Skip to content
SuccessHomeschool
Menu
Log inStart free

Year 8 · Mathematics · Algebra

Solving linear equations

Solve linear equations with one or more steps, including equations with brackets and with the pronumeral on both sides, and check solutions by substitution.

Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.

Learning goals

Learners will

  • Solve one-step and two-step linear equations using inverse operations.
  • Solve equations involving brackets and pronumerals on both sides.
  • Check a solution by substituting it back into the original equation.

Success looks like

  • I can solve an equation such as 3x + 7 = 22 and explain each step.
  • I can solve an equation with brackets, such as 5(x − 2) = 30.
  • I can solve an equation with x on both sides, such as 4x + 3 = 2x + 15.
  • I can check my answer by substitution.

The big idea

An equation is like a set of balanced scales: whatever is on the left has the same value as whatever is on the right. To keep the scales balanced, whatever you do to one side, you must do to the other.

To solve an equation, we want to get the pronumeral on its own. We do this by "undoing" the operations that have been applied to it, using inverse operations. The operations are undone in the reverse order to how they were applied.

For example, in 3x + 7 = 22, the x has been multiplied by 3 and then 7 has been added. To undo this, first subtract 7 from both sides (3x = 15), then divide both sides by 3 (x = 5).

When an equation has brackets, you can either expand the brackets first or divide both sides by the number outside the brackets. Both methods give the same answer, so choose whichever makes the numbers easier.

When the pronumeral appears on both sides, move all the pronumeral terms to one side first. It is usually easiest to subtract the smaller pronumeral term from both sides so that you keep a positive coefficient.

Always finish by checking: substitute your answer into the original equation and confirm that both sides give the same value. This habit catches most errors.

Worked examples

A two-step equation

Solve 3x + 7 = 22.

  1. Subtract 7 from both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check: 3 × 5 + 7 = 15 + 7 = 22. Correct.

Answer: x = 5

An equation with brackets

Solve 5(x − 2) = 30.

  1. Divide both sides by 5: x − 2 = 6.
  2. Add 2 to both sides: x = 8.
  3. Check: 5 × (8 − 2) = 5 × 6 = 30. Correct.

Answer: x = 8

Pronumerals on both sides

Solve 4x + 3 = 2x + 15.

  1. Subtract 2x from both sides: 2x + 3 = 15.
  2. Subtract 3 from both sides: 2x = 12.
  3. Divide both sides by 2: x = 6.
  4. Check: left side 4 × 6 + 3 = 27; right side 2 × 6 + 15 = 27. Both sides match.

Answer: x = 6

Practice check

Have a go, then check your answers. Each answer comes with an explanation.

  1. 1.Solve x + 9 = 4.
  2. 2.Solve 4x = 28.
  3. 3.Which is a correct first step for solving x/3 − 2 = 5?
  4. 4.Solve 2(x + 4) = 18.
  5. 5.Which equation has the solution x = 3?
  6. 6.Solve 6x − 5 = 31.
  7. 7.Solve 7x + 2 = 3x + 18.
  8. 8.Mia thinks of a number. She multiplies it by 4 and then subtracts 6. Her answer is 30. What was her number?
Answer guide for parents

  1. Solve x + 9 = 4.

    x = −5 — Subtract 9 from both sides: x = 4 − 9 = −5.

  2. Solve 4x = 28.

    x = 7 — Divide both sides by 4: x = 28 ÷ 4 = 7.

  3. Which is a correct first step for solving x/3 − 2 = 5?

    Add 2 to both sides — Adding 2 to both sides undoes the subtraction and keeps the equation balanced: x/3 = 7, so x = 21.

  4. Solve 2(x + 4) = 18.

    x = 5 — Divide both sides by 2 to get x + 4 = 9, then subtract 4 to get x = 5.

  5. Which equation has the solution x = 3?

    2x + 1 = 7 — Substitute x = 3: 2 × 3 + 1 = 7, which is true. The others give 8 ≠ 10, 1 ≠ 3 and 2 ≠ 8.

  6. Solve 6x − 5 = 31.

    x = 6 — Add 5 to both sides (6x = 36), then divide by 6 (x = 6).

  7. Solve 7x + 2 = 3x + 18.

    x = 4 — Subtract 3x from both sides (4x + 2 = 18), subtract 2 (4x = 16), then divide by 4 (x = 4).

  8. Mia thinks of a number. She multiplies it by 4 and then subtracts 6. Her answer is 30. What was her number?

    9 — Write the equation 4n − 6 = 30. Add 6 to get 4n = 36, then divide by 4 to get n = 9.

Watch out for

  • Doing an operation to only one side of the equation. Return to the balance-scales picture: if you take something off one side, you must take the same amount off the other.
  • Undoing operations in the wrong order, such as dividing by 3 before subtracting 7 in 3x + 7 = 22. Undo the operation applied last first.
  • Sign errors when moving terms, such as turning x + 9 = 4 into x = 13. Encourage writing the full step, such as "subtract 9 from both sides", rather than "moving" numbers across.

Tips for parents

  • Ask your learner to say aloud what they are doing to both sides at each step. Clear reasoning matters more than speed.
  • Insist on the substitution check for every answer. If the check fails, help them find the step where the balance was lost rather than giving the answer.
  • Use a simple balance with coins or buttons to model an equation such as 2 cups + 3 coins = 11 coins if your learner finds the abstract version difficult.

Go further

  • Write your own "think of a number" puzzles for a family member, then turn their answers back into equations and solve them.
  • Create an equation whose solution is a negative number or a fraction, and swap with someone to solve.
  • Investigate a phone or electricity plan with a fixed fee plus a cost per unit, and write an equation to find how many units give a particular bill.