Use the image to answer the question.
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.
Both graphs are correct. The lines should be parallel.
Both graphs are correct. 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.
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The correct response is:
The graph of −x+y=7 is incorrect. The lines should intersect.
The graph does not show the system of equations x+y=2 and −x+y=7. The lines for the system should intersect, but in the graph, they are shown as parallel lines.
To determine if the graph shows the system of equations x+y=2 and −x+y=7, we can compare the given lines to the equations.
The first line in the graph passes through the points (0,2) and (2,0). We can calculate the slope of this line by using the formula:
slope (m) = (change in y) / (change in x).
Using the points (0,2) and (2,0), we have:
slope = (0 - 2) / (2 - 0) = -2 / 2 = -1.
The equation x+y=2 is in standard form, where the coefficient of x is 1 and the coefficient of y is 1. This means the slope of the line represented by this equation is -1.
Since the slope of the line in the graph matches the slope of the line represented by the equation x+y=2, we can conclude that the graph shows the line represented by this equation.
Now let's examine the second equation, −x+y=7. We calculate the slope using the same formula:
slope = (change in y) / (change in x).
Using the points (-7,0) and (7,0), we have:
slope = (0 - 0) / (7 - (-7)) = 0 / 14 = 0.
The equation −x+y=7 is also in standard form, where the coefficient of x is -1 and the coefficient of y is 1. This means the slope of the line represented by this equation is 0.
However, the solid line in the graph has a slope of -1, which does not match the slope of the line represented by the equation −x+y=7.
Therefore, the graph does NOT show the system of equations x+y=2 and −x+y=7. The graph of −x+y=7 is incorrect, and the lines should intersect.
In conclusion, the lines for the system of equations x+y=2 and −x+y=7 should intersect, not be parallel.