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 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 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 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.
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 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 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 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.
Answers
Answered by
GPT-4o mini
To determine which graph corresponds to the system of equations \( y = 3x + 4 \) and \( -3x + y = 8 \), we first rewrite the second equation in slope-intercept form (y = mx + b) for easier comparison.
1. Start with the equation \( -3x + y = 8 \).
2. Rearranging gives \( y = 3x + 8 \).
Now we have the two equations:
- \( y = 3x + 4 \) (solid line)
- \( y = 3x + 8 \) (dashed line)
Both lines have the same slope (\( m = 3 \)), meaning they are parallel. Since parallel lines never intersect, this system of equations has no solution.
Now let's evaluate the options:
1. **First Option:** Solid line passes through \((-1, 3)\) and \((1, -3)\) - **not parallel** to dashed line through \((0, 8)\) and \((4, -4)\).
2. **Second Option:** Solid line passes through \((0, 4)\) and \((-3, -5)\) - **not parallel** to dashed line through \((0, 8)\) and \((-4, -4)\).
3. **Third Option:** Solid line passes through \((0, 4)\) and \((2, -2)\) - **not parallel** to dashed line through \((0, 8)\) and \((4, -4)\).
4. **Fourth Option:** Solid line passes through \((2, 2)\) and \((0, -4)\) - **not parallel** to dashed line through \((3, 1)\) and \((0, -8)\).
Since none of these equations match the system you provided (which should show parallel lines) and all represent systems with solutions, please check the conditions or clarify if there are specific conditions for no solution. From the interpretations provided, you should look for the graphs that clearly indicate the equations result in parallel lines.
1. Start with the equation \( -3x + y = 8 \).
2. Rearranging gives \( y = 3x + 8 \).
Now we have the two equations:
- \( y = 3x + 4 \) (solid line)
- \( y = 3x + 8 \) (dashed line)
Both lines have the same slope (\( m = 3 \)), meaning they are parallel. Since parallel lines never intersect, this system of equations has no solution.
Now let's evaluate the options:
1. **First Option:** Solid line passes through \((-1, 3)\) and \((1, -3)\) - **not parallel** to dashed line through \((0, 8)\) and \((4, -4)\).
2. **Second Option:** Solid line passes through \((0, 4)\) and \((-3, -5)\) - **not parallel** to dashed line through \((0, 8)\) and \((-4, -4)\).
3. **Third Option:** Solid line passes through \((0, 4)\) and \((2, -2)\) - **not parallel** to dashed line through \((0, 8)\) and \((4, -4)\).
4. **Fourth Option:** Solid line passes through \((2, 2)\) and \((0, -4)\) - **not parallel** to dashed line through \((3, 1)\) and \((0, -8)\).
Since none of these equations match the system you provided (which should show parallel lines) and all represent systems with solutions, please check the conditions or clarify if there are specific conditions for no solution. From the interpretations provided, you should look for the graphs that clearly indicate the equations result in parallel lines.
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