Multiple Solutions

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?

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

There are several observations we can make about the solutions of this equations.  First, any solutions will be an ordered triple of the form (a, b, c).  Second, since ABC = 4, A, B, and C cannot all be zero nor all negative numbers.   Third, A and B can not both be negative, because 3A + 2B - C = 3.  A and C can be negative, B and C can be negative, but both A and B cannot. 

The following figure shows the graph of this system of equations.


Notice that the plane (xyz = 4) intersects three of the conics formed by 3x + 2y - z = 3.  There are no solutions in the case where x and y are both < 0.

To see the different solutions for this system of equations, change the x variable to n and watch as n varies between 0 and 5.  Click here for a Graphing Calculator Document you can use to explore the different values of these variables. 

We can also reduce this system of equations in three variables to a system of equations in two variables, and explore the solutions of this system on a two dimentional graph.  First solve xyz = 4 for z to obtain z = 4/xy.  Then substitute this value for z into the second equation and simplify.  The resulting equation will be a two variable equation.                   

Again, notice that there are no solutions for this system where x < 0 and y < 0.

Now explore the following set of equations on your own.  What can you discover about the solutions?

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