Rearranging an equation is an algebraic skill which is required to solve real life problems in mathematics. You can change the subject of an equation by applying inverse operations. In this article we show you step-by-step examples on how to rearrange equations so you can do this with ease.

Rearranging Equations Guide

What Does ‘Rearranging an Equation’ Mean?

Rearranging an equation means changing the subject of the equation, rewriting the same equation with a new subject. The subject of the equation is simply the single variable which the rest of the equation is equal to.

The subject of the equation can be written on either the left or right of the equals sign.

‘Subject of the Equation’ Examples

• In the equation y = 5x + 7, y is the subject of the equation.
• In x = y² + y – 5, x is the subject of the equation.
• In 4a + 5b = c, c is the subject of the equation.

If we want to rearrange the equation y = 5x + 7 to make x the subject of the equation, we need to isolate x. The equation would now begin with x = ____.

In this example, rearranging to make x the subject of the equation would result in x = (y – 7) ÷ 5.

If you’re confused so far, don’t worry – we have plenty of examples to show how we did this!

How to Rearrange Equations

To rearrange an equation we need to look at which operations are being performed and do the opposite. Opposite operations are called inverse operations.

Operation	Symbol	Inverse Operation	Symbol
Addition	+	Subtraction	−
Subtraction	−	Addition	+
Multiplication	×	Division	÷
Division	÷	Multiplication	×
Squaring	²	Finding the square root	√
Finding the square root	√	Squaring	²
Cubing	3	Finding the cube root	3√
Finding the cube root	3√	Cubing	3

Examples

Example 1: One-Step Equation

In a one-step equation only one operation is being performed so you simply need to do one single inverse operation.

In this example the operation being performed on b was +5 so we simply -5 from both sides of the equation.

Example 2: One Step Equation

In this example the operation being performed on x was ÷ 7 so we simply × 7 on both sides of the equation.

Example 3: Two Step Equation

For a two-step equation, two operations are being performed so you need to perform two inverse operations. Don’t forget that you need to do these in the opposite order. The following example will show you how.

In this example, the operations on x are multiply by 5 and then add 8. To make x the subject we need to do the inverse operations in the reverse order. Ie. subtract 8 and then divide by 5.

Note the brackets around y – 8 show that all of y – 8 is being divided by 5.

Example 4: Two Step Equation

In this example, the operations on b are square root and then subtract 2. To make b the subject we need to do the inverse operations in the reverse order. Ie. plus 2 and then square.

Note the brackets around a + 2 show that we are squaring all of a + 2.

Example 5: Multi-Step Equations

Multi-step equations involve a minimum of two operations. The method for solving them remains the same by applying inverse operations; the only difference lies in the additional steps required.

In this example, the operations on y are multiply by 5, subtract b and then cube. To make y the subject we need to do the inverse operations in the reverse order. Ie. find the cube root, add b, and divide by 5.

Note the brackets around ∛z + b show that we are dividing all of ∛z + b by 5.

Example 6: Multi-Step Equations including x²

If an equation includes a term with x² (or any variable squared) we need to find the positive and negative square roots.

For example if x² = 16, x has two values as it could be 4 or -4. We can use the plus or minus sign ± to denote this. Eg. if x² = 16, x = ±4.

If we find the square root of an algebraic expression we also need to use the ± sign. See the example below.