Guide

Solving Linear Equations

A linear equation looks scary until you learn the one rule behind all of them. This guide teaches the step-by-step method with three worked examples: a basic equation, one with a check, and one with x on both sides. Plus the mistakes that cost students marks.

What Is a Linear Equation?

A linear equation is an equation where x appears to the power 1 only. No x², no square roots of x. Examples: 4x − 7 = 21, 3x + 5 = 20, 2x + 3 = x + 9. The goal is always the same: get x alone on one side so you can read off its value.

The One Rule

Whatever you do to one side, do to the other. That is the entire method. To free x, undo the operations around it in reverse order: addition and subtraction first, then multiplication and division.

Worked Example 1: 4x − 7 = 21

Solve 4x − 7 = 21
  1. Undo the −7 first: add 7 to both sides. 4x = 21 + 7 = 28.
  2. Undo the ×4: divide both sides by 4. x = 28 ÷ 4 = 7.
  3. Answer: x = 7

Worked Example 2: 3x + 5 = 20

Solve 3x + 5 = 20
  1. Subtract 5 from both sides: 3x = 20 − 5 = 15.
  2. Divide both sides by 3: x = 15 ÷ 3 = 5.
  3. Answer: x = 5

When x Is on Both Sides

First collect all the x terms on one side. Subtract the smaller x term from both sides, then solve as normal.

Solve 2x + 3 = x + 9
  1. Subtract x from both sides to gather x on the left: 2x − x + 3 = 9, so x + 3 = 9.
  2. Subtract 3 from both sides: x = 9 − 3 = 6.
  3. Answer: x = 6

Always Check Your Answer

Put your answer back into the original equation. For 4x − 7 = 21 with x = 7: 4(7) − 7 = 28 − 7 = 21. Correct. This ten-second check catches most sign errors, and teachers give marks for it.

Common Mistakes to Avoid

Doing it to one side only. Adding 7 to the left but not the right breaks the equation. Every step must touch both sides.

Undoing in the wrong order. In 4x − 7 = 21, dividing by 4 first gives a mess. Undo addition and subtraction before multiplication and division.

Sign errors when moving terms. Moving −7 across becomes +7. Moving +5 across becomes −5. Slow down on signs; they cause most lost marks.

Stopping before the check. One wrong sign anywhere flips the answer. Plugging x back in takes seconds and proves you are right.

Quick Practice

  1. 2x − 9 = 11 Answer: x = 10 (2x = 20)
  2. 5x + 3 = 28 Answer: x = 5 (5x = 25)
  3. 3x − 4 = x + 8 Answer: x = 6 (2x = 12)

Frequently Asked Questions

The highest power of x is 1. As soon as you see x², it is quadratic, not linear, and needs the quadratic formula instead.
Work in reverse order of operations: addition and subtraction before multiplication and division. In 4x − 7 = 21, add 7 first, then divide by 4.
Move all x terms to one side first by subtracting the smaller x term from both sides, then solve normally. In 2x + 3 = x + 9, subtract x from both sides to get x + 3 = 9.
Yes. If simplifying gives a false statement like 3 = 7, no value of x works. If it gives a true statement like −2 = −2, every value of x works. The Equation Solver detects both cases.
Substitute your x value back into the original equation. Both sides must come out equal. For x = 7 in 4x − 7 = 21: 28 − 7 = 21. Correct.
Sign errors. When a term crosses to the other side it changes sign: −7 becomes +7. Most wrong answers in linear equations come from this one slip.
Try it on the solver

Type any of these equations into the Equation Solver and watch each step appear.

Open Equation Solver

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