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Solving equations step by step

We solve an equation by getting x on its own on one side. We do the same thing on both sides, so that they stay equal. First we remove the term that is added or subtracted, then we divide by the number in front of x, and then we check.

Years 6–9 · Updated

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What is an equation?

The equals sign says that both sides have the same value. Solving the equation means finding out which number x is.

x + 7 = 12

Think of a set of balance scales. If we take the same amount from both pans, the scales still balance.

The goal: x on its own

We want x on its own.

We do the opposite of what is happening to x: we remove plus seven with minus seven, and we divide 3x by three. Always on both sides.

The order when several things happen to x

  • First remove what is added or subtracted (in 3x + 4 = 19 that is plus four).
  • Then divide by the number in front of x (here three).
  • Check the answer in the original equation.

If there is an x on both sides, we first collect all the x terms on one side, as in example 3.

Example

Example 1: x + 7 = 12

We want x on its own. Right now the left-hand side says x plus seven.

  1. We subtract seven from both sides, so that they stay equal.

    x + 7 − 7 = 12 − 7

  2. Seven minus seven is zero, so only x is left. Twelve minus seven is five.

    x + 0 = 5

  3. Now x is on its own.

    x = 5

5 + 7 = 12. Both sides have the same value, so x = 5 is correct.

Example 2: 3x + 4 = 19

We want x on its own. Two things are done to x: times three and plus four. We remove plus four first.

  1. We subtract four from both sides.

    3x + 4 − 4 = 19 − 4

  2. What is left is three x, which together make fifteen.

    3x = 15

  3. 3x is three equal parts. We divide both sides by three to get one part.

    3x ÷ 3 = 15 ÷ 3

  4. Three x divided by three is x. Fifteen divided by three is five.

    x = 5

3 · 5 + 4 = 15 + 4 = 19. The left-hand side is 19, the same as the right-hand side. That is correct.

Example 3: x on both sides, 5x − 2 = 2x + 10

We want all the x terms on one side. We start by removing 2x from the right-hand side.

  1. We subtract 2x from both sides.

    5x − 2 − 2x = 2x + 10 − 2x

  2. Now x is only on the left-hand side.

    3x − 2 = 10

  3. We add two to both sides, because −2 + 2 = 0.

    3x − 2 + 2 = 10 + 2

  4. What is left is three x, which make twelve.

    3x = 12

  5. We divide both sides by three.

    x = 4

Left-hand side: 5 · 4 − 2 = 18. Right-hand side: 2 · 4 + 10 = 18. They are equal, so x = 4 is correct.

Pitfalls

  • Changing only one side, for example subtracting seven on the left but not on the right.

    What we do to one side, we do to the other. Then the sides are still equal.

  • Dividing too early, so that 3x + 4 = 19 becomes x + 4 = 19 ÷ 3.

    If we divide one side by three, the whole side must be divided, including the four. So remove plus four first.

  • "Moving a term across and changing the sign" without knowing why, and changing the sign in the wrong place.

    Write out the step, for example 3x + 4 − 4 = 19 − 4. Then you can see what happens to each number.

Practise on your own

0 of 3 correct

  1. Solve the equation x + 9 = 15.

  2. Solve the equation 4x − 3 = 21.

  3. Solve the equation 7x + 5 = 3x + 25.

More walkthroughs with Fox

Common questions

What does it mean to solve an equation?

Finding the value of x that makes the left-hand side and the right-hand side equal. That value is called the solution.

Why do you do the same thing on both sides?

The sides have the same value to begin with. If we change them in exactly the same way, they still have the same value, so the equation still holds.

How do I know that my answer is right?

Put your answer in place of x in the original equation and work out both sides. If they are equal, the answer is right.

Can you move a term across and change its sign?

For terms that are added or subtracted it gives the same result, because it is a shortcut for doing the same thing on both sides. It does not work for numbers that are multiplied or divided. It helps to write out the step.

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