To find an equation of a line that is perpendicular to the given equation \(-4 + 3y = 12\), we first need to rewrite this equation in slope-intercept form (i.e., \(y = mx + b\)), where \(m\) is the slope.
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Rearranging the given equation:
\[ -4 + 3y = 12 \]
We can add 4 to both sides:
\[ 3y = 16 \]
Now, divide by 3:
\[ y = \frac{16}{3} \]
Since this line is horizontal, its slope is \(0\).
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Finding the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. However, since we have a horizontal line (slope of \(0\)), the slope of the perpendicular line will be undefined, which corresponds to a vertical line.
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Looking at the choices for vertical lines:
None of the options listed represent vertical lines. Let's break down each option to find the slope:
- \(y = 3x \times 2\) or \(y = 6x\) (slope = 6)
- \(y = -\frac{3}{4}x + 6\) (slope = -\(\frac{3}{4}\))
- \(y = \frac{4}{3}x + 6\) (slope = \(\frac{4}{3}\))
- \(y = 4x + 8\) (slope = 4)
Since none of the options represent a vertical line, it seems that none of them is correct for being perpendicular to the original horizontal line.
Conclusion: There is no provided option that correctly represents a line perpendicular to the initial equation of a horizontal line. A perpendicular line would be vertical, which is not included in your choices.