Use the following applet to explore systems of equations.  The applet shows the graph of a boy and girl moving between a house and a tree.  Watch as the graph representing their distance from the house versus time is drawn on the screen.  Look for the intersection of the two lines that represent the distance from the house for the children.

1. Print out the investigation.
2. Open the applet. (opens in a new window)
3. Run through a few exercises remembering that the solution to a system of equations is the point of intersection for the two lines.

You should have noticed that in the last two scenarios you encountered the two special cases. When the boy and girl started at the same place and moved at the same speed, they were together for the entire time. When you talk about the intersection of the graphs of their paths, you need to talk about the entire path—it was exactly the same. When both lines are the same line, there are infinitely many points of intersection.

When the boy and girl moved at the same speed but start in different locations, their paths will never meet. Whoever was ahead at the start will remain ahead throughout the entire trip. Because there is no intersection, there is no solution to the system.

Let's now look at some other examples.

Watch the following two videos to get a refresher on what you learned in Algebra 1 about solving equations by graphing:

Now, watch the problem clips for the first few examples of this lesson.

Reality is that most systems do not have points of intersection that fall very nicely on  graph paper, giving us integral solutions. In those cases, we either need to use a different method or we need the assistance of the intersection feature of a graphing calculator. Let's look at our graphing calculator now.

Be sure to follow all of the calculator steps as you work through the example on the next page.