## Simultaneous Equations With One a Quadratic

When solving ordinary – linear – simultaneous equations we multiply the equations by constant factors to make the coefficient of some variable the same in magnitude, then add or subtract the equations to eliminate that variable.

For example, solve

(1)

(2)

(1)*2-(2)*3 eliminatesto give

Substitution of this value ofinto (1) to find a gives

If one of the equations is a quadratic we may not be easily able to rearrange the equations to easily eliminate one of the variables and solve the equations. But we can rearrange one of the equations – usually the linear one - to make eitherorthe subject.

Example:

(1)

(2)

Rearrange (1) to makethe subject:and substitute this into (2) to get

To solve the last equation we can either factorise or use the quadratic formula.

By factorising:or

IffromIffrom

By using the quadratic formula:

hence

or 3.

As before , substitute these values ofback into (1) to obtain