To determine whether the lines given by the equations \( q: y = -\frac{78}{87}x - 11 \) and \( r: y = -\frac{87}{78}x + \frac{68}{13} \) are parallel or perpendicular, we need to look at their slopes.
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Identify the slopes of the lines:
- For line \( q \), the slope \( m_q = -\frac{78}{87} \).
- For line \( r \), the slope \( m_r = -\frac{87}{78} \).
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Simplify the slopes:
- The slope of line \( q \): \[ m_q = -\frac{78}{87} = -\frac{6}{7} \quad \text{(after dividing by 13)} \]
- The slope of line \( r \): \[ m_r = -\frac{87}{78} = -\frac{87}{78} = -\frac{87}{78} \quad \text{(already in simplest terms)} \]
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Check if the lines are parallel:
- Lines are parallel if their slopes are equal. Here, \[ \text{since} ; -\frac{6}{7} \neq -\frac{87}{78}, \text{ the lines are not parallel.} \]
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Check if the lines are perpendicular:
- Lines are perpendicular if the product of their slopes equals -1: \[ m_q \times m_r = \left(-\frac{6}{7}\right) \times \left(-\frac{87}{78}\right) \]
- Calculate the product: \[ = \frac{6 \times 87}{7 \times 78} = \frac{522}{546} \]
- Simplifying \( \frac{522}{546} = \frac{87}{91} \) (dividing both numerator and denominator by 6) which is not equal to -1.
Thus, since the slopes are neither equal (not parallel) nor do they satisfy the condition for being perpendicular, we conclude:
The lines \( q \) and \( r \) are neither parallel nor perpendicular.