To compare the given equations:
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First Equation: \(-\frac{1}{2}x + y = 8\)
- Rearranging to slope-intercept form \(y = mx + b\): \[ y = \frac{1}{2}x + 8 \]
- Here, the slope (\(m\)) is \(\frac{1}{2}\) and the y-intercept (\(b\)) is \(8\).
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Second Equation: \(\frac{1}{2}y = \frac{1}{2}x - 4\)
- Multiplying both sides by \(2\) to eliminate the fraction: \[ y = x - 8 \]
- Here, the slope (\(m\)) is \(1\) and the y-intercept (\(b\)) is \(-8\).
Now, let's analyze the given options:
A. The equations have the same slope: This is false because the slopes are \(\frac{1}{2}\) and \(1\), which are not the same.
B. The equations have no common characteristics: This is false. While their slopes and y-intercepts are different, they are both linear equations.
C. The equations have the same y-intercept or initial value: This is false. The y-intercepts are \(8\) and \(-8\), which are not the same.
D. The equations have the same slope, and are parallel lines: This is false because, as stated previously, their slopes are different.
Since all options appear to be incorrect, it seems like there could be a misunderstanding. Double-checking indicates that the slopes are indeed different, confirming there is no valid option among A, B, C, or D that aligns with the characteristics of the equations.
It appears that none of the provided statements are true based on the calculations. Would you like to modify or explore a different set of characteristics?