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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
       y = 4 - 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
y = 1

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.