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

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.

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/