To determine whether the pairs of lines are intersecting, perpendicular, or parallel, we need to analyze their slopes.
Pair A:
Equations:
- \( 2x - y = 7 \)
- \( 3x - 4y = -8 \)
Convert to slope-intercept form (y = mx + b)
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For \( 2x - y = 7 \): \[ -y = -2x + 7 \implies y = 2x - 7 \] (Slope, \( m_1 = 2 \))
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For \( 3x - 4y = -8 \): \[ -4y = -3x - 8 \implies y = \frac{3}{4}x + 2 \] (Slope, \( m_2 = \frac{3}{4} \))
Compare slopes:
- \( m_1 = 2 \) and \( m_2 = \frac{3}{4} \)
Since the slopes are different, the lines are intersecting.
Pair B:
Equations:
- \( y - 6 = \frac{1}{4}(x + 2) \)
- \( y = -4x - 10 \)
Convert to slope-intercept form:
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Rearranging \( y - 6 = \frac{1}{4}(x + 2) \): \[ y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5 \] (Slope, \( m_1 = \frac{1}{4} \))
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The second equation is already in slope-intercept form \( y = -4x - 10 \): (Slope, \( m_2 = -4 \))
Compare slopes:
- \( m_1 = \frac{1}{4} \) and \( m_2 = -4 \)
Since the slopes are different and \( m_1 \cdot m_2 = \frac{1}{4} \cdot (-4) = -1 \), the lines are perpendicular.
Pair C:
Equations:
- \( y = -\frac{2}{3}x + 7 \)
- \( -2x - 3y = 12 \)
Convert to slope-intercept form:
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The first line is already in slope-intercept form: (Slope, \( m_1 = -\frac{2}{3} \))
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Rearranging \( -2x - 3y = 12 \): \[ -3y = 2x + 12 \implies y = -\frac{2}{3}x - 4 \] (Slope, \( m_2 = -\frac{2}{3} \))
Compare slopes:
- \( m_1 = -\frac{2}{3} \) and \( m_2 = -\frac{2}{3} \)
Since the slopes are equal, the lines are parallel.
Summary:
- Pair A: Intersecting
- Pair B: Perpendicular
- Pair C: Parallel
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