Simultaneous equations involve two variables, each of whose value must be found.

The following method, in which one equation is substituted into the other, works best when one equation contains a single letter on its own.

**Example**

Solve simultaneously

3x - 2y = 14

y - 2x = -9

**Answer**

3x - 2y = 14 (a)

y - 2x = -9 (b)

We find the equation with the single letter, i.e. (b), which has a single y in it.

We rearrange this to read y = ...

y = 2x - 9 (b)

Substitute (b) into (a).

Wherever we see a y in (a) we will replace it with (2x - 9), remembering the brackets.

(a) becomes:

3x - 2(2x - 9) = 14

Multiply out the brackets and solve.

Be careful with the double negative signs.

*Notice that we now have an equation just in x.*

3x - 4x + 18 = 14

-x + 18 = 14

-x = -4

x = 4

Now use (b) to determine y.

y = 2x - 9

When x = 4, this gives:

y = 2×4 - 9 = 8 - 9 = -1

y = -1

Check by putting this into the other equation (a) to get 3 × 4 - 2 × -1 = 12 + 2 = 14

Solution is **x = 4, y = -1**