 # Multiple Solutions

A Problem From Project InterMath

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

ABC = 4            and          3A + 2B - C = 3.

What observations can you make about your results?

## Discussion:

That is a rather vague and open ended question about your results.

Note that the problem does not specify that A, B, and  C   should be integers.   Hence you should explore both possibilities.

For the moment, assume we want to find integer values.

Let    A = x, B = y,  and   c = z   so that we have equations    xyz = 4   and 3x + 2y -z = 3  that we can graph with some technology tool.  One such tool is Graphing Calculator 3.5 and an animation of the three dimensional figure.   The Blue surface has four distinct parts for xyz = 4.   The red surface is a plane cutting three of the four parts of the blue surface.   In the octant where all three of x, y, and z are positive, the intersection of the red plane and the blue corner piece seems to be somewhat like a parabola.   The intersection with the two pieces where  z is negative is harder to describe.

You might want to explore graphing the figures with your own software.

## A strategy Hint

Fix the value of  C  (or   z)   and explore the problem for locating the other two values.

Repeat for several values of  c. This strategy could use

Graphing

Algebra

Numerical exploration

For example,  if   z = 1,  our two-dimensional graph of   xy = 4   and 3x + 2y - 1 = 3 is at the right.   The red curve and the blue curve do not intersect.   This tells us there is no solution when  C =  1.

Make additional graphs for  C = 2, 3, 4, etc.  and   C= -1, -2, - 3, etc.

Alternatively, we could make spread sheet exporation for the different values of   C

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