How to Solve Linear Equations Step by Step (Algebra 1 Guide)

To solve a linear equation, you follow three moves in order: simplify both sides, move variable terms to one side and constants to the other, then divide by the coefficient of the variable. For example, 2x + 3 = 7 becomes 2x = 4, then x = 2. Every linear equation, no matter how messy it looks, reduces to that same pattern.

Below we walk through two full examples, spell out the method, and list the mistakes that cost students points on tests.

What a linear equation looks like

A linear equation is any equation where the variable (usually x) only appears to the first power — no squares, no square roots, no x in a denominator. The standard form is ax + b = c, where a, b, and c are numbers. The goal is always to get x alone on one side of the equals sign.

Worked example: solve 3x + 5 = 20

1. Simplify. Both sides are already simplified — no like terms to combine.
2. Move constants away from the variable term. Subtract 5 from both sides: 3x + 5 - 5 = 20 - 5, which gives 3x = 15.
3. Divide by the coefficient of x. Divide both sides by 3: x = 15 / 3 = 5.
4. Check. Plug back in: 3(5) + 5 = 15 + 5 = 20. ✓

The answer is x = 5.

The three-step method for any linear equation

1. Simplify each side separately. Distribute parentheses and combine like terms. Example: 2(x + 3) + x = 2x + 6 + x = 3x + 6.
2. Get all variable terms on one side. Add or subtract to move every x-term to the left and every constant to the right. Example: from 5x - 2 = 3x + 8, subtract 3x to get 2x - 2 = 8.
3. Get all constants on the other side. Add or subtract to isolate the variable term. Example: 2x - 2 = 8 becomes 2x = 10.
4. Divide by the coefficient. Divide both sides by the number multiplying x. Example: 2x = 10 becomes x = 5.
5. Check your solution. Substitute back into the original equation and confirm both sides are equal.

The golden rule of equations

Whatever you do to one side, you must do to the other. That single rule is what keeps the equation balanced through every step.

A harder worked example: solve 4(x - 2) + 3 = 2x + 9

1. Simplify. Distribute the 4 on the left: 4x - 8 + 3 = 2x + 9. Combine like terms: 4x - 5 = 2x + 9.
2. Move variable terms to the left. Subtract 2x from both sides: 4x - 2x - 5 = 9, giving 2x - 5 = 9.
3. Move constants to the right. Add 5 to both sides: 2x = 14.
4. Divide by the coefficient. Divide both sides by 2: x = 7.
5. Check. Left side: 4(7 - 2) + 3 = 4(5) + 3 = 23. Right side: 2(7) + 9 = 14 + 9 = 23. ✓

The answer is x = 7.

Common mistakes to avoid

• Forgetting to distribute to every term. 4(x - 2) is 4x - 8, not 4x - 2.
• Only doing an operation to one side. If you subtract 5 from the left, you must subtract 5 from the right too.
• Sign errors when moving terms. Moving +3 across the equals sign makes it -3. Track signs carefully.
• Dividing before isolating. Don’t divide by the coefficient until every non-variable term has already been moved away.
• Skipping the check. Plugging your answer back into the original equation catches almost every arithmetic slip.

Practice the method with a guided tutor

Reading a worked example is one thing; getting stuck mid-problem and needing a hint is another. LernOS’s guided tutor walks you through linear equations one line at a time, asks the right question when you’re stuck, and never just hands you the answer — so the method actually sticks. Sign up to try it on your next problem.