Answer
$(-25, 5)$ and $(-25, -5)$
Work Step by Step
Using the substitution method, we can isolate the variable $x$ from the first equation:
$x + y^{2} = 0$
$x = -y^{2}$
We use this expression to substitute into the second equation to solve for $y$:
$2(-y^{2}) + 5y^{2} = 75$
$-2y^{2} + 5y^{2} = 75$
Combine like terms:
$3y^{2} = 75$
Divide both sides by $3$:
$y^{2} = 25$
Take the square root of both sides:
$y = 5$ or $y = -5$
Now that we have the values for $y$, we can substitute these values into the first equation to solve for $x$:
$x + y^{2} = 0$
$x + 5^{2} = 0$ or $x + (-5)^{2} = 0$
$x + 25 = 0$ or $x + 25 = 0$
Solve for $x$:
$x = -25$
The values for $x$ for both points are $-25$.
Therefore, the points where the two equations intersect are $(-25, 5)$ and
$(-25, -5)$.
