To determine which of the given lines is parallel to the line \( y = 34x - 2 \), we need to identify the slope of that line. The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope.
In this case, the slope (\( m \)) of the line \( y = 34x - 2 \) is \( 34 \). For two lines to be parallel, they must have the same slope.
Now, let's rewrite each of the given options in slope-intercept form (\( y = mx + b \)) to find their slopes:
Option A:
\( 4x - 3y = 6 \)
Rearranging gives:
\(-3y = -4x + 6\)
Dividing everything by -3:
\( y = \frac{4}{3}x - 2 \)
Slope = \( \frac{4}{3} \)
Option B:
\( 3x - 4y = 12 \)
Rearranging gives:
\(-4y = -3x + 12\)
Dividing everything by -4:
\( y = \frac{3}{4}x - 3 \)
Slope = \( \frac{3}{4} \)
Option C:
\( 4x - 3y = 12 \)
Rearranging gives:
\(-3y = -4x + 12\)
Dividing everything by -3:
\( y = \frac{4}{3}x - 4 \)
Slope = \( \frac{4}{3} \)
Option D:
\( 3x - 4y = 8 \)
Rearranging gives:
\(-4y = -3x + 8\)
Dividing everything by -4:
\( y = \frac{3}{4}x - 2 \)
Slope = \( \frac{3}{4} \)
Now we compare the slopes:
- Option A: \( \frac{4}{3} \)
- Option B: \( \frac{3}{4} \)
- Option C: \( \frac{4}{3} \)
- Option D: \( \frac{3}{4} \)
None of the options have a slope of \( 34 \). Therefore, none of the lines given in A, B, C, or D are parallel to the line \( y = 34x - 2 \).
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