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### Equations with Fractions: 7y/9+2 = 2y/3

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Date: 11/23/97 at 21:38:27
From: Jackie
Subject: Equations with fractions

7y/9 + 2 = 2y/3

First I moved the variables to one side and the numbers to the other,
but I always ended up with -18/13.
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```
Date: 11/25/97 at 14:37:31
From: Doctor Mark
Subject: Re: Equations with fractions

Hi Jackie,

Well, you have the right idea to move the variables to one side and
the numbers to the other, but there is often another thing that you
should do first, and that is to *clear fractions*.  What that means is
to get rid of the fractions by multiplying both sides of the equation
by the Least Common Multiple of the bottoms of all the fractions you
see. Here, the fractions are 7y/9 and 2y/3, so the bottoms are 9 and
3, and the Least Common Multiple of these (3 and 9) is just 9.  So
before you do anything else, you should multiply both sides of the
equation by 9.  When we do that, we get:

([9][7y])/9 + [9](2) = ([9][2y])/3

Canceling, we find that ([9][7y])/9 = 7y, and that ([9][2y])/3 =
[3](2y) = 6y, and so the equation becomes:

7y + 18 = 6y.

Do you see why this is called "clearing fractions?"

Now bring the 6y over to the left, and the 18 over to the right [that
is, subtract 6y from both sides, and subtract 18 from both sides] to
get:

7y - 6y = - 18.

Now collect terms to get y = - 18.  You can check that this does in
fact solve the equation (make sure you use parentheses when you
substitute -18!).

I think the reason that you got y = - 18/13 was that you subtracted
6y from the right side (good), but *added* 6y to the left side (not so
good).  Or, if you didn't clear fractions first, you subtracted 2y/3
from the right side, but added it to the left side, which is the same
kind of mistake.

I try to avoid mistakes like that by thinking of the equals sign as a
kind of funny mirror. You know that if you hold your right hand up to
a mirror, what you see in the mirror is a left hand, that is, the
mirror changes the "handedness" of a hand. You can think of the equals
sign as a mirror sort of like that, except that instead of changing
right into left, it changes the sign of terms which are brought from
one side to the other. So in the example above, when you try to solve

7y + 18 = 6y,

you bring the 6y "through the mirror," to the left side, where it
becomes -6y (remember that the 6y really has a secret "+" sign in
front of it; that's why it becomes -6y when you bring it through the
mirror). Then you bring the + 18 through the mirror to the right side,
where it becomes -18, and we have:

7y - 6y = -18, as before.

Hope this helps.  Good luck.

-Doctor Mark,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Equations
Middle School Fractions

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