This week we learned how to graph systems.

Systems are two equations and the solutions to them are where the lines intersect.

There are three different types of systems we graphed:

Linear, Linear,

Linear, Quadratic

and Quadratic, Quadratic

Linear, Linear:

In linear-linear systems there can only be one or zero solutions, never more.

Since it is linear your equations will be in slope intercept form (y=mx+b).

Here is an example:

Now we just have to put these into a graph and find where they intersect:

Now we can see where the lines intersect and determine the solution is (-1,7).

If the lines do not intersect there is no solution.

Linear, Quadratic:

A linear-quadratic system can have 0, 1 or 2 solutions.

Here is an example:

If you put this into a graph you get:

You can then determine the solutions are (-1,5) and (3, 13).

If the line only intersects at one spot, there is one solution, and if at none then 0 solutions.

Quadratic, Quadratic:

A quadratic-quadratic system can have 0,1,2 or infinite solutions.

Here is one with two solutions:

Here is what it looks like when graphed:

When you look closer you can see the solutions are (-1.4, 7) and (1.4, 7).

There are 1 solutions if they intersect a one point and 0 if they do not touch.

There are infinite solutions if the parabolas land directly on top of each other.