Question

find the solution(s) to the system of equations. select all that apply.
math description
$y = x^2 - 2x - 3$
$y = 2x - 3$
(graph with a parabola $y = x^2 - 2x - 3$ and a line $y = 2x - 3$ intersecting)
a. $(4, 5)$
b. $(3, 0)$
c. $(-1, 0)$
d. $(0, -3)$

Explanation:

Step1: Set equations equal

Since \( y = x^2 - 2x - 3 \) and \( y = 2x - 3 \), set them equal: \( x^2 - 2x - 3 = 2x - 3 \)

Step2: Simplify equation

Subtract \( 2x - 3 \) from both sides: \( x^2 - 4x = 0 \)

Step3: Factor equation

Factor out \( x \): \( x(x - 4) = 0 \)

Step4: Solve for x

Set each factor to zero: \( x = 0 \) or \( x = 4 \)

Step5: Find corresponding y

• For \( x = 0 \): \( y = 2(0) - 3 = -3 \) (so \( (0, -3) \))
• For \( x = 4 \): \( y = 2(4) - 3 = 5 \) (so \( (4, 5) \))

Check other options:

• For \( (3, 0) \): \( y = 2(3) - 3 = 3

eq 0 \), so no.

• For \( (-1, 0) \): \( y = 2(-1) - 3 = -5

eq 0 \), so no.

Answer:

A. \( (4, 5) \), D. \( (0, -3) \)