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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

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

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

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

Both graphs are correct. The lines should be parallel.
Both graphs are correct. The lines should be parallel.
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1 answer

Let's analyze the system of equations:

  1. The first equation is \( x + y = 2 \).

    • This can be rewritten in slope-intercept form as \( y = -x + 2 \).
    • The slope is -1, and the y-intercept is 2.

  2. The second equation is \( -x + y = 7 \).

    • This can be rewritten in slope-intercept form as \( y = x + 7 \).
    • The slope is 1, and the y-intercept is 7.

Now, let's evaluate the lines based on their slopes:

  • The line \( x + y = 2 \) has a slope of -1.
  • The line \( -x + y = 7 \) has a slope of 1.

Since the slopes of the two lines are different (one is -1 and the other is 1), the lines are not parallel and will intersect at some point.

Based on this analysis, the correct response to the options presented is:

The graph of \( x + y = 2 \) is incorrect. The lines should intersect. The graph of \( -x + y = 7 \) is incorrect. The lines should intersect.