Solving Fractional Systems of Equations

Date: 06/08/98 at 19:16:56
From: Lisa Mannon
Subject: Algebra

I need help solving the following system of equation. The fractions 
have confused me:

   1/x + 3/y = 7
   4/x - 2/y = 1

Thanks.

Date: 06/08/98 at 20:20:44
From: Doctor Gary
Subject: Re: Algebra

You might want to "play" with the graph of this relation a bit.
It will come in handy later, when we're checking our answer.

When x is 1, 1/x is 1. Then 7 - 1 is 6, so 3/y must be 6. 
If 3/y is 6, then 3 is 6y, and y is 3/6 = 1/2.

When x is 1/2, y is  3/5
When x is 1/4, y is  1
When x is 1/8, y is -3

Because division by zero is undefined, there can be no points on 
either the x or y axis. So, since there are points in more than one 
quadrant, the graph is discontinuous.

When fractions are getting you down, try re-expressing the equation(s) 
by multiplying both sides by the denominator(s) - so long as we 
remember that neither x nor y can be zero. Then:

    1/x + 3/y = 7  

is equivalent to:

        y + 3x = 7xy
            3x = 7xy - y
            3x = y(7x-1)
   (3x)/(7x-1) = y

We can use the same re-expression technique on the second equation:

     4/x - 2/y = 1

Again, remembering thant neither x nor y can be equal to zero, we can 
re-express this as:

       4y - 2x = xy
           -2x = xy - 4y
           -2x = y(x-4)
    (-2x)(x-4) = y
    (2x)/(4-x) = y

Since two different expressions involving x are both equal to y, 
our solution(s) will be found when the expressions of x are equal 
in value:

   (3x)/(7x-1) = (2x)/(4-x)

Since multiplying both sides of this equation by both denominators 
will create a cubic equation, let's see if there's a shortcut.  
Remember that we can always multiply any expression by 1 without 
changing its value.

If we multiply (3x)/(7x-1) by 2/2 and multiply (2x)/(4-x) by 3/3, we 
can re-express our equation as:

   (6x)/(14x-2) = (6x)/(12-3x)

Since the numerators are equal (and, remember, nonzero), the fractions 
are equal only if the denominators are equal.

        14x - 2 = 12 - 3x
            17x = 14
              x = 14/17

Let's get those equations down here and see if x being equal to 14/17 
"works":

Using 1/x + 3/y = 7, when x is 14/17:

   1/x is equal to 17/14 
   3/y must be equal to 81/14, so y must be 14/27

Using 4/x -2/y = 1, when x is 14/17:

   4/x is equal to 68/14
   2/y must be equal to 54/14, so y must by 14/27

-Doctor Gary,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/