Algebra mathematics

Question

First, create 3 equations of the form  , where a, b, c, and d are constants (integers between – 5 and 5). For example,  . Perform row operations on your system to obtain a row-echelon form and the solution.

Go to the 3D calculator website GeoGebra and enter each of the equations.

After you have completed this first task, choose one of the following to complete your discussion post.

1. Reflect on what the graphs are suggesting for one equation, two equations, and three equations, and describe your observations. Think about the equation as a function  of x and y, for example,  in the example above. Geogebra automatically interprets this way, that is, like  , it isolates z in the equation.

2. What did the graphs show when you entered the second equation?

3. Give a simple description of the system

x=0

y=0

z=0

x = 0 can be seen as the constant function  . Of course, you can use GeoGebra to “observe” the system.

4. Give an example with 2 equations as simple as possible with 3 variables (at least 1 being non-linear; keeping z to the one power on both equations) and describe the potential of GeoGebra to study nonlinear systems.

Answer 1

I chose

x + 2y - z = -1

4x -3y + 3z = 5

-2x + y - 2z = -4

Since you only need to answer one of them...

Answering (1) The graphs below start off with the first equation, which when plotted on the x-y plane, appears to be a line. You can visualize z much more clearly as the 3D extension of this line in both + and - z directions, creating a plane. When you add more systems of equations, you get 3D planes intersecting. The points where the planes intersect is the solution between the systems of equations. If two equations are identical (have coefficients that are multiples of each other) then they will have infinitely many solutions.

Although they intersect in many places due to being planes, there are still one x,y, and z value in which they meet when you substitute those values into the system. You will only get planes with linear equations, you will not get quadratics unless you have an exponential of 2 or greater on your variables.