Question

graph the system of equations and determine the points of intersection.
y = -x + 7
y = x² - 4x - 3
(-2, 9) (9, -2) (0, 0)
(2, 5) (5, 2)

Explanation:

Step1: Set equations equal

Since $y = -x + 7$ and $y = x^2 - 4x - 3$, set them equal to each other:
$$-x + 7 = x^2 - 4x - 3$$

Step2: Rearrange to standard quadratic

Bring all terms to one side to form a quadratic equation:
$$x^2 - 3x - 10 = 0$$

Step3: Factor the quadratic

Factor the quadratic expression:
$$(x - 5)(x + 2) = 0$$

Step4: Solve for x-values

Set each factor equal to 0 and solve for $x$:
$x - 5 = 0 \implies x = 5$
$x + 2 = 0 \implies x = -2$

Step5: Find corresponding y-values

Substitute $x = 5$ into $y = -x + 7$:
$y = -5 + 7 = 2$
Substitute $x = -2$ into $y = -x + 7$:
$y = -(-2) + 7 = 9$

Answer:

$(-2, 9)$, $(5, 2)$