To compare the equations \(-\frac{1}{2}x + y = 8\) and \(\frac{1}{2}y = \frac{1}{4}x - 4\), we first need to rewrite both equations in slope-intercept form (y = mx + b) where m is the slope and b is the y-intercept.
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Rearranging the first equation:
\[ -\frac{1}{2}x + y = 8 \]
Add \(\frac{1}{2}x\) to both sides:
\[ y = \frac{1}{2}x + 8 \]
Now we can identify the slope (m) and y-intercept (b):
- Slope (m) = \(\frac{1}{2}\)
- y-intercept (b) = \(8\)
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Rearranging the second equation:
\[ \frac{1}{2}y = \frac{1}{4}x - 4 \]
Multiply both sides by 2 to isolate \(y\):
\[ y = \frac{1}{2}x - 8 \]
Now we can identify the slope (m) and y-intercept (b):
- Slope (m) = \(\frac{1}{2}\)
- y-intercept (b) = \(-8\)
Now we can compare the two equations:
- Both equations have the same slope: \(\frac{1}{2}\).
- The y-intercepts are different: \(8\) and \(-8\).
Now, let's analyze the choices provided:
A. The equations have the same slope - True B. The equations have no common characteristics - False (they have the same slope) C. The equations have the same y-intercept or initial value - False (they have different y-intercepts) D. The equations have the same slope, and are parallel lines - True (since they have the same slope but different y-intercepts).
Thus, the appropriate choices from the options given are A and D.