Final Exam

Part B

Multiple Solutions

Claudette Tucker

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

xyz = 4

3x + 2y - z = 3.

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 tasks. (Not, required, but you might want to consider how to explore this one with a spread sheet as well as with graphing tools.)

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

What do we know about the equations above?

It is a system of equation with three variables.

A solution must satisfy each equation.

A solution must be an ordered triple such as ( a, b, c).

The values of a, b, and c can not equal zero nor negative numbers.

The solutions can be displayed in a three-dimensional coordinate.

What do we want to show?

I want to show or find many solutions for the system of equations. How can I do this? Since a, b, and c can not equal zero, one could use trial and error to produce an ordered triple, such choosing values for a and b, b and c, or a and c. For instance, if a = 1 and b = 2 are substituted into the first equation, it yields that c = 2. When you plug the values into the second, it generates that 5 = 3. This is impossible because 5 and 3 are not equivalent. Thus this strategy of trial and error is not the best way to obtain many solutions that will satisfy the system of equations.

Since substituting variables fail to work continuously. Let's look at the equations graphically.

The graph shows that plane, the linear equation, and the conics are produced by xyz = 4. The plane intersects three of the hyperbolic-shaped conics. The points of intersection are solutions to the system of equations. There are infinitely many solutions, but there are not solutions when x and y are less than zero and z is greater than zero.

We need to find solutions. In the begining, we used the trial and error strategy. Perhaps, there is a better way get the necessary information. Let's change the variable z to n in the original equations and use graphing calculator to see possible values for the system of equations.

Second option for finding solutions:

Let's rewrite one of the equations as a function of two variables. We can rewrite the xyz = 4 as z = 4/(xy) and substitute z into the linear equation, z = 3x + 2y -3. Then we will have ...

This graph reveals the solutions of the system of equations. Notice there are no solutions where x and y are both negative, as we had predict earlier using three dimensional coordinate system.

Would you like to explore another system of equations?