Simultaneous Equations

Simultaneous equations and linear equations, after studying this section, you will be able to:

  • solve simultaneous linear equations by substitution
  • solve simultaneous linear equations by elimination
  • solve simultaneous linear equations using straight line graphs



Solving Simultaneous Equations Video


This method is called solution by substitution.





Solve the two simultaneous equations:

2y + x = 8 [1]

1 + y = 2x [2]


from [2] y = 2x -1  ← subtract 1 from each side

Substituting this value for y into [1] gives:


2(2x – 1) + x = 8


4x – 2 + x = 8  ← expand the brackets

5x – 2 = 8  ←tidy up

5x = 10  ←Add 2 to each side

x = 2  ←By dividing both sides by 5 the value of x is found.


Substitute the value of x into y = 2x – 1 gives

y = 4 - 1 = 3


So x = 2 and y = 3



  • Choose the one that is easiest!

The second method is called solution by elimination.


It works because of two properties of equations:

  • Adding two equations produces another valid equation:


Here we will manipulate one of the equations so that when it is combined with the other equation either the x or y terms will drop out. In this example the x term will drop out giving a solution for y. This is then substituted into one of the otiginal equations.


Label your equations so you know which one your are working with at each stage.


Equation [1] is 2y + x = 8

Equation [2] is 1 + y = 2x


Rearrange one equation so it is similar to the other.


[2] y – 2x = -1


also 2 x [1] gives 4y + 2x = 16 which we call [3]


[2] y – 2x = -1

[3] 4y +2x = 16


[2] + [3] gives 5y = 15

so y = 3


substituting y = 3 into [1] gives 1 + (3) = 2x

so 2x = 4, giving x = 2 and y = 3


Solving simultaneous linear equations using straight line graphs


There is only one point the two equations cross.


Note: Only draw a graph if the question asks you to, it is usually quicker to work out the point two simultaneous equtions cross algebraically.

Click here for more on solving equations.