Multiple Solutions

Final Project Part B

By Gloria L. Jones

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I).  In this write-up we will attempt to find as many solutions as possible for X, Y, and Z that satisfy both equations:

XYZ = 4

3X + 2Y – Z = 3

Given the two equations, letıs first observe the possibilities for the variables.

-We know that (0) is not a possibility for any of the variables in equation XYZ= 4.

-We also we know that a system of equation that contains three variables and two equations provides infinitely many solutions and therefore has no unique solution. _____________________________________________________________

Using the Graphics Calculator application, we are able to see graphically the intersections of the two equations when we substitute Z and solve for X and Y:

Many solutions can be obtained from this graphic application by assigning various values to X to derive the values of Y and Z.  Notice on the graph above that the conic curve does not appear in quadrant III, which might indicate that there is no solution for the system when all the variables are negative or when there is only one negative variable.  Or that X and Y can not be negative at the same time.  Letıs look at a chart of possible rounded values:

By assigning a value for X in the equation XYZ = 4, we are given the value for Y and by using Microsoft Excel, we are able to obtain the value for Z.

 X Y Z 1 1.40 2.86 2 0.50 4.00 3 0.20 6.67 4 0.10 10.00 5 0.06 13.33 6 0.04 16.67 8 0.02 25.00 -1 1.00 -4.00 -2 0.23 -8.70 -3 0.11 -12.12 -4 0.07 -14.93 -5 0.04 -18.18 -6 0.03 -22.22 -7 0.02 -28.57 -8 0.01 -50.00

This list of possible solutions to the system of equation could perhaps go on and on.  The asymptotes on the coordinate plane above suggest that the curves are limited between 10-11 depending on the axes???????

Another observation from a 3 dimensional view would look like this:

Our equation 3X + 2Y – Z= 3 is the plane in the graph and equation XYZ =4 is a conic surface.   Where the two surfaces intersect is where there are solutions.  Notice once again that in quadrant III there is no intersection.

Now for a view of the equation XYZ = 4 where Z is substituted from equation 3X+ 2Y– Z =3, we get XY(3X +2Y -3) =4 in 3 dimension:

Once again there is no intersection between the conic surface and the plane in quadrant III.  In fact, it appears an intersection only takes place in quadrant IV.  Which might indicate that when X is positive, Y and Z are positive and when X is negative, Y is positive and Z is negative.

II).  Take a moment and try to explore the following system of equations for solutions to XYZ using Graphics Calculator and Microsoft Excel:

2X² +XY +Z= 15

XYZ= 20

Good Luck!!!