**Part B: Multiple Solutions**

**1. Find as many solutions as possible for A, B, and C that satisfy
both equations:**

**ABC = 4**

**3A + 2B - C = 3.**

**What observations can you make about your results? Again, we discussed
some approaches to this one in class. Your task is to prepare a write-up that
explores this task. (Not required, but you might want to consider how to explore
this one with a spreadsheet as well as with graphing tools.).**

** 2. Create another set of equations that also yield a useful exploration.**

I began this problem by graphing the two equations using Graphing Calculator. The graph is shown from two angles:

By solving for **z** in the equation for the plane,
and substituting this value in for **z** in the first equation,
we get a graph that shows us the possible values of **x** and **y**.
By picking **(x,y)**-coordinate pair on this graph, we can plug
it in to get **z**. Giving us a solution to the equation. The solution
set to this system of equations results from finding **z** for
all such **(x,y)**-coordinate pairs. This graph gives us an idea
of what **x** and **y** values will satisfy this system
of equations:

When we expand the above equation, we notice that it is a quadratic
in **x**:

, so

Using the quadratic formula (or Maple ;) ), we find that:

We can now parametrize our solution set, where **t**
ranges over all real numbers by:

Another exploration could be to solve the following system of equations:

The graph of these two equations follows. With the addition of one more sphere, these equations could be useful explaining how triangulation and GPS work.