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Question

Which of the graphs with no solution correctly shows the system of equations y=3x+4
and −3x+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 3 comma negative 5 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis negative 4 comma negative 4 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 3 comma negative 5 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis negative 4 comma negative 4 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 1 comma 3 right parenthesis and left parenthesis 1 comma negative 3 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis 4 comma negative 4 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 1 comma 3 right parenthesis and left parenthesis 1 comma negative 3 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis 4 comma negative 4 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 2 comma negative 2 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis 4 comma negative 4 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 2 comma negative 2 right parenthesis. The dashed lines pass through left parenthesis 0 comma 8 right parenthesis and left parenthesis 4 comma negative 4 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 2 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis 3 comma 1 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.

1 answer

To determine which of the graphs with no solution correctly represents the system of equations \(y = 3x + 4\) and \(-3x + y = 8\), we need to analyze the relationships between the two lines represented by these equations.

  1. Convert the second equation to slope-intercept form:

    • The equation given is \(-3x + y = 8\). We can isolate \(y\): \[ y = 3x + 8 \]

  2. Identify the slopes and y-intercepts:

    • For \(y = 3x + 4\):
      • Slope (m) = 3
      • y-intercept = 4
    • For \(y = 3x + 8\):
      • Slope (m) = 3
      • y-intercept = 8

Since both lines have the same slope (3) but different y-intercepts (4 and 8), they are parallel and will never intersect. This means that the system of equations has no solution.

  1. Analyzing the Options:
    • We need to find the graph that depicts two parallel lines.
    • The correct graph will show two lines with the same slope of 3 that do not intersect.

Now let's evaluate the graphs based on the descriptions provided:

  • Graph Option 1: Solid line through \((0, 4)\) (y-intercept) and a dashed line through \((0, 8)\) - right criteria (parallel with y-intercept difference).
  • Graph Option 2: Solid line through points with inconsistent intercepts - incorrect criteria.
  • Graph Option 3: Similar to Graph Option 1 with correct points but different slopes indicated - also incorrect.
  • Graph Option 4: Describes solid and dashed that do not match slopes at all - definitely incorrect.

Conclusion

Based on the information, Graph Option 1 correctly shows the system of equations with no solution, as it depicts two parallel lines with slopes of 3 and different y-intercepts (4 and 8).