Answer
$(2, 3)$ and $(-2, 3)$
Work Step by Step
We can solve the first equation in terms of $x$ to isolate the $y$ variable:
$x^{2} - y = 1$
Subtract $x^{2}$ from both sides:
$-y = 1 - x^{2}$
Divide both sides by $-1$:
$y = x^{2} - 1$
Now that we have $y$ by itself, we can use it to substitute into the second equation:
$2x^{2} + 3(x^{2} - 1) = 17$
Distribute the terms:
$2x^{2} + 3x^{2} - 3 = 17$
Combine like terms:
$5x^{2} = 20$
Divide both sides by $5$.
$x^{2} = 4$
Take the square root of both sides:
$x = 2$ or $x = -2$
Now that we have the values for $x$, we can plug them into one of the equations to solve for $y$:
$2^{2} - y = 1$ or $(-2)^{2} - y = 1$
$4 - y = 1$ or $4 - y = 1$
$-y = -3$
Divide by $-1$:
$y = 3$
The points at which these two equations intersect are $(2, 3)$ and $(-2, 3)$.
