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.
$y = x^2 - 6x + 6$
$y = -2x + 3$
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: \( x^2 - 6x + 6 = -2x + 3 \)

Step2: Simplify the equation

Rearrange to standard quadratic form: \( x^2 - 6x + 2x + 6 - 3 = 0 \) → \( x^2 - 4x + 3 = 0 \)

Step3: Factor the quadratic

Factor: \( (x - 1)(x - 3) = 0 \)

Step4: Solve for \( x \)

Set each factor to zero: \( x - 1 = 0 \) → \( x = 1 \); \( x - 3 = 0 \) → \( x = 3 \)

Step5: Find corresponding \( y \) values

For \( x = 1 \): \( y = -2(1) + 3 = 1 \)
For \( x = 3 \): \( y = -2(3) + 3 = -3 \)
(Graphically, the parabola \( y = x^2 - 6x + 6 \) (vertex at \( x = 3, y = -3 \)) and line \( y = -2x + 3 \) intersect at \( (1,1) \) and \( (3,-3) \))

Answer:

The solution set is \( \{(1, 1), (3, -3)\} \)