Ask a New Question
Menu

Systems of Linear Equations Unit Test

5 of 145 of 14 Items

Question
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 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 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 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.

1 answer

To determine the correct graphs for the system of equations \(y = 2x + 4\) and \(-2x + y = 8\), it's essential to first identify their slopes and intercepts.

  1. Equation 1: \(y = 2x + 4\)

    • This is in slope-intercept form \(y = mx + b\) where \(m = 2\) (slope) and \(b = 4\) (y-intercept).
    • The line has a positive slope, indicating it rises as it moves from left to right.

  2. Equation 2: \(-2x + y = 8\) (Rearranging to slope-intercept form)

    • We rearrange this to get \(y = 2x + 8\).
    • Here, the slope \(m = 2\) and the y-intercept \(b = 8\).
    • This line also has a positive slope and rises as it moves from left to right.

Since both lines have the same slope (2) but different y-intercepts (4 and 8), they are parallel and will never intersect. This indicates that there is no solution to the system of equations.

The correct graph displaying parallel lines (i.e., having the same slope but different y-intercepts) would be one where:

  • One solid line represents \(y = 2x + 4\).
  • One dashed line represents \(y = 2x + 8\).

Looking at your description of the graphs, you should choose the option that clearly shows two parallel lines, where one has a solid line and the other has a dashed line, consistent with the y-intercepts of the equations.

In summary, the correct response will show the two lines with:

  • A solid line passing through points (0, 4) and (–2, 0) for the equation \(y=2x+4\).
  • A dashed line passing through points (0, 8) and (–4, 0) for the equation \(y=2x+8\).

Please review the options you provided to select the one with the correct lines as described.