Find all ordered pairs $(x,y)$ that satisfy both $\frac{3x-4y}{xy} = -8$ and $\frac{2x+7y}{xy} = 43$

Question

The following is listed under the "multiple variable" category of my Algebra I homework.

Find all ordered pairs $(x,y)$ that satisfy both $\frac{3x-4y}{xy} = -8$ and $\frac{2x+7y}{xy} = 43$

I can't wrap my head around what to do. Thanks in advance for your help.

Answer 1

$$\frac{3x-4y}{xy} = \frac{3}{y} - \frac{4}{x} = -8$$

$$\frac{2x+7y}{xy} = \frac{2}{y}+\frac{7}{x} = 43$$

Let $u = \frac{1}{x}$ and $v = \frac{1}{y}$.

Then,

$3v - 4u = -8$ and $2v + 7u = 43$

Solve these linear equations and replace $x$ and $y$.