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 graph shows that the lines for the system of equations x+y=2 and −x+y=7 are parallel.
The lines shown in the graph are parallel. The graph of x + y = 2 is correct and the solid line passes through (0, 2) and (2, 0). However, the graph of -x + y = 7 is incorrect as it should intersect with the other line. Hence, the correct answer is:
The graph of x + y = 2 is correct.
The graph of -x + y = 7 is incorrect.
The lines should intersect, not be parallel.
To determine if the graph shows the system of equations x+y=2 and -x+y=7, we need to analyze the lines on the graph and see if they match the given equations.
First, let's examine the solid line on the graph. It passes through the points (0, 2) and (2, 0). We can use these points to find the slope of the line using the formula:
slope = (change in y)/(change in x)
Using (0, 2) and (2, 0):
slope = (0 - 2)/(2 - 0) = -2/2 = -1
So, the slope of the solid line is -1. Now, let's check if the line satisfies the equation x+y=2:
Substituting the coordinates (0, 2):
0 + 2 = 2
2 = 2
The equation holds true for this point. Now, let's check the other point (2, 0):
2 + 0 = 2
2 = 2
Again, the equation is satisfied. Therefore, the solid line matches the equation x+y=2.
Now, let's examine the dashed line on the graph. It passes through the points (-7, 0) and (7, 0). We can again find the slope using these points:
slope = (0 - 0)/(7 - (-7)) = 0/14 = 0
The slope of the dashed line is 0. Now, let's check if this line satisfies the equation -x+y=7:
Substituting the coordinates (-7, 0):
-(-7) + 0 = 7
7 + 0 = 7
7 = 7
The equation is satisfied for this point. Now, let's check the other point (7, 0):
-7 + 0 = 7
-7 = 7
The equation is NOT satisfied for this point. Therefore, the dashed line does NOT match the equation -x+y=7.
Based on our analysis, the graph does NOT show the system of equations x+y=2 and -x+y=7.
Additionally, since the equations x+y=2 and -x+y=7 have different slopes (-1 and 0, respectively), the lines for the system should intersect, not be parallel.