Introducing Systems of Equations

Reference > Mathematics > Algebra > Systems of Equations

If you've been studying a first-year Algebra class, you've probably spent most of your year solving equations like these:

2x = 6
3z - 10 = 8
2(3w + 4) = 26
2z + 7 = z - 15
y² - 4y + 3 = 0
x² = 8x - 12

Each of these looks quite different, but they all have something in common: they each have only one variable. Another thing that they have in common is that they can all be solved (the quadratics have two solutions; all the others have just one solution). If an equation has more than one variable, however, it's very unlikely that you would be able to solve it. Take a look at this example, which is a linear equation:

x + y = 12

You could certainly solve for x (or y), but solving for x simply means getting x by itself, and everything else on the other side of the equation:

x = 12 - y.

Unfortunately, that doesn't tell you what x is; you'd need to know y in order to find out what x is!

As a matter of fact, there are an infinite number of solutions to this equation. Here are just a few (and you should be able to find more):

x = 6; y = 6
x = 5; y = 7
x = 1; y = 11
x =
1
2
; y =
23
2
x = -10; y = 22

An equation of this type is called indeterminate. That's one of those "nice" math words - it's nice because it sounds a lot like what it means. Indeterminate means cannot be determined. An equation is indeterminate if it's impossible to determine a solution.

A nice mathematical rule for you is that a single linear equation with two variables is indeterminate. In order to be solveable, you need to have two equations. If you have two variables and two equations, we refer to it as a system of equations in two unknowns. You'll see the word system throughout this unit, and any time you see that word, you should just think: more than one equation. I know, it's not as obvious as indeterminate. But you'll get used to it.

So here's a general rule: if you have more than one variable, you need more than one equation in order to solve. As a matter of fact, the rule is even more specific. If you have two variables, you need two equations, if you have three variables, you need three equations, and so forth. Later you'll see some examples where a system of two equations in two unknowns is still indeterminate, which is why this is a "general" rule. We'll get to the specifics of that later.

Let me give you an example of a system of two equations in two unkowns. We'll start with the equation I gave above, and then add another equation to go along with it.

x + y = 12
x - y = 2

It's possible to figure out what x and y have to equal. Even if you've never learned how to solve a problem like this, you might be able to figure out the answer by trial and error. How would you do it? Well, we've already made a list of some of the possibilities that satisfy the first equation; why not test those possibilities in the second equation until we find one that works?

The first solution we came up with x = 6 and y = 6. If we plug those into the second equation we get 6 - 6 = 2, which is definitely not true. So let's try the next one (x = 5; y = 7). 5 - 7 = 2. Is that true? No! However, if we switched the order (7 - 5 = 2) we'd get a true equation. That means the x and y that make both equations true are: x = 7; y = 2.

In theory, you can use that method to solve any system of equations, but let's get realistic; in most cases you would not want to try it! For example, you wouldn't want to try to solve the system below by trial and error...would you?

31x + 27y = 2435
17x - 11y = 845

Even if you want to solve this by trial and error, I sure don't!

Examples like this make it obvious that it would be nice to have algebraic methods that don't rely on trial-and-error for solving two equations with two variables. In this unit, we'll spend most of our time exploring just that question. We'll also talk about indeterminate and inconsistent systems. After we've worked on all that for awhile, we'll look at how to solve systems of equations that have fractions, and also systems of three equations (with three unknowns). Along the way, we'll also tackle some word problems involving multiple variables.