Solving Equations

From: Anonymous
Date: Fri, 28 Jun 1996 9:22:58 -0400 (EDT) 
Subject: Find k

I don't understand this problem. Can you help me?

Find k, if 7/x+y = k/x+2 = 11/2-y

Date: Fri, 28 Jun 1996 13:33:35 -0400 (EDT)
From: Dr. Anthony
Subject: Re: Find k

If we take first and third of these 7/(x+y) = 11/(2-y) 

then 14-7y = 11x + 11y or 11x +18y = 14

Now take the 2nd and third expressions to k/(x+2) = 11/(2-y) 

then 2k - ky = 11x + 22 or 11x + ky = 2k - 22 

So we have two equations 11x + ky = 2k-22 
and 11x +18y = 14

Subtract	(k-18)y = 2k - 36
(k-18)y = 2(k-18)

So either y=2 or k = 18. But y=2 puts a zero in the denominator of 
the last expression, and would require x = -2 and k indeterminate, 
so we require instead k = 18.

Check that this gives first and second expression consistent with 
first and third.

We showed that the first and third led to 11x + 18y = 14, so now 
try the first and second with k=18

7x + 14 = 18x + 18y

14 = 11x + 18y so the result is consistent.

-Doctor Anthony, The Math Forum