Question
which of the graphs with no solutions correctly shows the system of equations
y = 2x + 4 and -2x + y = 8
A. 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
B. 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
C. 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.
D. 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.
y = 2x + 4 and -2x + y = 8
A. 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
B. 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
C. 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.
D. 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.
Answers
Answered by
GPT-4o mini
To determine which graph correctly represents the system of equations \( y = 2x + 4 \) and \( -2x + y = 8 \) with no solutions, we first need to analyze these equations.
1. **Convert the second equation into slope-intercept form (y = mx + b)**:
\[
-2x + y = 8 \implies y = 2x + 8
\]
2. **Identify the slopes and intercepts**:
- The line from the first equation, \( y = 2x + 4 \), has a slope of 2 and a y-intercept of 4.
- The line from the second equation, \( y = 2x + 8 \), has the same slope of 2 but a different y-intercept of 8.
3. **Conclusion**: Both lines have the same slope but different y-intercepts. Therefore, they are parallel and will never intersect, implying there are no solutions to this system of equations.
4. **Identify the characteristics of the lines you need**:
- Both lines must be parallel.
- Their respective y-intercepts must be different (4 for the solid line and 8 for the dashed line).
Now, looking at the options provided:
- **Option A**: The solid line has points (2, 0) and (0, 4), while the dashed line has points (-4, 0) and (0, -8). The solid line \( y = 2x + 4 \) is correct, but the dashed line does not correspond to \( y = 2x + 8 \).
- **Option B**: The solid line goes through points (0, 4) and (-2, 0), which is correct for \( y = 2x + 4 \). The dashed line goes through points (4, 0) and (0, -8), not correct for \( y = 2x + 8 \).
- **Option C**: The solid line goes through points (0, 4) and (-2, 0), correct for \( y = 2x + 4 \). The dashed line goes through points (-4, 0) and (0, 8). This is correct for \( y = 2x + 8 \) because it has the correct y-intercept at positive 8.
- **Option D**: The solid line goes through (-2, 0) and (0, -4) which is incorrect for \( y = 2x + 4 \).
Thus, the correct choice for the graph that shows the system of equations with no solutions, both lines being parallel, is:
**C.**
1. **Convert the second equation into slope-intercept form (y = mx + b)**:
\[
-2x + y = 8 \implies y = 2x + 8
\]
2. **Identify the slopes and intercepts**:
- The line from the first equation, \( y = 2x + 4 \), has a slope of 2 and a y-intercept of 4.
- The line from the second equation, \( y = 2x + 8 \), has the same slope of 2 but a different y-intercept of 8.
3. **Conclusion**: Both lines have the same slope but different y-intercepts. Therefore, they are parallel and will never intersect, implying there are no solutions to this system of equations.
4. **Identify the characteristics of the lines you need**:
- Both lines must be parallel.
- Their respective y-intercepts must be different (4 for the solid line and 8 for the dashed line).
Now, looking at the options provided:
- **Option A**: The solid line has points (2, 0) and (0, 4), while the dashed line has points (-4, 0) and (0, -8). The solid line \( y = 2x + 4 \) is correct, but the dashed line does not correspond to \( y = 2x + 8 \).
- **Option B**: The solid line goes through points (0, 4) and (-2, 0), which is correct for \( y = 2x + 4 \). The dashed line goes through points (4, 0) and (0, -8), not correct for \( y = 2x + 8 \).
- **Option C**: The solid line goes through points (0, 4) and (-2, 0), correct for \( y = 2x + 4 \). The dashed line goes through points (-4, 0) and (0, 8). This is correct for \( y = 2x + 8 \) because it has the correct y-intercept at positive 8.
- **Option D**: The solid line goes through (-2, 0) and (0, -4) which is incorrect for \( y = 2x + 4 \).
Thus, the correct choice for the graph that shows the system of equations with no solutions, both lines being parallel, is:
**C.**
There are no human answers yet. A form for humans to post answers is coming very soon!