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Which statement best describes the system of equations {y=2x+13y=2x+1 ?(1 point)

Responses

The equations have different slopes, but they share the same y-intercept, (0,2). Thus, the system has one solution at (0,2).
The equations have different slopes, but they share the same y -intercept, left parenthesis 0 comma 2 right parenthesis . Thus, the system has one solution at left parenthesis 0 comma 2 right parenthesis .

Both equations have a slope of 2, 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 2, but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel.

The equations have different slopes and different y-intercepts. Thus, the system has one solution at (2,13).
The equations have different slopes and different y -intercepts. Thus, the system has one solution at left parenthesis 2 comma 13 right parenthesis .

Both equations have a slope of 2, and they share the same y-intercept. Thus, the system has an infinite number of solutions because they are identical lines.

1 answer

To analyze the given system of equations \( y = 2x + 13 \) and \( y = 2x + 1 \):

  1. Slopes: Both equations have the same slope of 2. This indicates that the lines are parallel.

  2. Y-Intercepts: The first equation has a y-intercept of 13 (the point (0, 13)), while the second equation has a y-intercept of 1 (the point (0, 1)). Therefore, they do not share the same y-intercept.

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

The correct description of the system is:

"Both equations have a slope of 2, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel."

This matches the response:

"Both equations have a slope of 2, but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel."