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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.

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

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.

1 answer

To determine whether the system of equations \( x + y = 2 \) and \( -x + y = 7 \) is represented correctly on the graph and whether the lines should be parallel, let's analyze the equations:

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

  2. The second equation, \( -x + y = 7 \), can be rewritten as: \[ y = x + 7 \] This line has a slope of 1.

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

Now, looking at the graph:

  • The solid line passing through (0, 2) and (2, 0) represents \( x + y = 2 \), which is correctly represented.
  • The dashed line passing through (-7, 0) and (7, 0) represents a horizontal line, which does not match \( -x + y = 7 \). A correct representation for \( -x + y = 7 \) should be an upward slope line, not horizontal.

Therefore, the graph of \( -x + y = 7 \) is incorrect, and the lines should intersect rather than be parallel.

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