Solving a linear equation means finding the value of x that makes both sides equal. The method: do the same thing to both sides until x is alone. This guide walks one-step, two-step, and equations with brackets — with the three error patterns that lose the most marks.

The core principle: balance

An equation like 3x + 5 = 14 is a pair of scales. The = sign means "balanced". Anything you do to one side, you must do to the other — otherwise the scales tip and the equation breaks.

One-step equations

Example: x + 5 = 12

x has a + 5 attached. To remove it, do the opposite: subtract 5 from both sides.

x + 5 = 12        (start)
x + 5 - 5 = 12 - 5     (subtract 5 from both sides)
x = 7            (simplify)

Example: 3x = 15

x has a × 3 attached. To remove it, divide both sides by 3.

3x = 15           (start)
3x ÷ 3 = 15 ÷ 3   (divide both sides by 3)
x = 5             (simplify)

Two-step equations

Example: 3x + 5 = 14

Two things attached to x: a × 3 and a + 5. Always undo the + or − first, then the × or ÷.

3x + 5 = 14
3x + 5 - 5 = 14 - 5     (subtract 5 from both sides)
3x = 9                  (simplify)
3x ÷ 3 = 9 ÷ 3          (divide both sides by 3)
x = 3                   (simplify)

Example: 2x - 3 = 11

2x - 3 = 11
2x - 3 + 3 = 11 + 3     (add 3 to both sides — opposite of subtract)
2x = 14
2x ÷ 2 = 14 ÷ 2
x = 7

Equations with brackets

Example: 2(x + 4) = 18

Two options. Option A: expand first.

2(x + 4) = 18
2x + 8 = 18              (expand brackets — see /blog/expanding-brackets-ks3)
2x = 10                  (subtract 8)
x = 5                    (divide by 2)

Option B: divide by 2 first (works because the whole left side has a factor of 2).

2(x + 4) = 18
x + 4 = 9                (divide both sides by 2)
x = 5                    (subtract 4)

Both work. Option A is more general; Option B is faster when the right side divides cleanly.

Equations with x on both sides

Example: 3x + 2 = x + 10

Get all x's on one side, all numbers on the other.

3x + 2 = x + 10
3x - x + 2 = 10           (subtract x from both sides)
2x + 2 = 10
2x = 8                    (subtract 2 from both sides)
x = 4                     (divide by 2)

The three error patterns

Error 1: Sign flip on the move

❌ 3x - 5 = 10 → 3x = 10 - 5 = 5 ✅ 3x - 5 = 10 → 3x = 10 + 5 = 15

Fix: write the operation as its own step: 3x - 5 + 5 = 10 + 5. The extra line prevents the slip.

Error 2: Forgetting "both sides"

❌ 3x = 15 → x = 15 (forgot to divide by 3) ✅ 3x = 15 → 3x ÷ 3 = 15 ÷ 3 → x = 5

Fix: never apply an operation to one side without the other.

Error 3: Skipping verification

After you find x = 5, plug it back in: 3(5) + 5 = 20 ≠ 14. Wait — let me redo that example… 3(5) + 5 = 15 + 5 = 20. The correct example was 3x + 5 = 14 → x = 3, and 3(3) + 5 = 9 + 5 = 14 ✓.

Fix: always check the answer by substituting back. 60-second insurance.

Practice problems

  1. x + 7 = 12 → x = 5
  2. 4x = 20 → x = 5
  3. 2x + 3 = 13 → x = 5
  4. 5x - 4 = 21 → x = 5
  5. 3(x + 2) = 21 → x = 5
  6. 4x + 1 = 2x + 9 → x = 4

How Professor Pi teaches this

Pi never gives the final value of x. The hint ladder for 3x + 5 = 14:

  • L1: "What's attached to x in this equation?"
  • L2: "Right — a × 3 and a + 5. To isolate x, undo each. Which do we undo first — the + or the ×?"
  • L3: "Subtract 5 from both sides. What does that give you?"
  • L4: Worked twin for 2x + 7 = 13, then you do 3x + 5 = 14.

You get to x = 3 yourself, every time.

FAQ

What is the balance method for solving equations?

An equation is like scales: whatever you do to one side you must do to the other to keep them equal.

How do you solve a two-step equation like 3x + 5 = 14?

First remove the +5 by subtracting 5 from both sides (3x = 9), then divide both sides by 3 (x = 3).

What's the difference between one-step and two-step equations?

One-step needs one operation to isolate x. Two-step needs two — usually subtract/add first, then divide/multiply.

Pedagogy from Professor Pi at aitutors.me. Updated 20 May 2026.