Question

question 9 of 25
on a piece of paper, graph this system of equations.
y = x - 2
y = x² - 6x + 8
then determine which answer choice matches the graph you drew and identify the solutions to the system.

a. solutions: (2, 0) and (3, -1)
graph with a blue line and a red parabola
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b. solutions: (-2, 0) and (2, 0)
graph with a blue line and a red parabola
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c. solutions: (2, 0) and (4, 2)
graph with a blue line and a red parabola
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d. solutions: (2, 0) and (5, 3)
graph with a blue line and a red parabola
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Explanation:

Step1: Set the equations equal

Since \( y = x - 2 \) and \( y = x^2 - 6x + 8 \), we set \( x - 2 = x^2 - 6x + 8 \).

Step2: Rearrange into quadratic equation

\( x^2 - 6x + 8 - x + 2 = 0 \) simplifies to \( x^2 - 7x + 10 = 0 \).

Step3: Factor the quadratic

\( x^2 - 7x + 10 = (x - 2)(x - 5) = 0 \).

Step4: Solve for x

Setting each factor to zero: \( x - 2 = 0 \) gives \( x = 2 \); \( x - 5 = 0 \) gives \( x = 5 \).

Step5: Find corresponding y-values

For \( x = 2 \), \( y = 2 - 2 = 0 \). For \( x = 5 \), \( y = 5 - 2 = 3 \). So the solutions are \( (2, 0) \) and \( (5, 3) \), which matches option D.

Answer:

D. Solutions: (2, 0) and (5, 3)