Use the image to answer the question.

Question

A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line passes through left parenthesis negative 7 comma 0 right parenthesis and left parenthesis 7 comma 0 right parenthesis.

Does the graph show the system of equations x+y=2
and −x+y=7
? Should the lines for the system be parallel?

(1 point)
Responses

Both graphs are correct. The lines should be parallel.

The graph of x+y=2
is incorrect. The lines should intersect.

The graph of −x+y=7
is incorrect. The lines should intersect.

The graph of −x+y=7
is incorrect. The lines should be parallel.

Answer 1

To analyze the given equations:

1. The equation \(x + y = 2\) can be rewritten in slope-intercept form as \(y = -x + 2\). This line has a slope of -1.
2. The equation \(-x + y = 7\) can be rewritten as \(y = x + 7\). This line has a slope of +1.

Since the slopes of the two lines are different (-1 and +1), the lines are not parallel and should intersect at some point.

Based on the given graph:

• The solid line has the correct intercepts corresponding to \(x + y = 2\).
• The dashed line does not correspond to the equation \(-x + y = 7\); instead, it is a horizontal line (a slope of 0), which is incorrect.

So, the correct conclusion based on the graph is:

The graph of \(-x + y = 7\) is incorrect. The lines should intersect.