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

Question

?(1 point)
Responses

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.
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.

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.

Answer 1

To analyze the given system of equations:

1. \( y = 2x + 13 \)
2. \( y = 2x + 1 \)

Both equations are in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• The slope (\( m \)) for both equations is 2, meaning they are parallel lines.
• The y-intercept for the first equation is 13, and for the second equation, it is 1. Since these intercepts are different, the lines do not coincide.

Since the lines are parallel and have different y-intercepts, the system of equations has no solutions (they do not intersect).

Therefore, the correct statement 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.