To determine which lines are parallel to the line given by the equation \( y = \frac{3}{4}x - 2 \), we need to identify the slope of the line.
The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. From the given equation \( y = \frac{3}{4}x - 2 \), we can see that the slope \( m \) is \( \frac{3}{4} \).
For a line to be parallel, it must have the same slope. Therefore, we need to check the slopes of the lines represented by the equations given in the possible answers.
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For \( 3x - 4y = 8 \):
- Rearranging to the slope-intercept form: \[ -4y = -3x + 8 \implies y = \frac{3}{4}x - 2 \]
- The slope is \( \frac{3}{4} \).
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For \( 3x - 4y = 12 \):
- Rearranging to the slope-intercept form: \[ -4y = -3x + 12 \implies y = \frac{3}{4}x - 3 \]
- The slope is \( \frac{3}{4} \).
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For \( 4x - 3y = 6 \):
- Rearranging to the slope-intercept form: \[ -3y = -4x + 6 \implies y = \frac{4}{3}x - 2 \]
- The slope is \( \frac{4}{3} \).
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For \( 4x + 3y = 8 \):
- Rearranging to the slope-intercept form: \[ 3y = -4x + 8 \implies y = -\frac{4}{3}x + \frac{8}{3} \]
- The slope is \( -\frac{4}{3} \).
Now, we can summarize:
- The lines 1 and 2 have the same slope \( \frac{3}{4} \) as \( y = \frac{3}{4}x - 2 \), which means these lines are parallel.
- The lines 3 and 4 have different slopes (\( \frac{4}{3} \) and \( -\frac{4}{3} \)), so they are not parallel to the given line.
Final Answer: The lines that are parallel to \( y = \frac{3}{4}x - 2 \) are:
- \( 3x - 4y = 8 \)
- \( 3x - 4y = 12 \)