Answer
The solutions are $(0, -1)$ and $(2, -3)$.
Work Step by Step
We will use substitution to solve this system of equations. We substitute one of the expressions for $y$, which would mean that we are going to set the two equations equal to one another to solve for $x$ first:
$-x^2 + x - 1 = -x - 1$
We want to move all terms to the left side of the equation.
$-x^2 + x - (-x) - 1 + 1 = 0$
Combine like terms:
$-x^2 + 2x = 0$
Divide both sides by $-1$:
$x^2 - 2x = 0$
Factor out any common terms:
$x(x - 2) = 0$
Use the Zero-Product Property by equating each factor to $0$, then solve each equation.
First factor:
$x = 0$
Second factor:
$x - 2 = 0$
$x = 2$
Now that we have the two possible values for $x$, we can plug them into one of the original equations to find the corresponding $y$ values. Let's use the second equation:
$y = -x - 1$
Substitute the solution $0$ for $x$:
$y = -0 - 1$
Subtract to solve:
$y = -1$
Let's solve for $y$ using the other solution, $x = 2$:
$y = -2 - 1$
Subtract to solve:
$y = -3$
The solutions are $(0, -1)$ and $(2, -3)$.
