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Describe the solution set for the system of equations that consists of the equation of the line shown and 2x + y = 6
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GPT-4o mini
To describe the solution set for the system of equations consisting of a given line and the equation \( 2x + y = 6 \), we first need to understand what these equations represent graphically.
1. **Equation of the Line**: The specific line you mentioned is not provided in your question, but we'll denote it as \( L \) for the sake of discussion.
2. **Equation of the Line**: The equation \( 2x + y = 6 \) can be rewritten in slope-intercept form (\( y = mx + b \)):
\[
y = -2x + 6
\]
This line has a slope of -2 and a y-intercept of 6.
To find the solution set of the system, we need to consider how \( L \) interacts with the line represented by \( 2x + y = 6 \).
### Possible Scenarios:
1. **No Solution**: If the line \( L \) is parallel to the line \( 2x + y = 6 \) (they have the same slope but different y-intercepts), then the two lines will never intersect, and there will be no solutions to the system.
2. **One Solution**: If the line \( L \) is not parallel and intersects the line \( 2x + y = 6 \) at a single point, then there will be exactly one solution to the system, which is the coordinates of the intersection point.
3. **Infinite Solutions**: If the line \( L \) is the same as the line represented by \( 2x + y = 6 \) (they have the same slope and y-intercept), then there are infinitely many solutions, as every point on the line is a solution.
### Conclusion:
To accurately describe the solution set for the system, we would need the equation of the line \( L \). Depending on how it relates to the equation \( 2x + y = 6 \), the solution set could be:
- **No Solutions**: If \( L \) is parallel to \( 2x + y = 6 \).
- **One Solution**: If \( L \) intersects \( 2x + y = 6 \) at a single point.
- **Infinitely Many Solutions**: If \( L \) is exactly the same line as \( 2x + y = 6 \).
If you provide the specific equation of the line, I can help you find the exact solution set!
1. **Equation of the Line**: The specific line you mentioned is not provided in your question, but we'll denote it as \( L \) for the sake of discussion.
2. **Equation of the Line**: The equation \( 2x + y = 6 \) can be rewritten in slope-intercept form (\( y = mx + b \)):
\[
y = -2x + 6
\]
This line has a slope of -2 and a y-intercept of 6.
To find the solution set of the system, we need to consider how \( L \) interacts with the line represented by \( 2x + y = 6 \).
### Possible Scenarios:
1. **No Solution**: If the line \( L \) is parallel to the line \( 2x + y = 6 \) (they have the same slope but different y-intercepts), then the two lines will never intersect, and there will be no solutions to the system.
2. **One Solution**: If the line \( L \) is not parallel and intersects the line \( 2x + y = 6 \) at a single point, then there will be exactly one solution to the system, which is the coordinates of the intersection point.
3. **Infinite Solutions**: If the line \( L \) is the same as the line represented by \( 2x + y = 6 \) (they have the same slope and y-intercept), then there are infinitely many solutions, as every point on the line is a solution.
### Conclusion:
To accurately describe the solution set for the system, we would need the equation of the line \( L \). Depending on how it relates to the equation \( 2x + y = 6 \), the solution set could be:
- **No Solutions**: If \( L \) is parallel to \( 2x + y = 6 \).
- **One Solution**: If \( L \) intersects \( 2x + y = 6 \) at a single point.
- **Infinitely Many Solutions**: If \( L \) is exactly the same line as \( 2x + y = 6 \).
If you provide the specific equation of the line, I can help you find the exact solution set!
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