gj3ddemo.html

3D Demo of Row Operations

In the section on elementary row operations, graphs were drawn at each step to show what the row operations did to the graphs of the lines represented in the augmented matrices. We noted that while the lines changed, their point of intersection did not. In this section we will demonstrate the same thing for linear systems with three variables.

We will work with the following system.

( i ) 2x + 4y + 6z = 14

( ii ) -3x + y + 2z = -3

( iii ) 4x - 2y + z = 7

Each equation has a plane as a graph. When there is a unique solution, the solution is the one point that is on all three planes. As an example imagine that two of the planes are adjacent walls in a room and the third plane is the floor of the room. The intersection is the unique point that is the corner of the room. In most examples the planes that we start with are not at right angles like the walls and floor in a room so it is more difficult to see the intersection point on the graph. Eventually, as we apply the row operations to the example above, we will get three planes meeting in a point just as adjacent walls and the floor do.

Here is the augmented matrix for this system.

This is the graph of the three planes in the system. Can you tell what the solution is yet?

We now start the Gauss Jordan method to solve this system.

Multiply row 1 by 1/2

We next graph the system of equations that corresponds to the last augmented matrix. Mutiplying an equation by a nonzero number does not change the graph as the next picture indicates.