To find the equation of a line parallel to $3y = 9 - 4x$ that passes through the point $(6, -9)$, follow these steps:

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Step 1: Rewrite the given line in slope-intercept form

The given equation is: $3y = 9 - 4x$ Divide through by 3: $y = -\frac{4}{3}x + 3$ This is in the form $y = mx + b$, where the slope $m = -\frac{4}{3}$.

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Step 2: Use the slope for the parallel line

Since parallel lines have the same slope, the new line will also have a slope of $-\frac{4}{3}$.

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Step 3: Use the point-slope formula

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Here, $(x_1, y_1) = (6, -9)$ and $m = -\frac{4}{3}$. Substituting these values: $y - (-9) = -\frac{4}{3}(x - 6)$ $y + 9 = -\frac{4}{3}x + 8$

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Step 4: Simplify the equation

Subtract 9 from both sides: $y = -\frac{4}{3}x - 1$

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Final Equation

The equation of the parallel line is: $y = -\frac{4}{3}x - 1$

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Would you like a detailed explanation of any step or additional examples? Here are some related questions to consider:

1. What is the process for finding a perpendicular line instead of a parallel one?
2. How do you determine the distance between two parallel lines?
3. Can you derive the general form of the equation from the final result?
4. How can you graph both lines to visualize their parallelism?
5. What happens to the slope when the line is horizontal or vertical?

Tip: Always rewrite the given equation in slope-intercept form ($y = mx + b$) to easily identify the slope.