Inside Collection: Advanced Algebra II: Activities and Homework

Summary: This module provides sample which develop concepts related to using a matrices to solve linear equations.

I’m sure you remember our whole unit on solving linear equations…by graphing, by substitution, and by elimination. Well, now we’re going to find a new way of solving those equations…by using matrices!

Oh, come on…why do we need another way when we’ve already got three?

Glad you asked! There are two reasons. First, this new method can be done entirely on a calculator. Cool! We like calculators.

Yeah, I know. But here’s the even better reason. Suppose I gave you three equations with three unknowns, and asked you to do *that *on a calculator. Think you could do it? Um…it would take a while. How about four equations with four unknowns? Please don’t do that. With matrices and your calculator, all of these are just as easy as two. Wow! Do those really come up in real life? Yes, all the time. Actually, this is just about the only “real-life” application I can give you for matrices, although there are also a lot of other ones. But solving many simultaneous equations is incredibly useful. Do you have an example? Oh, look at the time! I have to explain how to do this method.

So, here we go. Let’s start with a problem from an earlier homework assignment. I gave you this matrix equation:

The first thing you had to do was to rewrite this as two equations with two unknowns. Do that now. (Don’t bother solving for

The point is that *one matrix equation* is the same, in this case, as *two simultaneous equations.* What we’re interested in doing is doing that process in *reverse*: I give you simultaneous equations, and you turn them into a matrix equation that represents the same thing. Let’s try a few.

Write a single matrix equation that represents the two equations:

Now, let’s look at three equations:

**a.**Write a single matrix equation that represents these three equations.**b.**Just to make sure it worked, multiply it out and see what three equations you end up with

OK, by now you are convinced that we can take simultaneous linear equations and rewrite them as a single matrix equation. In each case, the matrix equation looks like this:

where

Take the equation

Now, we have

Now, we’re multiplying

We’re done! We have now solved for the matrix

So, what good is all that again?

Oh, yeah…let’s go back to the beginning. Let’s say I gave you these two equations:

You showed in #2 how to rewrite this as one matrix equation

Solve those two equations for

*Did it work? *We find out the same way we always have—plug our

Check your answer to #7.

Now, solve the three simultaneous equations from #3 on your calculator, and check the answers.

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Comments:"This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"