# Lines

When looking at the paths across a plane or landscape, it may be helpful to calculate where they cross. This can be done visually obviously, as seen below:

However, we also have techniques to find these points algebraically. Although they may be in different formats, the equations used for each technique below are the same as those seen above:

**Equal Values Method:**

If two equations are equal to the same term, then we can set them equal to each other using the Equal Values Method as seen below:**Substitution:**

If, instead, one equation has an isolated variable, we can replace this variable in the other equation:

**Elimination:**

Finally, if both equations are structured in the same format, we can add/subtract them to eliminate variables:

# Parabolas

Obviously, not all curves are lines, so let’s look at the following visually first:

To solve algebraically, we will set up using the Equal Values Method then explore the ways to solve for the points of intersection. Once again, each method uses the same equations seen in the graph above.

**Factoring**

Sometimes a quadratic can be broken down into its factors. If we end up with a quadratic equal to zero, then factoring allows us to find the two possible x values that make the equation true:

**Completing the Square**

This uses factoring, but rather than two potentially distinct factors, we adjust the quadratic so that factoring produces two identical factors:

**Quadratic Formula**

Finally, there is a formula that works for any quadratic. For one of the research tasks, you will actually derive this, but it is as follows:

Check out the following research tasks that arise from solving for POI: