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Linear Equations

Find the mystery number: whatever you do to one side of an equation, do to the other. One-step and two-step equations, Year 8 style.

  • Solve one-step equations using inverse operations
  • Solve two-step equations like 3x + 2 = 14
  • Check answers by substituting back

The mystery number

An equation like x + 5 = 12 is a puzzle: which number, plus 5, gives 12? Solving it means finding that mystery number. (Here it's 7 — but let's learn a method that works every time, not just guessing.)

Definition: solving an equation

Solving an equation means finding the value of the unknown (usually x) that makes the equation true.

The golden rule

Imagine the equals sign as the pivot of a perfectly balanced see-saw. If you change one side, you must change the other side in exactly the same way — or it tips over.

The golden rule

Whatever you do to one side, do to the other.

One-step equations

In x + 5 = 12, the + 5 is getting in the way of x. Undo it with its inverse operation — subtraction — on both sides:

Maths
x + 5 = 12
x + 5 − 5 = 12 − 5
x = 7
Worked example

Solve x − 4 = 9.

The inverse of "minus 4" is "plus 4":

Maths
x − 4 = 9
x − 4 + 4 = 9 + 4
x = 13

Multiplication and division work the same way — they're inverses of each other:

Maths
x/3 = 9
x = 9 × 3
x = 27

Two-step equations

3x + 2 = 14 has two layers wrapped around x: multiply by 3, then add 2. Undo them in reverse order — addition/subtraction first:

Maths
3x + 2 = 14        subtract 2 from both sides
3x = 12            divide both sides by 3
x = 4
Always check

Put your answer back into the original equation. 3(4) + 2 = 14 ✓ — correct! Checking takes seconds and catches most mistakes.

Try it yourself

Type any linear equation below and watch each balancing step unfold.

Linear equation solverType any linear equation in x — it is solved step by step, entirely in your browser.

Press Solve to see each step — then cover the steps and try the next one yourself.

Practice

Solve: x + 9 = 20.
Subtract 9 from both sides.

x = 11. Check: 11 + 9 = 20 ✓

Solve: 5x = 35.
Divide both sides by 5.

x = 7. Check: 5 × 7 = 35 ✓

Solve: 2x − 3 = 11.
Add 3 first, then divide by 2.

2x = 14, so x = 7. Check: 2(7) − 3 = 11 ✓

I think of a number, multiply it by 4 and subtract 6. The answer is 26. What was my number?
Write it as 4x − 6 = 26, then solve.

4x = 32, so x = 8.

Quick check

Linear equations — quick check

To solve x + 7 = 15, you should…

Undo +7 with −7 on both sides: x = 8.

Solve: 4x + 1 = 21.

Subtract 1: 4x = 20. Divide by 4: x = 5.

Why do we check our answer?

Substituting back verifies the solution actually works in the original equation.
Key takeaways
  • Solving means finding the x that makes the equation true.
  • Golden rule: do the same thing to both sides.
  • Undo with inverse operations: +/− cancel each other, ×/÷ cancel each other.
  • For two-step equations, undo addition/subtraction before multiplication/division.
  • Always check by substituting back.
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