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Linear equations for AQA GCSE Maths

Linear equations

This page covers the following topics:

1. Solving linear equations
2. Linear equations with brackets
3. Equivalent expressions

To find the unknown in a linear equation the balancing method can be used. In the balancing method we either add or subtract the same value on each side of the equation so that only the unknown is left on one side of the equal sign.

Solving linear equations

To solve a linear equation that contains a set of brackets first use the balance method to ensure that the brackets are alone on one side of the equation. Then divide the whole equation by the value in front of the brackets and then use the balance method to solve the rest of the equation.

Linear equations with brackets

If two expressions can be rearranged to be exactly the same, they are called equivalent expressions. Various operations can be used to prove or check that two expressions are equivalent, including balancing method, expanding brackets and factorisation.

Equivalent expressions

1

Solve 3(w + 2) = 21 for w.

Divide both sides of the equation by 3 to get w + 2 = 7.
Then subtract 2 from both sides of the equation to get w = 5.

w = 5

Solve 3(w + 2) = 21 for w.

2

Are expressions 3x + 1 โˆ’ 7x and 6 โˆ’4x โˆ’ 5 equivalent? Show your working.

3x + 1 โˆ’ 7x = 1 โˆ’ 4x
6 โˆ’4x โˆ’ 5 = 1 โˆ’ 4x
1 โˆ’ 4x = 1 โˆ’ 4x

yes

Are expressions 3x + 1 โˆ’ 7x and 6 โˆ’4x โˆ’ 5 equivalent? Show your working.

3

Adam has improved his running score which is now 16 seconds less than triple the initial time. Is it the same as doubling his running time and reducing it by 11 seconds? Provide a reasoning for your answer.

The first change in Adamโ€™s running time can be expressed as 3x โˆ’ 16.
The second change in Adamโ€™s running time can be expressed as 2x โˆ’ 11.
These two expressions are not equivalent, thus the effects of the statements are not the same.

no, 3x โˆ’ 16 โ‰  2x โˆ’ 11

Adam has improved his running score which is now 16 seconds less than triple the initial time. Is it the same as doubling his running time and reducing it by 11 seconds? Provide a reasoning for your answer.

4

Solve 3w + 3 = โˆ’7. Give your answer as an improper fraction.

First subtract 3 from both sides to get 3w = โˆ’10.
Then divide both sides by 3 to get w = โˆ’10/3.

w = โˆ’10/3

Solve 3w + 3 = โˆ’7. Give your answer as an improper fraction.

5

Solve the equation 4x + 3 = 23.

First subtract 3 from both sides to get 4x = 20.
Then divide both sides by 4 to get x = 5.

x = 5

Solve the equation 4x + 3 = 23.

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