Question
Solve the system
xy=-64 \\\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{4}
Solve the system
xy=-64 \\\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{4}
Let
a=xy
b=x+y
Then,
\dfrac{1}{x}-\dfrac{1}{y} =\dfrac{y-x}{xy} =\dfrac{1}{4}
Square both sides of the equation
\dfrac{(x-y)^2}{x^2y^2} = \dfrac{1}{16}
\dfrac{(x+y)^2-4xy}{x^2y^2} = \dfrac{1}{16}
Plugging a,b gives
Substitute the value of a in (1) and solve for b
b = 0
Then we get the system
\begin{cases} xy=-64 \\ x+y = 0 \end{cases}
Solving for x,y yields
\begin{cases} x=8 \\ y = -8 \end{cases} \quad\text{or}\quad \begin{cases} x=-8 \\ y = 8 \end{cases}