# Homework Help: Is there any simpler methods to solve this?

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1. Jan 19, 2015

### sooyong94

1. The problem statement, all variables and given/known data
I was asked to solve this equation:
${(x-\frac{1}{y})}^{2} -(y-\frac{1}{x})(x-\frac{1}{y})=9x$
$x-y=1$

2. Relevant equations
Simultaneous equations, factor theorem and quadratic formula

3. The attempt at a solution
I know I could have solved for x in the second equation, and substitute it into the first equation. However, the algebra becomes incredibly messy and the resulting equation becomes hard to solve. I see there is a very specific pattern in the first equation, though I can't really figure it out.

2. Jan 19, 2015

### ehild

Bring the terms in x-1/y and y-1/x to common denominator. Then factorize the left hand side.

3. Jan 20, 2015

### sooyong94

Now I have
$(\frac{xy-1}{y})(\frac{x^2 y-x-xy^2 +y}{xy})=9x$

Then how should I proceed?

Update: I finally managed to solve it. Thanks!

Last edited: Jan 20, 2015
4. Jan 20, 2015

### ehild

How did you proceed? I thought of
$\frac{(xy-1)^2}{y^2} -\frac{xy-1}{x} \frac{xy-1}{y} = \frac{(xy-1)^2}{y} \left(\frac{1}{y}-\frac{1}{x}\right)$, using x-y=1 substituting xy=u and solving for u first.

5. Jan 20, 2015

### sooyong94

I factored the equation, and substituted x-y for 1. Then I got (1-1/xy)(1-1/xy)=9, and (1-1/xy)^2 =9, and find an equation in terms of xy. Then I use the equation x-y=1 again to solve for x and y.

6. Jan 20, 2015

### ehild

Nice, it was what I had in my mind :)