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### Equations: the "Elimination" Method

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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
```

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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
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
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.

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/
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Associated Topics:
High School Basic Algebra
Middle School Algebra

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