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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

1.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.

1.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.

1.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.

1.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 of the given graphs shows a system of equations with no solution, we first need to analyze the two equations:

  1. \( y = 2x + 4 \)
  2. \( -2x + y = 8 \)

We can rewrite the second equation in slope-intercept form (y = mx + b) to make it easier to compare slopes:

Starting with the second equation: \[ -2x + y = 8 \] Add \( 2x \) to both sides: \[ y = 2x + 8 \]

Now, we have:

  1. \( y = 2x + 4 \) (slope = 2, y-intercept = 4)
  2. \( y = 2x + 8 \) (slope = 2, y-intercept = 8)

Both lines have the same slope (2), which means they are parallel. Since the y-intercepts are different (4 and 8), these lines will never intersect and therefore represent a system of equations with no solution.

Now, to select the correct graph showing parallel lines, we look for the following characteristics:

  • Both lines should be drawn parallel to each other.
  • The solid line must represent \( y = 2x + 4 \) and pass through points such as (0, 4) and (2, 0).
  • The dashed line must represent \( y = 2x + 8 \) and pass through points such as (0, 8) and any point corresponding to that line.

Among the options given, look for:

  • A solid line passing through (0, 4) (for \( y = 2x + 4 \)).
  • A dashed line passing through (0, 8) or another point consistent with \( y = 2x + 8 \).

Based on this analysis, select the graph that meets these criteria, which will indicate the correct representation of a system of equations with no solution (parallel lines).