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How to Solve Linear Equations Step by Step

By the Math Solver Free Editorial Team · Last reviewed: October 2026

A linear equation is an equation where the unknown (usually called x) appears only to the first power — no x², no square roots. If you can solve these confidently, you've unlocked the foundation of all algebra.

The goal is always the same: isolate x on one side of the equals sign. Whatever you do to one side, you must do to the other. That's the entire game.

The 4-step method

  1. Simplify each side — clear brackets and combine like terms.
  2. Move the x-terms to one side — add or subtract terms from both sides.
  3. Move the numbers to the other side — same idea, opposite side.
  4. Divide to isolate x — divide both sides by the coefficient of x.

Worked example 1: 2x + 5 = 13

Step 1: Nothing to simplify — move straight on.

Step 2: The x-term is already alone on the left.

Step 3: Subtract 5 from both sides: 2x + 5 − 5 = 13 − 5 → 2x = 8.

Step 4: Divide both sides by 2: x = 8 ÷ 2 → x = 4.

Check: substitute back: 2(4) + 5 = 8 + 5 = 13 ✓

Worked example 2 (with fractions): x/3 + 2 = 7

Step 3: Subtract 2 from both sides: x/3 = 5.

Step 4: Multiply both sides by 3: x = 15.

Check: 15/3 + 2 = 5 + 2 = 7 ✓

Fractions scare people, but the steps are identical — clearing the denominator (multiplying both sides by 3) is just step 4 with multiplication instead of division.

Worked example 3 (x on both sides): 5x − 3 = 2x + 9

Step 2: Subtract 2x from both sides: 3x − 3 = 9.

Step 3: Add 3 to both sides: 3x = 12.

Step 4: Divide by 3: x = 4.

Check: 5(4) − 3 = 17; 2(4) + 9 = 17 ✓

Common mistakes to avoid

  • Forgetting to apply an operation to both sides. If you subtract 5 from the left, you must subtract 5 from the right too.
  • Sign errors when moving terms. Moving +2x to the other side makes it −2x. Write the sign change explicitly until it's automatic.
  • Dividing only part of the expression. In (2x + 8) ÷ 2, BOTH terms get divided: x + 4, not x + 8.
  • Skipping the check. Thirty seconds of substitution catches most errors — always do it in exams.

Frequently asked questions

What is a linear equation in simple terms?

An equation where the variable appears only to the first power, like 3x + 2 = 11. Its graph is always a straight line.

How do I check my answer?

Substitute your x value back into the original equation. If both sides are equal, your answer is correct.

What if x is on both sides?

Collect the x-terms on one side first, then proceed as normal — see worked example 3 in this guide.

Is there a faster way than doing it by hand?

For homework and practice, doing it by hand builds the skill. To check your work instantly, use our free Algebra Calculator — it shows every step.

Try it yourself

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