Question

on the set of axes below, solve the following system of equations graphically and state the coordinates of all points in the solution set.
( y = -x^2 - 10x - 16 )
( y = -2x - 4 )
you can move the parabola by dragging the dots. graph the line by clicking twice.

Explanation:

Step1: Set the equations equal

Since both equations equal \( y \), set them equal:
\( -x^2 - 10x - 16 = -2x - 4 \)

Step2: Simplify the equation

Move all terms to one side:
\( -x^2 - 10x - 16 + 2x + 4 = 0 \)
\( -x^2 - 8x - 12 = 0 \)
Multiply by -1:
\( x^2 + 8x + 12 = 0 \)

Step3: Factor the quadratic

Factor \( x^2 + 8x + 12 \):
\( (x + 2)(x + 6) = 0 \)

Step4: Solve for \( x \)

Set each factor to zero:
\( x + 2 = 0 \) or \( x + 6 = 0 \)
\( x = -2 \) or \( x = -6 \)

Step5: Find corresponding \( y \)

For \( x = -2 \):
\( y = -2(-2) - 4 = 4 - 4 = 0 \)
For \( x = -6 \):
\( y = -2(-6) - 4 = 12 - 4 = 8 \)

Answer:

The solution set is \( \{(-2, 0), (-6, 8)\} \)