Question

question 5 of 25
on a piece of paper, graph this system of equations.
y = x - 2
y = x² - 5x + 6
then determine which answer choice matches the graph you drew and identify the solutions to the system.

Explanation:

Step1: Find intersection by solving equations

Set \( x - 2=x^{2}-5x + 6 \). Rearrange to \( x^{2}-6x + 8 = 0 \).

Step2: Factor the quadratic equation

Factor \( x^{2}-6x + 8 \) as \( (x - 2)(x - 4)=0 \).

Step3: Solve for x

Set each factor to zero: \( x - 2 = 0 \) gives \( x = 2 \); \( x - 4 = 0 \) gives \( x = 4 \).

Step4: Find corresponding y - values

For \( x = 2 \), \( y=2 - 2 = 0 \). For \( x = 4 \), \( y=4 - 2 = 2 \).

Answer:

The solutions to the system are \( (2,0) \) and \( (4,2) \). To graph: \( y = x - 2 \) is a straight line with slope 1 and y - intercept - 2. \( y=x^{2}-5x + 6=(x - 2)(x - 3) \) is a parabola opening upwards with roots at \( x = 2 \) and \( x = 3 \), vertex at \( x=\frac{5}{2},y=(\frac{5}{2})^{2}-5\times\frac{5}{2}+6=-\frac{1}{4} \). The intersection points are \( (2,0) \) and \( (4,2) \).