To determine which of the given lines is parallel to the line \(y = 34x - 2\), we need to find the slope of the original line.
The equation \(y = 34x - 2\) is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is \(34\).
Two lines are parallel if they have the same slope. Therefore, we need to convert the given equations into slope-intercept form (if they are not already) and compare their slopes.
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For \(3x - 4y = 8\): \[ -4y = -3x + 8 \implies y = \frac{3}{4}x - 2 \quad \text{(slope is } \frac{3}{4}\text{)} \]
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For \(4x - 3y = 12\): \[ -3y = -4x + 12 \implies y = \frac{4}{3}x - 4 \quad \text{(slope is } \frac{4}{3}\text{)} \]
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For \(4x - 3y = 6\): \[ -3y = -4x + 6 \implies y = \frac{4}{3}x - 2 \quad \text{(slope is } \frac{4}{3}\text{)} \]
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For \(3x - 4y = 12\): \[ -4y = -3x + 12 \implies y = \frac{3}{4}x - 3 \quad \text{(slope is } \frac{3}{4}\text{)} \]
Now, we compare the slopes:
- The slope of \(y = 34x - 2\) is \(34\).
- The slope of \(3x - 4y = 8\) is \(\frac{3}{4}\).
- The slope of \(4x - 3y = 12\) is \(\frac{4}{3}\).
- The slope of \(4x - 3y = 6\) is \(\frac{4}{3}\).
- The slope of \(3x - 4y = 12\) is \(\frac{3}{4}\).
None of the given lines have a slope of \(34\), which means none of them are parallel to the line \(y = 34x - 2\).
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