To compare the two equations \(-\frac{1}{2}x + y = 8\) and \(\frac{1}{2}y = \frac{1}{4}x - 4\), we will first 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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For the first equation: \(-\frac{1}{2}x + y = 8\)
Rearranging: \[y = \frac{1}{2}x + 8\] Here, the slope \(m_1 = \frac{1}{2}\) and the y-intercept \(b_1 = 8\).
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For the second equation: \(\frac{1}{2}y = \frac{1}{4}x - 4\)
Multiply both sides by 2 to eliminate the fraction: \[y = \frac{1}{2}x - 8\] Here, the slope \(m_2 = \frac{1}{2}\) and the y-intercept \(b_2 = -8\).
Now we can analyze the slopes and y-intercepts:
- Slopes: Both slopes are the same (\(m_1 = m_2 = \frac{1}{2}\)).
- Y-Intercepts: The y-intercept of the first equation is \(8\) and the second equation is \(-8\).
Now, let's evaluate the options:
A. The equations have the same slope: True (both have a slope of \(\frac{1}{2}\)).
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 (they have the same slope and different y-intercepts, indicating they are parallel lines).
Since both statements A and D are true, but the question asks for one answer, the best choice that succinctly captures the relationship is:
D. the equations have the same slope, and are parallel lines.