Ask a New Question
Menu

page 19 of 19

One Solution, No Solution, or Many Solutions Quick Check
3 of 53 of 5 Items

Question
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=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 solve this problem, let’s break down the given equations and their graphical representations.

  1. Equations:

    • The first equation is \(x + y = 2\).
    • The second equation is \(-x + y = 7\).

  2. Slope and Intercept:

    • For the equation \(x + y = 2\):
      • Rearranging gives \(y = -x + 2\), which has a slope of \(-1\).
    • For the equation \(-x + y = 7\):
      • Rearranging gives \(y = x + 7\), which has a slope of \(1\).

  3. Determining Line Characteristics:

    • The first line (from \(x + y = 2\)) has a slope of \(-1\).
    • The second line (from \(-x + y = 7\)) has a slope of \(1\).
    • Since the slopes are different (\(-1\) and \(1\)), these lines will intersect at some point, which means there is one solution for the system.

  4. Graph Interpretation:

    • According to the problem, the graph shows the lines as parallel.
    • Parallel lines would imply they do not intersect, indicating there would be no solution to the system of equations.

  5. Conclusion:

    • Since the two lines are not parallel (due to different slopes), the graphs provided in the question must be incorrect.
    • Therefore, the correct answer from the responses would be:
    • The graph of \(-x+y=7\) is incorrect. The lines should intersect.

This response indicates that the representations shown in the graph do not correctly reflect the nature of the equations provided.