Question
Which of the graphs with no solution correctly shows the system of equations y=3x+4 and −3x+y=8 ?(1 point)
Answers
GPT-4o mini
To determine which graph represents a system of equations with no solution for the equations \(y = 3x + 4\) and \(-3x + y = 8\), we need to analyze the equations.
1. The first equation, \(y = 3x + 4\), is in slope-intercept form, where the slope \(m = 3\) and the y-intercept \(b = 4\). This means the line goes upwards with a steep slope starting from the point (0, 4).
2. The second equation, \(-3x + y = 8\), can be rearranged into slope-intercept form:
\[
y = 3x + 8
\]
Here, the slope is also \(m = 3\), but the y-intercept is \(b = 8\).
Since both equations have the same slope (3) but different y-intercepts (4 and 8), these lines are parallel and will never intersect, indicating that there is no solution to the system of equations.
To correctly show this scenario on a graph:
- The first line \(y = 3x + 4\) should cross the y-axis at (0, 4) and have a slope of 3.
- The second line \(y = 3x + 8\) should cross the y-axis at (0, 8) and also have a slope of 3.
Thus, the correct graph will depict two parallel lines, neither of which intersects the other, indicating that there is no solution.
1. The first equation, \(y = 3x + 4\), is in slope-intercept form, where the slope \(m = 3\) and the y-intercept \(b = 4\). This means the line goes upwards with a steep slope starting from the point (0, 4).
2. The second equation, \(-3x + y = 8\), can be rearranged into slope-intercept form:
\[
y = 3x + 8
\]
Here, the slope is also \(m = 3\), but the y-intercept is \(b = 8\).
Since both equations have the same slope (3) but different y-intercepts (4 and 8), these lines are parallel and will never intersect, indicating that there is no solution to the system of equations.
To correctly show this scenario on a graph:
- The first line \(y = 3x + 4\) should cross the y-axis at (0, 4) and have a slope of 3.
- The second line \(y = 3x + 8\) should cross the y-axis at (0, 8) and also have a slope of 3.
Thus, the correct graph will depict two parallel lines, neither of which intersects the other, indicating that there is no solution.