Simultaneous equations - BBC Bitesize (2024)

Introducing simultaneous equations

We have looked at how to solve equations with one unknown value, but what happens when there are two unknown values?

An equation with two unknown values will have infinitely many solutions.

Example

\(x + y = 10\) has infinitely many values for \(x\) and \(y\) for which this is true, such as:

\(x=1\), \(y=9\)

or \(x=10\), \(y=0\)

or \(x=100\), \(y=-90\), etc.

But you can use two equations together, if they have the same two unknowns, to make one equation that has only one solution.

\(x\) and \(y\) values can be found which will solve both of the original equations at the same time or simultaneously.

A pair of equations like this are called simultaneous equations - because you are trying to solve them both with the same values for \(x\) and \(y\).

Substitution method

Example

Solve the simultaneous equations:

\(y = 2x\)

\(x + y = 6\)

One way to solve them is by using the substitution method.

Begin by labelling the equations (1) and (2):

\(y = 2x\) (1)

\(x + y = 6\) (2)

Equation (1) tells you that \(y = 2x\), so substitute this value of \(y\) into the second equation, ie replace \({y}\) with \({2x}\).

\(x + 2x = 6\)

\(3x = 6\)

\(x = 2\)

This gives you the value of \(x\), but what is the value of \(y\)?

Equation (1) tells you that \(y = 2x\), so \(y\) must be \(4\).

You can check your answer in the other equation - in this case, (2).

\(2 + 4 = 6\)

This is right, so the values are correct.

The solution of the equations is therefore:

\(x=2\), \(y=4\)

Algebraic method

You can solve simultaneous equations by adding or subtracting the two equations in order to end up with an equation with only one unknown value.

This is known as the algebraic method.

Example

Solve the simultaneous equations:

\(2x + y = 9\) (1)

\(3x - y = 1\) (2)

Add the two equations together and you will find that the \(y\) disappears:

\(3x + 2x + y - y = 9 + 1\)

This can be simplified to:

\(5x = 10\)

\(x = 2\)

Substituting this value of \(x\) in (1) gives:

\(4 + y = 9\)

\(y = 5\)

Check in (2):

\(6 - 5 = 1\) (which is correct)

So the solution is:

\(x=2\), \(y=5\)

Sometimes equations need to be altered, by multiplying throughout, before being able to eliminate one of the variables (letters).

Example

Solve the simultaneous equations:

\({2x} + {3y} = {9}\) (1)

\({3x} + {y} = {10}\) (2)

Neither the \(x\) nor the \(y\) will be eliminated by adding or subtracting these equations as they stand.

By multiplying the second equation by \({3}\) throughout, both equations will then include \({3y}\), which will allow us to continue with the solution.

(2)\(\times~{3}\) gives \({9x} + {3y} = {30}\) (3)

Don’t forget to multiply the right hand side by \(3\) as well.

Now, looking at equations (1) and (3), as they both include ‘\(+{3}{y}\)’, we will subtract one equation from the other.

Subtract equation (1) from equation (3) and you’ll find that the \(y\) disappears:

\((9x+3y)-(2x+3y)=30-9\)

This can be simplified to:

\({7x} = {21}\)

\({x} = {3}\)

Substituting this value of \(x\) in (1) gives:

\({2}\times{3}+{3y}={9}\)

\({6}+{3y}={9}\)

\({3y}={3}\)

\({y}={1}\)

Check in (2):

\({9} + {1} = {10}\) (which is correct)

So the solution is:

\({x}={3}\), \({y}={1}\)

As well as solving simultaneous equations algebraically, you can solve them by rearranging them into the form of \(y = mx+c\) so that you can plot them as straight line graphs.

When you have plotted the graphs, you can work out the values of \(x\) and \(y\) that solve the equations.

Simultaneous equations

Example

Use the graphical method to solve the simultaneous equations:

\(y = 2x\)

\(2x + y = 8\)

Start by rearranging the two equations to get them in the form:

\(y = mx + c\)

In this case you only need to rearrange the second equation. The two equations are now:

\(y = 2x\)

\(y = -2x + 8\)

Plotting the graph

We now have two equations of straight line graphs, which we can plot.

When we do this we can look at where the two lines cross (the point of intersection).

The values of \(x\) and \(y\) at this point are the solutions of the simultaneous equations.

The solution of this pair of simultaneous equations is:

\(x = 2\), \(y = 4\)

Where next?

Discover more about this topic from Bitesize.

Algebra