Compare the equations −1/2 x + y = 8

Question

and 1/2 y = 1/4 x − 4
. Identify the true statement.(1 point)
Responses

The equations have the same slope and y
-intercept.
The equations have the same slope and y -intercept.

The equations have the same y
-intercept, or initial value.
The equations have the same y -intercept, or initial value.

The equations have the same slopes and are parallel lines.
The equations have the same slopes and are parallel lines.

The equations have no common characteristics.

Answer 1

Let's first rewrite both equations in slope-intercept form (y = mx + b) to compare their slopes and y-intercepts.

1. For the first equation: \[ -\frac{1}{2}x + y = 8 \] We can isolate \(y\): \[ y = \frac{1}{2}x + 8 \] Here, the slope (m) is \(\frac{1}{2}\) and the y-intercept (b) is \(8\).

2. For the second equation: \[ \frac{1}{2}y = \frac{1}{4}x - 4 \] Multiplying both sides by \(2\) to eliminate the fraction gives: \[ y = \frac{1}{2}x - 8 \] Here, the slope (m) is also \(\frac{1}{2}\) and the y-intercept (b) is \(-8\).

Now we can compare:

• Both equations have the same slope, which is \(\frac{1}{2}\).
• The y-intercepts are different: one is \(8\) and the other is \(-8\).

Based on this analysis, the correct response is:

The equations have the same slopes and are parallel lines.