Linear-Quadratic Systems

In the relations and functions unit you looked at solving systems of linear equations by graphing, elimination and substitution. Now we will look at systems of equations where we mix linear equations with quadratic equations, or systems made up solely of quadratic equations. The only solving method we will look at here is substitution, but your teacher may make you solve these systems using graphing as well.

Example

Solve the following system using substitution:

Solution

As previously stated, we will solve this system using the method of substitution. Both of the equations above have the y variable isolated. If the right hand side of both equations is equal to y, then logically they must be equal to one another. Therefore we can set up the following equation:

This is a quadratic equation, which you learned to solve in the equations and inequalities unit. The solution is shown below:

Remember that this is a system of equations and has 2 variables - don't forget the y! Using the second equation from the system and each value for x that we found we find the following values for y:

We now have two solutions: (-2,0) and (1, -3). Note that these solutions are where the graphs of the two functions in the system intersect. A graph is shown below to illustrate this point. The points of intersection are marked.

Now you can work on the questions in your booklet, or you can head back to the Quadratic Functions main page.