Which statement best describes the system of equations {y=2x−132x−y=13

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 negative 13 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.

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)
.

Answer 1

Let's analyze the system of equations given:

1. The first equation is \( y = 2x - 13 \).
2. To analyze the second equation \( 2x - y = 13 \), we can rearrange it to slope-intercept form (\( y = mx + b \)):
   • Starting with \( 2x - y = 13 \), we can isolate \( y \): \[ -y = -2x + 13 \quad \text{(subtracting } 2x \text{ from both sides)} \] \[ y = 2x - 13 \quad \text{(multiplying both sides by -1)} \]

Now we can see both equations:

• \( y = 2x - 13 \)
• \( y = 2x - 13 \) (second equation rearranged)

Both equations are identical, which means they represent the same line.

Thus, the correct interpretation of the system of equations is:

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.