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

Question

-the graph 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
-Both graphs are correct. The lines should be parallel

Answer 1

Let's analyze the two equations you provided:

1. First equation: \( x + y = 2 \)
   This can be rearranged to \( y = -x + 2 \). This line has a slope of \(-1\).

2. Second equation: \( -x + y = 7 \)
   This can be rearranged to \( y = x + 7 \). This line has a slope of \(+1\).

To determine the relationship between the two lines:

• The slopes of the two equations are \(-1\) and \(1\), which are not equal. Therefore, the lines are not parallel.
• Since the slopes are different, the lines will intersect at some point.

Now, based on this analysis:

• The graph of the first equation \( x + y = 2 \) should intersect the graph of the second equation \( -x + y = 7 \).
• If the graphs are represented correctly, then they should not be parallel.

Thus, if the graph shows the two lines as parallel or incorrect in representation, you should choose one of the following options:

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

Both lines being parallel is incorrect, as they do intersect due to different slopes. Therefore, the correct answer would refer to one or both graphs as incorrect based on their representation of the equations.