Answer
The solutions are $(0, 1)$ and $(-3, -2)$.
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:
$x^2 + 4x + 1 = x + 1$
We want to move all terms to the left side of the equation.
$x^2 + 4x - x + 1 - 1 = 0$
Combine like terms:
$x^2 + 3x = 0$
Factor out common terms:
$x(x + 3) = 0$
Use the Zero-Product Property by equating each factor to $0$, then solve each equation.
First factor:
$x = 0$
Second factor:
$x + 3 = 0$
Subtract $3$ from each side of the equation:
$x = -3$
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:
$y = (0) + 1$
Add to solve:
$y = 1$
Let's solve for $y$ using the other solution for $x$:
$y = (-3) + 1$
Add to solve:
$y = -2$
The solutions are $(0, 1)$ and $(-3, -2)$.
