To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we first need to convert each equation into the slope-intercept form \( y = mx + b \), where \( m \) represents the slope.
Pair 1:
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Equation 1: \( y - 6 = \frac{1}{4}(x - 2) \)
We can rewrite this in slope-intercept form: \[ y - 6 = \frac{1}{4}x - \frac{1}{2} \] \[ y = \frac{1}{4}x + 5.5 \] Slope \( m_1 = \frac{1}{4} \)
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Equation 2: \( y = -4x - 10 \)
Slope \( m_2 = -4 \)
To check if they are parallel or perpendicular:
- Two lines are parallel if their slopes are equal: \( m_1 = m_2 \).
- Two lines are perpendicular if the product of their slopes is \(-1\): \( m_1 \cdot m_2 = -1 \).
Calculating the product: \[ \frac{1}{4} \cdot (-4) = -1 \] Since their slopes satisfy the condition for being perpendicular, these two lines are perpendicular.
Pair 2:
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Equation 1: \( 2x - y = 7 \)
To get the slope-intercept form: \[ -y = -2x + 7 \] \[ y = 2x - 7 \] Slope \( m_1 = 2 \)
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Equation 2: \( 3x - 4y = -8 \)
Rearranging: \[ -4y = -3x - 8 \] \[ y = \frac{3}{4}x + 2 \] Slope \( m_2 = \frac{3}{4} \)
Checking for parallel or perpendicular:
- Product of slopes: \[ 2 \cdot \frac{3}{4} = \frac{3}{2} \] Since the slopes are not equal and their product is not \(-1\), these lines are intersecting.
Pair 3:
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Equation 1: \( y = -\frac{2}{3}x + 7 \)
Slope \( m_1 = -\frac{2}{3} \)
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Equation 2: \( -2x - 3y = 12 \)
Rearranging: \[ -3y = 2x + 12 \] \[ y = -\frac{2}{3}x - 4 \] Slope \( m_2 = -\frac{2}{3} \)
Checking for parallel or perpendicular:
- Product of slopes: \[ -\frac{2}{3} \cdot -\frac{2}{3} = \frac{4}{9} \] Since the slopes are equal, these lines are parallel.
Summary:
- First pair: Perpendicular
- Second pair: Intersecting
- Third pair: Parallel