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.
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?
