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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 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.

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.

1 answer

To analyze the given system of equations:

  1. The first equation is \( x + y = 2 \). This is a linear equation whose graph is a straight line that will intersect the y-axis at (0, 2) and the x-axis at (2, 0).

  2. The second equation is \( -x + y = 7 \), which can be rearranged to \( y = x + 7 \). This has a y-intercept at (0, 7) and will intersect the x-axis at (-7, 0).

Now, we check the relationships between the lines:

  • The two lines represent different slopes:
    • The slope of the line \( x + y = 2 \) is -1 (since \( y = -x + 2 \)).
    • The slope of the line \( -x + y = 7 \) is 1 (since \( y = x + 7 \)).

Given that they have different slopes, the lines should intersect. Therefore, they should not be parallel.

Considering this analysis, we can determine that:

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

So the correct response is:

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