Click here to return to all Mathslearn guides


GCSE: Linear Equation Solving Part 1

We will be adding more basic notes for equation solving in the Key Stage 3 section.

This will provide those who need more detail on the fundamental ideas to really get to grips with the concepts.

We will add it as a link to the GCSE section when it is ready

For the moment, these provide a suitable set of exemplar examples and act as a check list for ideas which you may wish to grasp to access the higher grades at GCSE




1)           Basic equation

2)          Involving brackets

3)          Terms on both sides

4)          Fractions



Basic equation: Remember you use inverse functions




Add 2 to both sides


Divide by 5 (The coefficient)




Subtract 4 from each side


Divide by 7




Add 4 to both sides


Divide by 11.

This answer does not simplify

Leave as a fraction




Subtract 7 from both sides


Divide by 13

This answer does not simplify

Leave as a fraction







Subtract 4 from both sides


Divide by 2


Involving brackets

Key: It will nearly always help to expand the bracket first.





Subtract 35 from both sides.


Divide by 5


This answer does not simplify






Add 35 to both sides


Divide by 10


Leave as a fraction





Ignore the +13 at first


Work out +8+13=+21


Subtract 21 from both sides


Divide both sides by 12



Terms on both sides: This will revise many of the concepts met in the first two sections.

The aim is to get it to look like one of the basic examples from section 1.







If we subtract 2x from both sides, look what happens. It becomes a simpler equation.


Now, take 3 from each side


Now divide by 2





Now we will add 4x to both sides


Add 7 to both sides


Divide by 15





Multiply out both brackets



Subtract 6x


Subtract 20


Divide by 4





Add x to each side



Add 1 to each side


Divide by 2











Expand the bracket first, ignore all the other terms for the moment


Simplify the 4x + 3x


For ease re-write, so the largest number of xs is on the left. (Makes it easier


Subtract 7x from each side


Subtract 9 from each side


Divide by 5



Equations with fractions: Part 1



Multiply both sides by 3






Multiply by 7



Divide by 2





Multiply by x




Divide by 8





Multiply by 3x




Divide by 30



Equations with fractions: Part 2











So this is the same as:



The denominators are 4,2 & 8

Find the Lowest Multiple (LCM) of these numbers. This is 8.

Re-write all the fractions with this as their denominator.




If we now multiply by 8, the denominators all cancel.


Subtract 3


Divide by 2


This is the same as:




LCM of 7, 4 & 2 is 28


Multiply by 28


Add 7


Divide by 12

Equations with fractions: part 3


Re-write as:





So this is the same as










LCM of 5, 1 and 3 is 15


Multiply by 15







Subtract 99


Divide by 3