Answer
The solutions are $(0, -3)$ and $(-2, -11)$.
Work Step by Step
To solve this system of equations, we set the two equations equal to one another because they are both equal to the same number, $y$:
$-x^2 + 2x - 3 = 4x - 3$
Subtract $4x$ from both sides to gather the variables on the left side of the equation:
$-x^2 + 2x - 3-4x = 4x - 3-4x$
$-x^2 -2x - 3 = -3$
Add $3$ to both sides of the equation to move the constants to the right side of the equation:
$-x^2 - 2x - 3+3 = -3+3$
$-x^2 - 2x = 0$
Factor out a $-x$ from the left side of the equation:
$-x(x + 2) = 0$
According to the Zero-Product Property, if the product of two factors $a$ and $b$ equals zero, then either $a$ is zero, $b$ is zero, or both equal zero. Therefore, we can set each factor equal to zero then solve each equation,
First factor:
$-x = 0$
$x = 0$
Second factor:
$x + 2 = 0$
Subtract $2$ from each side of the equation to solve for $x$:
$x = -2$
Now that we have the values for $x$, we can plug these value into either of the original equations to find the values for $y$. Let us use the second equation:
For $x=0$:
$y = 4(0) - 3$
$y = 0 - 3$
$y = -3$
For $x=-2$:
$y = 4(-2) - 3$
$y = -8 - 3$
$y = -11$
Thus, the solutions are $(0, -3)$ and $(-2, -11)$.
