To determine if the lines given by the equations \( y = -\frac{1}{4}x \) and \( y = 4x - 5 \) are parallel, perpendicular, or neither, we need to look at their slopes.
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Finding the slope of the first line \( y = -\frac{1}{4}x \):
- The slope \( m_1 \) of this line is \(-\frac{1}{4}\).
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Finding the slope of the second line \( y = 4x - 5 \):
- The slope \( m_2 \) of this line is \(4\).
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Comparison of the slopes:
- Two lines are parallel if their slopes are equal, i.e., \( m_1 = m_2 \).
- Two lines are perpendicular if the product of their slopes is \(-1\), i.e., \( m_1 \cdot m_2 = -1\).
Now, let's check:
- The slopes are \( m_1 = -\frac{1}{4} \) and \( m_2 = 4 \).
Calculating the product of the slopes: \[ m_1 \cdot m_2 = -\frac{1}{4} \cdot 4 = -1 \]
Since the product of the slopes is \(-1\), the lines are perpendicular.
Conclusion: The lines \( y = -\frac{1}{4}x \) and \( y = 4x - 5 \) are perpendicular.
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