Multiple Solutions

Final Assignment completed by Johnie Forsythe

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

What observations can you make about your results? Again, we discussed some approaches to this one in class. Your task is to prepare a write-up that explores this task. (Not required, but you might want to consider how to explore this with a spreadsheet as well as with graphing tools.).

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

I will begin this exploration by changing the variables:

A = X

B = Y

C = Z

This does not change the equation in any significant way.

I will graph each equation using Graphing Calculator.

   

Now, let's graph this system of equations on the same coordinate plane.

 

The equation xyz=4 produces a four hyperbolic shaped conics.

The equation 3x+2y-z=3 produces a plane.

The solutions to the system of equations are the points of intersection.

 

Click here to view a Graphing Calculator file of the points of intersection when z varies.

Looking at the graph of the system of equations, we see that the plane intersects only 3 out of the 4 hyperbolic shaped conics. What may be going on??

Let's investigate this question further.........

What happens when we rewrite one of the equations as a function of two variables, and then substitute the dependent variable into the second equation.

The graph of the equation looks like the following:

 

This graph represents the infinite solutions for the system of equations. We can see that there is no solution were both x and y values are less than zero (or negative).

Recall the 3D graphical representation: this explains why the plane intersects only 3 out of the 4 hyperbolic shaped conics.

 

Here is another system of equations to explore:

Click here to see a 3D graphical representation.

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