Can Factoring Simplify Complex Algebraic Equations?

[sooyong94]

Homework Statement

I was asked to solve this equation:
${(x-\frac{1}{y})}^{2} -(y-\frac{1}{x})(x-\frac{1}{y})=9x$
$x-y=1$

Homework Equations

Simultaneous equations, factor theorem and quadratic formula

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.

 

[ehild]

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

 

[sooyong94]

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

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!

 

[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.

 

[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.

 

[ehild]

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