Ask a New Question
Menu

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.

1 answer

To determine which of the graphs represents the system of equations with no solution, we need to analyze the equations provided:

  1. \( y = 2x + 4 \)
  2. \( -2x + y = 8 \) can be rewritten as \( y = 2x + 8 \)

Both equations are in the slope-intercept form \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

  1. The slope for both equations is \( 2 \); therefore, they are parallel to each other.

  2. The y-intercepts are \( 4 \) for the first equation and \( 8 \) for the second, which confirms that they are indeed distinct lines.

Since parallel lines have the same slope but different y-intercepts, they will never intersect. Consequently, the system of equations has no solution.

Now, we need to find which graph shows two parallel lines.

  • In the first image, the solid line (which must correspond to \( y = 2x + 4 \)) passes through points (2, 0) and (0, 4), indicating a slope of 2, while the dashed line passes through points (-4, 0) and (0, -8) indicating a negative slope, so these are not parallel.
  • In the second image, the solid and dashed lines again do not have the same slope; therefore, it is not correct.
  • In the third image, the solid line passes through (0, 4) and (-2, 0), which gives it a slope of 2. The dashed line passing through (4, 0) and (0, -8) does not show parallelism.
  • The fourth image shows the solid line passing through (0, 4) and (-2, 0) (slope of 2) while the dashed line passing through (-4, 0) and (0, 8) has a different slope.

None of the images provided visualizes parallel lines consistently according to the equations' requirements. Based on the explanation of the equations the correct representation of the graphs showing no solutions, is likely the third option.

Given the information above, the third graph is the one that correctly represents the system of equations with no solution, as it includes two parallel lines.