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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 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 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 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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must choose 1st 2nd 3rd or 4th option

1 answer

To determine which graph represents a system of equations with no solution, we first need to analyze the two equations given:

  1. The first equation is \( y = 2x + 4 \).

    • This is in slope-intercept form (y = mx + b), where the slope \( m \) is 2 and the y-intercept \( b \) is 4.
    • The line passes through the point \( (0, 4) \) and has a positive slope.

  2. The second equation is \( -2x + y = 8 \).

    • We can rearrange this into slope-intercept form: \[ y = 2x + 8 \]
    • This line also has a slope of 2 but a different y-intercept \( (0, 8) \).

Since both lines have the same slope (2), they are parallel. Parallel lines do not intersect, which means that the system of equations has no solution.

Now, we can look at the options provided:

  1. The first graph has a solid line (from the first equation) passing through points \( (0, 4) \) and \( (-2, 0) \) and a dashed line (from the second equation) passing through \( (4, 0) \) and \( (0, -8) \).
  2. The second graph has a solid line passing through \( (2, 0) \) and \( (0, 4) \) and a dashed line through \( (-4, 0) \) and \( (0, -8) \).
  3. The third graph has the solid line passing through \( (-2, 0) \) and \( (0, -4) \), and the dashed line passing through \( (-4, 0) \) and \( (0, -8) \).
  4. The fourth graph has a solid line passing through \( (0, 4) \) and \( (-2, 0) \) and a dashed line passing through \( (-4, 0) \) and \( (0, 8) \).

The key condition is that the lines must be parallel. Observing the slopes and points of interception carefully, the first and fourth options appear to have the properties of parallel lines, but only the fourth option correctly positions the equations:

  • Solid line: \( y = 2x + 4 \) (correct intercepts).
  • Dashed line: \( y = 2x + 8 \) (also parallel, different intercepts).

Therefore, the answer is the fourth option.