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.

Question

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.

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

The graph of −x+y=7
is incorrect. The lines should be parallel.

Answer 1

To determine whether the graphs of the equations \(x+y=2\) and \(-x+y=7\) are correct and whether the lines should be parallel, we need to analyze the equations.

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

   • This is a linear equation that can be transformed into slope-intercept form: \[ y = -x + 2 \]
   • The slope of this line is \(-1\), and it has a y-intercept at \( (0, 2) \), which matches the given solid line that passes through points \((0, 2)\) and \((2, 0)\).

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

   • This can also be rewritten in slope-intercept form: \[ y = x + 7 \]
   • The slope of this line is \(1\), and it has a y-intercept at \( (0, 7) \). However, the description states that the dashed line passes through points \((-7, 0)\) and \((7, 0)\), which indicates a horizontal line at \(y = 0\).

From the analysis:

• The solid line is indeed correct for the equation \(x + y = 2\).
• The dashed line does not represent the equation \(-x + y = 7\); instead, it represents the horizontal line at \(y = 0\).

Now, to answer whether the lines should be parallel:

• The two lines \(x + y = 2\) and \(-x + y = 7\) are not parallel. They have different slopes:
  • Slope of \(x + y = 2\) is \(-1\)
  • Slope of \(-x + y = 7\) is \(1\)

Conclusion: The correct response is:

• The graph of \(x + y = 2\) is correct.
• The graph of \(-x + y = 7\) is incorrect. The lines should intersect, not be parallel.