Solving Two-Step Linear Equations

This guide has everything you need on solving two-step linear equations.

The key principle of solving equations is to perform inverse operations. We need to identify what operations we have in each equation, then undo them by performing the inverse. We aim to have the variable (letter) left on one side of the equation and a constant (numerical value) on the other. If you find it tricky to identify operations, it may help to look back at the forming expressions revision guide to make sure you are comfortable with what each operation looks like in algebraic form. 

The best method to develop is the balancing method as it lets you systematically undo operations in equations of increasing difficulty. It is based on the idea of a balance beam. If a beam is balanced on a pivot, then we can conclude that the total value either side is equal. 

Furthermore, if we add or subtract the same on each side of the pivot, it will remain balanced. If we perform the same proportional changes (multiplication or division) to all terms then it will also remain balanced. 

In summary: 

  • Any amount of addition or subtraction should be the same on each side of the pivot (โ€˜=โ€™ sign). 
  • Any multiplication or division should be applied to ALL terms in the equation. 

How to Solve 2-Step Linear Equations

Example 1
Solve 2๐‘ฅ + 3 = 7.

When faced with 2-step equations, as a rule, it is generally best to reverse any additions or subtractions before reversing multiplications or divisions. This will avoid creating unnecessary fractions or decimals in the equation. 

The visual aid may help you to appreciate the context of your workings. 

2๐‘ฅ + 3 = 7
โ€“ 3โ€“ 3
2๐‘ฅ = 4
รท 2รท 2
๐‘ฅ = 2
Example 2
Solve 3๐‘ฅ + 5 = 12.

You will often find solutions to equations are not integers. In these cases, it is safest to leave your solutions as a fraction, unless specifically directed otherwise. 

3๐‘ฅ + 5 = 12
โ€“ 5โ€“ 5
3๐‘ฅ = 7
รท 3 รท 3 
๐‘ฅ = \frac{7}{3} \
Example 3
Solve 4(๐‘ฅ + 5) = 32.

When brackets are involved, it is usually easiest to expand them first before trying to balance. 

4(๐‘ฅ + 5) = 32
4๐‘ฅ + 20 = 32
โ€“ 20โ€“ 20
4๐‘ฅ = 12
รท 4รท 4
๐‘ฅ = 3

Testing Your Knowledge

Example 1
Solve 2๐‘ฅ + 3 = 7.

Test yourself on Solving Linear Equations.

Now that you know what you need to know about solving two-step linear equations, you’re probably ready for more revision. You can find more of our blogs here! You can also subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too.

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