__Solving Systems of Equations
(without graphs)__

So you've learned how to solve systems of equations *with*
graphs, but that takes an awful lot of work, paper, and time. Wouldn't it
be simpler if you could do it without all the hassle (gosh, I feel like I'm
selling a product on a low-budget commercial *_*)?

Well there is a way...two ways actually. The first one, my personal favorite, is on this page. The other is on the next page.

It is called the **substitution method.**

Say you have two equations...let's use ones from previous pages: x + y = 4, and 2x + y = 7.

First you work in one equation, solving for one variable in terms of another. Let's solve for y in terms of x in the first equation (you can do whichever variable in whichever equation you want, but you should experiment, and start recognizing which is the simplest one).

x + y = 4

__-x -x
__

Now this is where the substitution comes in. We substitute this value for y in the second equation.

2x + y = 7

2x + (4 - x) = 7

2x + 4 - x = 7

Combine like terms...

x + 4 = 7

-4 -4

**x = 3**

Now you've solved for one variable, x, but you still need y. So...substitute your value for x into either equation and solve that linear equation to find y.

x + y = 4

3 + y = 4

__-3 -3
__

Hooray, you've done it! The solution to this equation is x
= 3, y = 1. And you did it without a graph! These are called **
independent equations**. They have one common solution.

-----

Let's try another one. x + y = 4 and 2x + 2y = 8

x + y = 4

x = 4 - y

2x + 2y = 8

2(4 - y) + 2y = 8

8 - 2y + 2y = 8

8 = 8

Uh-oh, we got a constant equals a constant. It's a true
statement, but where did we go wrong? The answer is: nowhere. If you
remember graphing this (or if you do it on your own), you know that these
equations coincide. There are infinitely many solutions that will satisfy
the system of equations. A rule of thumb (love that phrase!) is that if
you get identity, that is an equation that is always true (constant = constant),
there are infinitely many solutions. The graphs will coincide. These
are called **dependent equations.** You can sometimes see these at a
glance (for example, each term in one equation is multiplied by two, and that is
the second equations, as in this case).

-----

How about one more... x + y = 4 and x + y = 5

x + y = 4

x = 4 - y

x + y = 5

4 - y + y = 5

4 = 5

What??? Now this *has* to be wrong, you think.
And you are right, it is wrong. But we haven't *done* anything wrong.
If you get a false statement (a constant = a different constant), the equations
have no common solutions. The lines will be parallel. These are
called **inconsistent equations.**

Click
here to go to the next page, the elimination method.

Click here
to go back to the index.