Answer
$(-3, 0)$ and $(2, 5)$
Work Step by Step
We use the substitution method to solve this system of equations. We can use the second equation to isolate one variable, $x$:
$x - y + 3 = 0$
$x - y = -3$
$x = y - 3$
Now we have an expression to substitute for $x$. We can use this expression to substitute for $x$ in the first equation to get the values for $y$:
$(y - 3)^{2} + y = 9$
Use the foil method to distribute the terms:
$(y - 3)(y - 3) + y = 9$
$(y^{2} - 3y - 3y + 9) + y = 9$
$y^{2} - 6y + 9 + y = 9$
Combine like terms:
$y^{2} - 5y + 9 = 9$
Subtract $9$ from both sides:
$y^{2} - 5y = 0$
Factor out a $y$ from the left-hand side:
$y(y - 5) = 0$
Set each factor equal to $0$ and solve for $y$:
$y = 0$ or $y = 5$
Now we use the values we just got for $y$ to substitute into the second equation to get the values for $x$:
$x - (0) + 3 = 0$ or $x - (5) + 3 = 0$
Combine like terms:
$x + 3 = 0$ or $x - 2 = 0$
Solve for $x$:
$x = -3$ or $x = 2$
The points where the two equations intersect are $(-3, 0)$ and $(2, 5)$.
