Question
Statement best describe the system of equations for y = 3x + 7 x = -3x - x the equations have different slopes in different y intercepts does the system has one solution at negative 3, 7 both equations have a slope of negative 3 but they do not share the same y intercept that's the system has no solution because the lines are pretty well
Answers
GPT-4o mini
Let's analyze the given equations step by step.
1. You have the first equation:
\[
y = 3x + 7
\]
Here, the slope is 3 and the y-intercept is 7.
2. For the second equation:
\[
x = -3x - x
\]
This is equivalent to simplifying it:
\[
x = -4x
\]
Adding \(4x\) to both sides gives:
\[
5x = 0 \implies x = 0
\]
This represents a vertical line where \(x = 0\).
Since the equations represent different types of lines (one is a linear function with a positive slope and the other is a vertical line), let's summarize our findings:
- The first equation has a slope of 3 and a y-intercept of 7, meaning it rises steeply as \(x\) increases.
- The second equation \(x = 0\) is a vertical line along the y-axis.
### Conclusion about the system:
1. **Different slopes**: The slope of the first line is 3, while the second line does not have a slope (as vertical lines are undefined). Therefore, they do not have the same slope.
2. **Different y-intercepts**: The first equation intersects the y-axis at 7, while the second does not intersect the y-axis at all since it is a vertical line.
3. **One solution**: Since the first equation is a sloped line and the second is a vertical line, they will intersect at exactly one point. If you plug \(x = 0\) into the first equation to find \(y\):
\[
y = 3(0) + 7 = 7
\]
So the point of intersection is (0, 7), not (-3, 7).
4. **No solution**: It is incorrect to say there is no solution because the two lines do intersect at one unique point.
So, the correct statement about the system of equations is that it has **one solution** at the point (0, 7), with the lines having different slopes and y-intercepts.
1. You have the first equation:
\[
y = 3x + 7
\]
Here, the slope is 3 and the y-intercept is 7.
2. For the second equation:
\[
x = -3x - x
\]
This is equivalent to simplifying it:
\[
x = -4x
\]
Adding \(4x\) to both sides gives:
\[
5x = 0 \implies x = 0
\]
This represents a vertical line where \(x = 0\).
Since the equations represent different types of lines (one is a linear function with a positive slope and the other is a vertical line), let's summarize our findings:
- The first equation has a slope of 3 and a y-intercept of 7, meaning it rises steeply as \(x\) increases.
- The second equation \(x = 0\) is a vertical line along the y-axis.
### Conclusion about the system:
1. **Different slopes**: The slope of the first line is 3, while the second line does not have a slope (as vertical lines are undefined). Therefore, they do not have the same slope.
2. **Different y-intercepts**: The first equation intersects the y-axis at 7, while the second does not intersect the y-axis at all since it is a vertical line.
3. **One solution**: Since the first equation is a sloped line and the second is a vertical line, they will intersect at exactly one point. If you plug \(x = 0\) into the first equation to find \(y\):
\[
y = 3(0) + 7 = 7
\]
So the point of intersection is (0, 7), not (-3, 7).
4. **No solution**: It is incorrect to say there is no solution because the two lines do intersect at one unique point.
So, the correct statement about the system of equations is that it has **one solution** at the point (0, 7), with the lines having different slopes and y-intercepts.