denominator, , goes into the LCD like any other.

Clearing Fractions and Decimals Together

If an equation somehow has both fractions and decimals, convert the decimals to fractions first (, , ), then clear everything with one LCD. Mixing the two clearing methods in one equation — multiplying part of it by for the decimals and part by the LCD for the fractions — is where sign and scale errors breed.

Revisiting Identities and Contradictions

Occasionally, after clearing fractions and simplifying, the variable cancels out entirely. If a true statement remains (like ), the equation is an identity and every real number is a solution. If a false statement remains (like ), it is a contradiction with no solution. Both outcomes are complete, correct answers — see Solutions and Solution Sets for the full treatment. Clearing fractions does not change which of these you get; it just makes the cancellation easier to see.

Worked Example A: Fractions on One Side

Solve .

Step 1 — the denominators are and ; they share no factor, so the LCD is . Multiply every term by :

Step 2 — combine like terms:

Step 3 — divide by :

Answer: . Check in the original: . ✓

Worked Example B: Fractions on Both Sides, with a Whole Number

Solve .

Step 1 — the denominators are , , and ; the LCD is . Multiply every term by , including the :

Step 2 — gather the variable on the left, the constants on the right:

Step 3 — divide by :

Answer: . A fractional answer is perfectly fine — clearing the denominators guarantees fraction-free work, not an integer solution. The dropped is the classic mistake here; it must be multiplied by just like every fraction.

Worked Example C: A Binomial Numerator

Solve .

Step 1 — the LCD of and is . Multiply every term by , keeping each numerator in parentheses:

Step 2 — distribute:

Step 3 — combine like terms:

Step 4 — solve:

Answer: .

Worked Example D: A Proportion

Solve .

Step 1 — the LCD of and is . Multiply both sides by :

Step 2 — distribute both sides:

Step 3 — solve:

Answer: . Because this equation is one fraction equal to one fraction, cross-multiplication gives the same line directly — cross-multiplying is just “multiply both sides by both denominators” for the special two-fraction case.

Worked Example E: A Fractional Coefficient Form

Solve .

Step 1 — the denominators are , , , and ; the LCD is . Multiply every term by :

Step 2 — solve:

Answer: .

Worked Example F: An Identity

Solve .

Step 1 — the LCD of , , and is . Multiply every term by :

Step 2 — simplify:

The variable cancelled and left a true statement. Answer: an identity — every real number is a solution, .

Common Mistakes to Avoid

• Multiplying only the fractions by the LCD. Every term gets multiplied, including whole numbers like the in Worked Example B.
• Dropping the parentheses on a binomial numerator. , which is , not .
• Using the product of the denominators when a smaller LCD exists. It still works, but the numbers get large fast — and have LCD , not .
• Forgetting a denominator on the other side of the equals sign. The LCD has to account for every fraction in the whole equation, left and right.
• Distributing the LCD before cancelling the denominator. Cancel first (), then attach the to the numerator.
• Mishandling a minus sign in front of a fraction with a binomial numerator. is , not .
• Expecting an integer answer. Clearing fractions makes the work fraction-free; the solution can still be a fraction.

Where This Shows Up Later

• Solving for a Variable. Formulas with fractional coefficients, like , are rearranged by clearing the fraction first — the same opening move as here.
• Rational Expressions and rational equations. The LCD step is identical; the new wrinkle is that a variable in the denominator forces you to check every answer for extraneous solutions.
• Systems of equations. A system with fractional coefficients is almost always cleared to integers first, one equation at a time, before elimination or substitution.
• Quadratic equations with fractions. Multiplying through by the LCD is still step one; it just leaves a quadratic instead of a linear equation.
• Work-rate word problems. “Together they finish in hours” produces an equation like , cleared with exactly this method.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Quick Reference

Once the denominators are gone, every remaining step comes straight from Linear Equations. Rearranging a formula that contains fractions uses this same first move — see Solving for a Variable. When a variable lands in a denominator, Rational Expressions picks up with the extra extraneous-solution check. The rest of the Algebra lessons are there when you want more.