Discrete Mathematics: Solving for x in a System of Equations

[tennesseewiz]

Homework Statement

Suppose a, b, and c are integers and x, y, and z are nonzero real numbers that satisfy the following equations:

xy/(x+y)=a
xz/(x+z)=b
yz/(y+z)=c

Is x rational? If so, express it as a ratio of two integers.

Homework Equations

I substituted a lot of equations and I know I need x to equal something. What I got was:

x= abcx/(acx+bcx-abx-abc)

However I don't know how to solve for x (my algebra skills suck... don't ask me how I made it to discrete mathematics...)

The Attempt at a Solution

I know that if I solve for x, I can basically work out the problem on my own.

 

[bel]

The equation with "c" in it relates "y" and "z", perhaps if you multiply the equation with "a" and the equation with "b" together, you could find some place to substitute some expression with "c" in for some combination of "y" and "z".

 

[EnumaElish]

Take the reciprocal of both sides. The expression you will have on the right can be separated into four terms. The x will cancel out in the first 3 terms. In the 4th term abc will cancel out, leaving -1/x. Now add +1/x to both sides, you will have 2/x on the left. (On the right, -1/x will cancel out.) Then divide both sides by 2. Everything on the right are integers, which means 1/x is rational. If 1/x is rational, so is x.

 

[tennesseewiz]

Oh my gosh, thanks! It's sad that I got the answer now after I had to turn in my homework, but at least I understand it now! :D