To find a line that is parallel to the given equation \(-4x + 3y = 12\), we first need to determine the slope of that line. We can rewrite the equation in slope-intercept form (y = mx + b).
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Start with the original equation: \[ -4x + 3y = 12 \]
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Solve for y: \[ 3y = 4x + 12 \] \[ y = \frac{4}{3}x + 4 \]
The slope \(m\) of the line is \(\frac{4}{3}\).
Since parallel lines have the same slope, the equation of a line parallel to the given line will also have a slope of \(\frac{4}{3}\).
Now, let's look at the options provided:
- \(y = \frac{4}{3}x + 6\) (this has the correct slope)
- \(y = -\frac{4}{3}x + 4\) (this has a negative slope, not parallel)
- \(y = 3x + 12\) (this slope is 3, not parallel)
- \(y = 4x + 8\) (this slope is 4, not parallel)
The only equation that is parallel to the original line is: \[ \text{y = } \frac{4}{3}x + 6 \]
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