Simultaneous equations usually refers to equations of the form

(1)

(2)

We solve these be equating the coefficients of or and then eliminating that term. For example, in this case we can multiply (1) by 3 to get

then subtract (2)

and from (1),

Our problem here is to solve equations such as

(1)

(2)

The general approach is to rearrange the linear equation (2) to make the subject say, then substitute the rearranged equation into the quadratic to find a quadratic equation in which we solve to find then substitute back into the linear equation to find For the above example:

Replace the in with to get

We expand the brackets and simplify this expression:

We can factorise and solve the last expression.

If we use (2) to find and if we use (1) to find

Example:

(1)

(2)

Make the subject of (2), then substitute the rearranged equation into (1) to find a quadratic equation in which we solve to find then substitute this into the (2) to find

Replace the in with to get

We expand the brackets and simplify this expression:

If we use (2) to find and if we use (1) to find