To determine whether the given pairs of lines are parallel, perpendicular, or intersecting, we can first convert each line into the slope-intercept form (y = mx + b), where m is the slope.
Pair 1:
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Line 1: \(2x - y = 7\)
Rearranging: \[ -y = -2x + 7 \ y = 2x - 7 \]
- Slope (m1) = 2
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Line 2: \(3x - 4y = -8\)
Rearranging: \[ -4y = -3x - 8 \ y = \frac{3}{4}x + 2 \]
- Slope (m2) = \(\frac{3}{4}\)
Comparison:
- Are the slopes equal? No, \(2 \neq \frac{3}{4}\) (not parallel).
- Are the slopes negative reciprocals? No, \(2 \cdot \frac{3}{4} \neq -1\) (not perpendicular).
Conclusion: Lines intersect.
Pair 2:
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Line 1: \(y = -\frac{2}{3}x + 7\)
- Slope (m1) = \(-\frac{2}{3}\)
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Line 2: \(-2x - 3y = 7\)
Rearranging: \[ -3y = 2x + 7 \ y = -\frac{2}{3}x - \frac{7}{3} \]
- Slope (m2) = \(-\frac{2}{3}\)
Comparison:
- Are the slopes equal? Yes, \(-\frac{2}{3} = -\frac{2}{3}\) (parallel).
Conclusion: Lines are parallel.
Pair 3:
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Line 1: \(y - 6 = \frac{1}{4}(x + 2)\)
Rearranging: \[ y - 6 = \frac{1}{4}x + \frac{1}{2} \ y = \frac{1}{4}x + \frac{13}{2} \]
- Slope (m1) = \(\frac{1}{4}\)
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Line 2: \(y = -4x - 10\)
- Slope (m2) = \(-4\)
Comparison:
- Are the slopes equal? No, \(\frac{1}{4} \neq -4\) (not parallel).
- Are the slopes negative reciprocals? Yes, \(\frac{1}{4} \cdot (-4) = -1\) (perpendicular).
Conclusion: Lines are perpendicular.
Summary:
- Pair 1: Intersecting
- Pair 2: Parallel
- Pair 3: Perpendicular