10.14 Solving 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:

$\left[\begin{array}{cc}9& -6\\ 18& 9\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}9\\ -3\end{array}\right]$

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.

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:

$\mathrm{AX}=B$

where $A$ is a big square matrix, and $X$ and $B$ are column matrices. $X$ is the matrix that we want to solve for—that is, it has all our variables in it, so if we find what $X$ is, we find what our variables are. (For instance, in that last example, $X$ was $\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ .) So how do you solve something like this for $X$ ? Time for some matrix algebra! We can’t divide both sides by $A$ , because we have not defined matrix division. But we can do the next best thing.

We’re done! We have now solved for the matrix $X$ .

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:

$3x+y=\mathrm{–2}$

$6x–2y=12$

You showed in #2 how to rewrite this as one matrix equation $\mathrm{AX}=B$ . And you just found in #6 how to solve such an equation for $X$ . So go ahead and plug $A$ and $B$ into your calculator, and then use the formula to ask your calculator directly for the answer!