To determine if the given lines are parallel, we need to rewrite each equation in the slope-intercept form (y = mx + b), where m represents the slope. If two lines have the same slope, they are parallel.
Let's analyze each pair of lines:
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Lines: \(2x + 2y = 2\) and \(2y = -2x + 3\)
First equation: \[ 2y = -2x + 2 \implies y = -x + 1 \quad \text{(slope = -1)} \]
Second equation: \[ 2y = -2x + 3 \implies y = -x + \frac{3}{2} \quad \text{(slope = -1)} \]
Both lines have the same slope \(-1\), so they are parallel.
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Lines: \(y = 3x + 2\) and \(y + 3x = -5\)
First equation: \[ y = 3x + 2 \quad \text{(slope = 3)} \]
Second equation: \[ y = -3x - 5 \quad \text{(slope = -3)} \]
The slopes are different (3 and -3), so they are not parallel.
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Lines: \(y = 2x - 3\) and \(6x + 3y - 9 = 0\)
First equation: \[ y = 2x - 3 \quad \text{(slope = 2)} \]
Second equation: \[ 3y = -6x + 9 \implies y = -2x + 3 \quad \text{(slope = -2)} \]
The slopes are different (2 and -2), so they are not parallel.
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Lines: \(y - 2x + 6 = 0\) and \(4y = 8x + 1\)
First equation: \[ y = 2x - 6 \quad \text{(slope = 2)} \]
Second equation: \[ y = 2x + \frac{1}{4} \quad \text{(slope = 2)} \]
Both lines have the same slope \(2\), so they are parallel.
Summary:
- The lines \(2x + 2y = 2\) and \(2y = -2x + 3\) are parallel.
- The lines \(y = 3x + 2\) and \(y + 3x = -5\) are not parallel.
- The lines \(y = 2x - 3\) and \(6x + 3y - 9 = 0\) are not parallel.
- The lines \(y - 2x + 6 = 0\) and \(4y = 8x + 1\) are parallel.