Answer
$(2, -2)$ and $(-2, 2)$
Work Step by Step
We can use the substitution method. Let's use the second equation to isolate one of the variables, $y$:
$x + y = 0$
$y = -x$
Use this expression for $y$ to substitute into the first equation:
$x^{2} + (-x)^{2} = 8$
$x^{2} + x^{2} = 8$
Combine like terms:
$2x^{2} = 8$
Divide each side by $2$:
$x^{2} = 4$
Take the square root of both sides:
$x = 2$ or $x = -2$
Now that we have the values for $x$, we can use these values to substitute into the second equation:
$(2) + y = 0$ or $(-2) + y = 0$
$y = -2$ or $y = 2$
The points where the two equations intersect are $(2, -2)$ and $(-2, 2)$.
