# Simultaneous linear and quadratic equations

**Simultaneous linear and quadratic equations**

In situations where one of the equations given is a quadratic, and the other is linear, the following steps are taken:

*y* = *x²* + 1 (equation a)

*y* = *x* + 3 (equation b)

Equation a and b can be joined together as they both equal y.

*x²* + 1 = *x* + 3

All terms are then collected in the following format:

*ax² + bx + c = 0 *

*x²* – *x –* 2 = 0* (*factorise)

*(x + *1)(*x –* 2) = 0

*x = –*1 or *x = 2*

To work out the values of *y* using the solutions deduced from *x* it is substituted as follows:

*y* = *x* + 3

when *x = –*1, *y* = 2

when *x = 2*, *y* = 5

**Using graphs to solve equations**

The equations above can also be visualised graphically in the below graph to show the solutions.

The solution is found by finding the points of intersection of the two formulas. However, this is very difficult to draw accurately, therefore it is a good way of demonstrating a solution but when solving the method the algebraic method should be used.