Answer
$(0, -3)$ and $\left(\frac{1}{3}, -\frac{31}{9}\right)$
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:
$-x^2 - x - 3 = 2x^2 - 2x - 3$
We want to move all terms to the left side of the equation.
First, we will subtract $2x^2$ from both sides of the equation:
$-3x^2 - x - 3 = -2x - 3$
Add $2x$ to both sides of the equation:
$-3x^2 + x - 3 = -3$
Add $3$ to both sides of the equation:
$-3x^2 + x = 0$
Factor out common terms:
$x(-3x + 1) = 0$
Set each factor equal to $0$.
First factor:
$x = 0$
Second factor:
$-3x + 1 = 0$
Subtract $1$ from each side of the equation:
$-3x = -1$
Divide each side of the equation by $-3$:
$x = \frac{1}{3}$
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 first equation:
$y = (-0)^2 - (0) - 3$
Evaluate the exponent first:
$y = 0 - 0 - 3$
Add or subtract from left to right:
$y = -3$
Let's solve for $y$ using the other solution for $x$:
$y = (\frac{1}{3})^2 - (\frac{1}{3}) - 3$
Evaluate the exponent first:
$y = \frac{1}{9} - (\frac{1}{3}) - 3$
Multiply next:
$y = \frac{1}{9} - \frac{1}{3} - 3$
Find equivalent fractions with common denominators to add the fractions together:
$y = \frac{1}{9} - \frac{3}{9} - \frac{27}{9}$
Add or subtract from left to right:
$y = -\frac{31}{9}$
The solutions are $(0, -3)$ and $\left(\frac{1}{3}, -\frac{31}{9}\right)$.
