Multiple Solutions

#### by

#### Rachael Brown

The problem we were given is to find all possible
solutions to the system:

You will notice that this is a system of two
equations and three unknowns. That makes this a little challenging!

To start, let's think about the signs of our
variables. If we look at the first equation it appears that they
all have to be positive or two of them can be negative. How does
this impact the second equation? It doesn't seem to have much
of an impact. Maybe we should graph them and see...

Here is the graph of xyz = 4:

The arrow pointing up is the positive y axis.
The arrow to the right is the positive x axis. The z axis is positive
as it moves towards us (the green line in the bottom left corner).
From this graph we can see that we get four distinct pieces. The
one in the center of the picture is all three positive. The bottom
right graph is x positive and y and z negative. The bottom left
graph is x and y negative and z positive. The graph in the back
is y positive and z and x negative. So we were right about our
values.

Now let's look at the second equation: 3x +
2y - z = 3:

This graph is a plane. Its not as interesting
as the last one! From the graph we see that x ,y and z are positive
and negative. If I look at the graph from a different perspective
that is more clear:

Now our job is to find where these to graphs
intersect. Let's graph them both together:

It looks like our solution will never be with
x and z negative! However the other three sections of the red
graph seem to be part of the solution. In order to get a better
look at the solution, let's solve for z and substitute it in the
second equation:

These graphs represent the solutions to the
system. There are infinitely many solutions and the solutions
need to lie on one of these three pieces of the graph. I spent
some time trying to figure out the equations or something close
to the equations for these pieces. They look like parts of hyperbolas
but the guessing and checking seemed to be fruitless.

Here is another interesting system of equations:

Click here
to open up a Graphing Calculator file with this system in it.
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