Maths

Solving equations

In an equation, each letter stands for a missing number. To solve an equation, find the values of the missing numbers.

Using inverses

The best way to solve an equation is by using 'inverses', or undoing what the equation is doing.

To use this method to solve equations remember that:

  • adding and subtracting are the inverse - or opposite - of each other
  • multiplying and dividing are the inverse of each other

When you use this method you must perform the same action on both sides.

Solve the equation: x - 6 = 9

To get x on its own, we need to add 6. If you add 6 to one side of the equation, you need to add 6 to the other side of the equation.

  • x - 6 + 6 = 9 + 6
  • x = 15
Question

Solve the equation 2y = 6 to find the value of y.

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Answer

To get y on its own, we need to divide it by 2. As you already know, you must always perform the same operation on both sides of the equation.

  • 2y ÷ 2 = 6 ÷ 2
  • y = 3

The following is an example of a typical exam question:

Solve the equation: 2a + 3 = 7

Sometimes you need to use the inverses method more than once to solve an equation.

Here is how to solve 2a + 3 = 7 using inverses.

  • 2a + 3 = 7
  • Undo the + 3 by subtracting 3
  • 2a + 3 - 3 = 7 - 3
  • 2a = 4
  • Then undo the multiply by 2 by dividing by 2 - again on both sides
  • 2a ÷ 2 = 4 ÷ 2

The answer is: a = 2

Now try this question.

Unknowns on both sides of the equation

Sometimes you will be asked to solve an equation with unknowns on both sides of the equation.

Remember that whatever you do to one side you must also do to the other.

Question

Solve the equation 3b + 4 = b + 12, and find the value of b.

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Answer

First, you need to get all the b terms on the same side of the equation.

Subtract b from both sides.

3b - b + 4 = 12

Then simplify.

2b + 4 = 12

Subtract 4 from both sides.

2b = 8

To find the value of b, divide both sides by 2.

b = 4

Try another question in the activity below.

Equations with brackets

If an equation has brackets in it, one method of solving it is to multiply out the brackets first, for example:

Solve the equation: 3(b + 2) = 15

  • 3(b + 2) = 15
  • Multiply out the brackets. Remember, everything inside the brackets gets multiplied by 3.
  • 3 × b + 3 × 2 = 15
  • When you have multiplied out the brackets you get: 3b + 6 = 15
  • Next, undo the + 6. In other words, do the inverse and subtract 6 from both sides.
  • 3b + 6 - 6 = 15 - 6
  • So 3b = 9

Therefore, to find out what b is you need to do the inverse of multiplying by 3 which is dividing by 3.

So b = 3

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