## Simultaneous Equations With One a Quadratic

Simultaneous equations usually refers to equations of the form

(1)

(2)

We solve these be equating the coefficients ofor 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 inwhich we solve to findthen substitute back into the linear equation to findFor the above example:

Replace thein with to get

We expand the brackets and simplify this expression:

We can factorise and solve the last expression.

Ifwe use (2) to findand ifwe use (1) to find

Example:

(1)

(2)

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

Replace thein with to get

We expand the brackets and simplify this expression:

If we use (2) to findand ifwe use (1) to find