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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=7 is incorrect. The lines should intersect.
The graph of negative x plus y equals 7 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=2 is incorrect. The lines should intersect.
The graph of x plus y equals 2 is incorrect. The lines should intersect.

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

1 answer

To examine the correctness of the graph based on the equations given, we need to analyze the lines represented by the equations \(x + y = 2\) and \(-x + y = 7\).

  1. Line \(x + y = 2\):

    • This is a linear equation with a slope of \(-1\) (since \(y = -x + 2\)).
    • The two points provided on this line, \( (0, 2) \) and \( (2, 0) \), can be used to find its slope and confirm that these points lie on this line.

  2. Line \(-x + y = 7\):

    • This can be rewritten as \(y = x + 7\), which also has a slope of \(1\).
    • The dashed line given, which includes points such as \( (-7, 0) \) and \( (7, 0) \), doesn’t align with \(y = x + 7\) as these points indicate a horizontal line (specifically \(y = 0\)) rather than a line with slope 1.

Next, let's look at the relationship between the two lines:

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

Because the slopes are different (\(-1\) for the first line and \(1\) for the second line), the two lines must intersect. Consequently, they should not be parallel.

Based on this analysis, we can conclude:

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

The correct response to select is: "The graph of \(-x + y = 7\) is incorrect. The lines should intersect."