To determine which line is parallel to the line given by the equation \(y = \frac{3}{4}x - 2\), we need to analyze the slope of the original line. The slope of the line in slope-intercept form \(y = mx + b\) is \(m\).
In this case, the slope \(m\) of the line \(y = \frac{3}{4}x - 2\) is \(\frac{3}{4}\). Two lines are parallel if they have the same slope.
Now, let's convert the equations provided in the responses to slope-intercept form (if necessary) to find their slopes:
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For \(3x - 4y = 12\): \[ -4y = -3x + 12 \ y = \frac{3}{4}x - 3 \] (Slope = \(\frac{3}{4}\))
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For \(4x - 3y = 6\): \[ -3y = -4x + 6 \ y = \frac{4}{3}x - 2 \] (Slope = \(\frac{4}{3}\))
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For \(3x - 4y = 8\): \[ -4y = -3x + 8 \ y = \frac{3}{4}x - 2 \] (Slope = \(\frac{3}{4}\))
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For \(4x - 3y = 12\): \[ -3y = -4x + 12 \ y = \frac{4}{3}x - 4 \] (Slope = \(\frac{4}{3}\))
Now we can compare the slopes:
- Lines from responses 1 and 3 have the same slope of \(\frac{3}{4}\), so they are parallel to the original line \(y = \frac{3}{4}x - 2\).
Thus, the correct answers are:
- 3x - 4y = 12
- 3x - 4y = 8
You would select either of those lines for the response indicating which are parallel to the given line. If only one option can be selected, you can choose either of the two since they both have the same slope.