Answer
$y=-5+i\sqrt{15}$, $x=5+i\sqrt{15}$
$y=-5-i\sqrt{15}$, $x=5-i\sqrt{15}$
Work Step by Step
Working from the first equation, $x-y=10$, we make $x$ the subject of the formula, $x=10+y$
We then sub this modified version of the first equation into the second equation , $xy=-40$ to get $(10+y)y=-40$.
Expanding this equation and moving all thew terms to the left hand side, we get: $y^{2}+10y+40=0$
The method used in the introduction then uses the quadratic method to solve for the equation. Therefore, we get:
$y=\frac{-10\pm\sqrt{10^{2}-4(1)(40)}}{2(1)}\\=-5\pm\frac{\sqrt{100-160}}{2}\\=-5\pm\frac{\sqrt{-60}}{2}\\=-5\pm\frac{\sqrt{(60)(-1)}}{2}\\=-5\pm\frac{2\sqrt{15}\sqrt{-1}}{2}\\=-5\pm i\sqrt{15}$
From this, we can deduce the corresponding values of $x$.
When $y=-5+i\sqrt{15}$, $x=10+(-5+i\sqrt{15})\\=5+i\sqrt{15}$
when $y=-5-i\sqrt{15}$, $x=10+(-5-i\sqrt{15})\\=5-i\sqrt{15}$
