Up next are those pesky systems of three equations with three variables. We are going to work on solving them by linear combination. I've decided not to cover matrices in this class because of the time it would take to teach it to a level that had any meaning to the students. I also don't think that they have much application up through Calculus (at least, not enough to justify taking time away from some of the other critical topics).
The class will start with a review of 2x2 systems, including systems with no solution or an infinite number of solutions. They will solve these algebraically and graphically; I want to see if they can draw their own conclusions first about what algebraic results like "0 = 3" or "0 = 0" imply graphically.
After this, I'll do direct instruction on these ideas, followed by a demonstration of how to solve a 3x3 system by linear combination. The steps I'll give them:
- Decide which variable is easiest to eliminate.
- Eliminate that variable from one pair of equations.
- Eliminate the same variable from a different pair of equations.
- Write the resulting equations as a 2x2 system and solve it.
- If the 2x2 has 0 or infinitely many solutions, the same is true about the 3x3 and you can stop there.
- See if you can use division to simplify the new equations.
- Plug in the solutions to one of the original equations and solve for the third variable.
- Check by plugging in the solution to the other two original equations.
After that, it's just a matter of practice grinding through a few systems. Homework will be more of the same, with a +10 point mega bonus extra credit problem (solving a 4x4 system).
In the following class, we will use the Grapher application in our Mac lab to see the planes that are produced by an equation with x, y, and z, and to understand what an ordered triple means. We will also be able to see how planes can intersect and so forth, and maybe play around with some more 3-D graphing. Grapher is a great application - check it out if you can.