Equations: the "Elimination" Method

Date: 11/20/96 at 19:12:23
From: collin
Subject: Easier Ways to Solve Two Equations

Solve the system by using the substitution method    

      5x - 3y = -36
      3x -  y = -10  

This is what I did, but it seems so involved.  Is there an easier way?

                     y = -3x - 10

      5x - 3(-3x - 10) = -36
       5x + 9x + 30    = -36
                   14x = -66
                     x = -66/14
                     x = -33/7

So

     y = -3(-33/7) -10
     y = 29/7

Date: 04/06/97 at 14:25:52
From: Doctor Sydney
Subject: Re: Easier Ways to Solve Two Equations

Dear Collin, 

Hello!  Thanks for writing.  You did a good job solving the problem 
you wrote about... you have made just one small error, which I will 
point out below.  

Sometimes the substitution method CAN be involved, as you note.  
Sometimes solving two equations with two unknowns or variables can be 
done in a variety of ways.  I must say that the substitution method 
may not be the best in this particular case.  It can certainly be 
done, but it does seem kind of involved!  However, it is good to 
practice using the substitution method even on problems that seem 
messy, because it sharpens your skills at doing it!

In this case I would use the "elimination" method which you might have 
already learned about.  I'll explain the "elimination" method in 
detail below.  

First, though, let's look at your solution to the problem.  I have 
marked where you make the mistake.

>       ...
>       y = -3x - 10

********    OOPS!  y = 3x + 10   ********

BUT, ASSUMING THAT Y=-3X - 10, YOU DID A FINE JOB ON THE REST OF IT!

So, your work is fine except that you made a sign error.  Go through 
the problem again using the equation y = 3x + 10 to find the right 
values for x and y.  

In general, if you want to check your answers to these kinds of 
problems, plug the values you get for both x and y into both equations 
to make sure they work. You must make sure that your value works for 
BOTH equations. If your answer does not work for either of your 
equations, it is not the right answer. 

For instance, look at the values you got for x and y above:  you got 
that x = -33/7 and y = 29/7.  Now plug those values into your second 
equation 3x - y = -10.  You get 3(-33/7) - (29/7) = -10. But 3(-33/
7) - (29/7) = -128/7 = 18.29, and 18.29 is not equal to 10, so these 
values x = -33/7 and y = 29/7 must not be right.    

Now, as promised, I will explain how to use a method other than 
substitution to solve this system of equations.  This is called the 
"elimination" method by some people. 

First, let's write our system of equations down again:

    5x -3y = -36
    3x -y = -10
            
Now, our first step is to choose one of the variables (either x or y), 
and then once we have chosen a variable (say we choose x), we multiply 
each equation by something so that the coefficients of that variable 
are the same in both equations.  It doesn't matter whether you choose 
x or y in this first step, although sometimes choosing one over the 
other will make it easier for you later.  After practicing this 
method, you will get a feel for which variable to choose.  For now, 
let's choose the variable x.  Now we want to multiply each of the two 
equations by a number such that the coefficient of x is the same in 
each equation.  One way to do this is to multiply the first equation 
by 3 and the second by 5.  

times 3:   15x - 9y = -108
times 5:   15x - 5y = -50

Now you subtract the second equation from the first.  Doing this gives 
you a new equation with just one variable!  Now you can see why we 
wanted the coefficients of the x variable to be the same... when we 
subtract equations from one another, the x term goes away.  

We are left with the following equation:  

Subtract:       -4y = -58

Solve for y to get:

                  y = -58 / -4 = (14 and 1/2)

Now that you have a y value, you can substitute this value in for 
either of your two original equations to get another equation that has 
only an x variable.  Solve this equation to figure out what x is.

Again, you should check your answer by making sure your x and y values 
work for BOTH equations.  

Solving simultaneous equations is nice in this way, because you check 
your answer more easily when doing these problems than you can in 
dealing with other kinds of problems.  

I hope that this helps.  If you need more help, please do write back! 

-Doctors D. and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/