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Which of the graphs with no solution correctly shows the system of equations y=2x+4

and −2x+y=8
?(1 point)
Responses

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 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: 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 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.

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 negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: 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 negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.

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 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: 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 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.

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 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Image with alt text: 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 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
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1 answer

To determine which graph shows the system of equations \( y = 2x + 4 \) and \( -2x + y = 8 \) with no solutions, we first need to analyze the two equations.

  1. Convert the second equation: \[ -2x + y = 8 \implies y = 2x + 8 \]

  2. Compare the equations:

    • The first equation is \( y = 2x + 4 \).
    • The second equation is \( y = 2x + 8 \).

Both equations have the same slope of \( 2 \) (indicating they are parallel), but different y-intercepts (4 and 8, respectively). This means that these lines will never intersect.

  1. Look at the options presented: Since we are looking for a graph where both lines are parallel and do not intersect, we need to select the option where:
    • The solid line represents \( y = 2x + 4 \) (must pass through \( (0, 4) \) and have a slope of 2).
    • The dashed line represents \( y = 2x + 8 \) (must pass through \( (0, 8) \) and also have a slope of 2).

From the descriptions of the graphs, look for:

  • One line that passes through \( (0, 4) \) (solid line) and extends upwards with a slope of \( 2 \).
  • The other line that is parallel, passing through \( (0, 8) \) (dashed line).

Therefore, the graph that shows the correct representation of the system of equations with no solution will be the one where both lines are parallel, one with \( y \)-intercept at 4, and the other at 8.

After analyzing the provided options, select the one with these characteristics.