# Solving simultaneous equations algebraically

## 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:

(1)

(2)

Add the two equations together and you will find that the disappears:

This can be simplified to:

Substituting this value of in (1) gives:

Check in (2):

(which is correct)

So the solution is:

,

Question

Solve the simultaneous equations

Firstly use labels for the equations:

(1)

(2)

To make the s disappear we can subtract equation (2) from equation (1):

This simplifies to:

Substitute in (1):

Check in (2):

(which is correct)

So the solution is:

,

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

### Example

Solve the simultaneous equations:

(1)

(2)

Neither the nor the will be eliminated by adding or subtracting these equations as they stand.

By multiplying the second equation by throughout, both equations will then include , which will allow us to continue with the solution.

(2) gives (3)

Don’t forget to multiply the right hand side by as well.

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

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

This can be simplified to:

Substituting this value of in (1) gives:

Check in (2):

(which is correct)

So the solution is:

,

Question

Solve the simultaneous equations:

Firstly use labels for the equations:

(1)

(2)

To make the s disappear we need to multiply equation (2) by before adding to equation (1):

(2) gives (3)

By adding equation (1) and equation (3) and you will find that the disappears:

This can be simplified to:

Substitute in (1):

Check in (2):

(which is correct)

So the solution is:

,