Answer
$(-4,-6)$ and $(4,6)$
Work Step by Step
Step 1. Rewrite the equations.
$y=\frac{24}{x}$
$2x^2-y^2+4=0$
Step 2. Substitute the first equation to the second one and solve for $x$.
$2x^2-(\frac{24}{x})^2+4=0$
$2x^2-\frac{576}{x^2}+4=0$ Multiply by $x^2$
$2x^4+4x^2-576=0$ Factorize
$(2x^2-32)(x^2+18)=0$ Note that $x^2+18>0$
$2x^2-32=0$
$x^2=16$
$x=\pm 4$
Step 3. Solve for $y$.
For $x=-4$, $y=\frac{24}{-4}=-6$
For $x=4$, $y=\frac{24}{4}=6$
Step 4. Conclude the solutions.
The solutions are $(-4,-6)$ and $(4,6)$.
