Which statement best describes the system of equations {y=−3x+7y=−3x−7

Question

?(1 point)
Responses

The equations have different slopes, but they share the same y-intercept, (0,−3). Thus, the system has one solution at (0,−3).
The equations have different slopes, but they share the same y -intercept, left parenthesis 0 comma negative 3 right parenthesis . Thus, the system has one solution at left parenthesis 0 comma negative 3 right parenthesis .

Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel.
Both equations have a slope of negative 3 , but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel.

Both equations have a slope of −3, and they share the same y-intercept. Thus, the system has an infinite number of solutions because they are identical lines.
Both equations have a slope of negative 3 , and they share the same y -intercept. Thus, the system has an infinite number of solutions because they are identical lines.

The equations have different slopes and different y-intercepts. Thus, the system has one solution at (−3,7).
The equations have different slopes and different y -intercepts. Thus, the system has one solution at left parenthesis negative 3 comma 7 right parenthesis .

Answer 1

To analyze the system of equations \(y = -3x + 7\) and \(y = -3x - 7\):

1. Both equations are in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
2. The slope of both equations is \(-3\), indicating that the lines are parallel.
3. The y-intercept of the first equation is \(7\) (at the point \((0, 7)\)), and the y-intercept of the second equation is \(-7\) (at the point \((0, -7)\)).

Since the two lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, there are no solutions to the system of equations.

Thus, the statement that best describes the system of equations is:

"Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel."

So, the correct choice would be:

"Both equations have a slope of negative 3, but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel."