To find an equation of a line that is parallel to the given line \(-4x + 3y = 12\), we first need to determine the slope of the given line.
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Convert the equation to slope-intercept form (y = mx + b):
- Start with the given equation: \[ -4x + 3y = 12 \]
- Solve for \(y\): \[ 3y = 4x + 12 \] \[ y = \frac{4}{3}x + 4 \]
From this, we can see that the slope (\(m\)) of the line is \(\frac{4}{3}\).
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Lines that are parallel have the same slope: Therefore, any line that is parallel to the given line will also have a slope of \(\frac{4}{3}\).
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Write the equation of the parallel line: Let's say we want the parallel line to pass through a specific point, for example, the point \((0, 0)\). Using the slope-intercept form, we can write:
\[ y = \frac{4}{3}x + b \]
Since it passes through the origin \((0, 0)\), we can substitute \(x = 0\) and \(y = 0\) to find \(b\): \[ 0 = \frac{4}{3}(0) + b \Rightarrow b = 0 \]
After this, the equation of the line parallel to the original line that passes through the origin is:
\[ y = \frac{4}{3}x \]
If you want a parallel line through a different point, you would just substitute that point into the equation to find the appropriate \(b\) (y-intercept) and formulate the equation accordingly. For instance, to find a parallel line through the point \((1, 2)\), you would do the following:
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Start with the slope: \[ y - 2 = \frac{4}{3}(x - 1) \]
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Rearranging gives: \[ y - 2 = \frac{4}{3}x - \frac{4}{3} \] \[ y = \frac{4}{3}x + 2 - \frac{4}{3} \] \[ y = \frac{4}{3}x + \frac{2}{3} \]
This gives you another equation for a line parallel to the original line, but you can choose any point to get your desired parallel line.