Simultaneous equations are used when you need to calculate the value of two or more unknown quantities. These are called simultaneous equations and when asked to solve them you must find values of the unknowns which satisfy all the given equations at the same time. You will need to learn how to solve Simultaneous equations as they are very common in the National 5 maths courses and exams.

We have used example Simultaneous equations from BBC Bite Size, which is also a great resource for you to learn with. Please also watch the video below to get our explanation of Simultaneous equations.

A simple example of a simultaneous equation:

Some equations have more than one unknown number so they can have more than one solution. For example:

2x + y = 10

This could be solved in a number of ways:

x = 1 and y = 8

x = 2 and y = 6

x = 3 and y = 4
To be able to solve an equation like this, another equation needs to be used alongside it. That way it is possible to find the only pair of values that solve both equations at the same time. These are known as simultaneous equations
An example of this is:
3x + y = 11 and
2x+ y = 8

The unknowns of x and y have the same value in both equations. This fact can be used to help solve the two simultaneous equations at the same time and find the values of x and y.

Solving simultaneous equations by elimination

The most common method for solving simultaneous equations is the elimination method which means one of the unknowns will be removed from each equation. The remaining unknown can then be calculated. This can be done if the coefficient of one of the letters is the same, regardless of sign.

Example

The solution of the pair of simultaneous equations
3x + 2y = 36, and 5x + 4y = 64
is x = 8 and y = 6. This is easily verified by substituting these values into the left-hand sides to obtain the values on the right. So x = 8, y = 6 satisfy the simultaneous equations.

How to solve simultaneous equations

There are many ways of solving simultaneous equations. Perhaps the simplest way is elimination. This is a process that involves removing or eliminating one of the unknowns to leave a single equation that involves the other unknown. The method is best illustrated by an example.

Example – Solve the simultaneous equations

3x + 2y = 36 (1)
5x + 4y = 64 (2)

Solution – Notice that if we multiply both sides of the first equation by 2 we obtain an equivalent equation
6x + 4y = 72 (3)

Solving simultaneous equations by substitution.

Substitution works by using one of the equations to get an expression of the form ‘y = …’ or ‘x = …’ and substituting this into the other equation. This gives an equation with just one unknown, which can be solved in the usual way. This value is then substituted in one or other of the original equations, giving an equation with one unknown.

Examples of solving 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

NOTE:

It is a good idea to label each equation. It helps you explain what you are doing − and may gain you method marks.
This value of x can be substituted into equation [1] or [2], or into the expression for y: y = 2x − 1.
Choose the one that is easiest!
As a check, substitute the values back into each of the two starting equations.

Solving simultaneous equations by multiplying.

It works because of two properties of equations:

Multiplying (or dividing) the expression on each side by the same number does not alter the equation.
Adding two equations produces another valid equation:
e.g. 2x = x + 10 (x = 10) and x − 3 = 7 (x also = 10).
Adding the equations gives 2x + x − 3 = x + 10 + 7 (x also = 10).

The object is to manipulate the two equations so that, when combined, either the x term or the y term is eliminated (hence the name) − the resulting equation with just one unknown can then be solved:

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 original equations.

Label your equations so you know which one you 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

NOTE:

    • It is a good idea to label each equation. It helps you explain what you are doing − and may gain you method marks.
    • This value of x can be substituted into equation [1] or [2], or into the expression for y: y = 2x − 1.
    • Choose the one that is easiest!
    • As a check, substitute the values back into each of the two starting equations.

Solving simultaneous equations by elimination.

It works because of two properties of equations:

Multiplying (or dividing) the expression on each side by the same number does not alter the equation.
Adding two equations produces another valid equation:
e.g. 2x = x + 10 (x = 10) and x − 3 = 7 (x also = 10).
Adding the equations gives 2x + x − 3 = x + 10 + 7 (x also = 10).

The object is to manipulate the two equations so that, when combined, either the x term or the y term is eliminated (hence the name) − the resulting equation with just one unknown can then be solved:

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 original 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 equations using straight line graphs

The 2 lines represent the equations ‘4x – 6y = -4’ and ‘2x + 2y = 6’.

There is only one point the two equations cross.

Because the graphs of 4x – 6y = 12 and 2x + 2y = 6 are straight lines, they are called linear equations.

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

Thanks to RevisionsMaths.com for providing these examples.

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