To determine which line is parallel to the line given by the equation \( y = 34x - 2 \), we need to find the slope of the given line. The equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope.
From the equation \( y = 34x - 2 \), we can see that the slope \( m \) is 34. For lines to be parallel, they must have the same slope.
Now let's convert the equations of the lines provided in the responses to slope-intercept form and identify their slopes.
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For \( 4x - 3y = 6 \): \[ -3y = -4x + 6 \quad \Rightarrow \quad y = \frac{4}{3}x - 2 \] Slope = \( \frac{4}{3} \)
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For \( 3x - 4y = 12 \): \[ -4y = -3x + 12 \quad \Rightarrow \quad y = \frac{3}{4}x - 3 \] Slope = \( \frac{3}{4} \)
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For \( 4x - 3y = 12 \): \[ -3y = -4x + 12 \quad \Rightarrow \quad y = \frac{4}{3}x - 4 \] Slope = \( \frac{4}{3} \)
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For \( 3x - 4y = 8 \): \[ -4y = -3x + 8 \quad \Rightarrow \quad y = \frac{3}{4}x - 2 \] Slope = \( \frac{3}{4} \)
Now, let's compare the slopes:
- Slopes are \( \frac{4}{3} \) (for two equations) and \( \frac{3}{4} \) (for two equations).
Only the lines corresponding to \( 4x - 3y = 6 \) and \( 4x - 3y = 12 \) have the same slope \( \frac{4}{3} \).
Since \( 34 \neq \frac{4}{3} \) and \( \frac{3}{4} \) is also far from \( 34 \), none of these lines is parallel to \( y = 34x - 2 \).
Thus, none of the given lines is parallel to \( y = 34x - 2 \).
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