To solve this problem, we need to write the equation of a line that is parallel to $5x - 4y = 4$ and passes through the point $(-9, -6)$. Here's the process:

1. Find the slope of the given line: Rewrite the given equation $5x - 4y = 4$ into slope-intercept form $y = mx + b$: $-4y = -5x + 4 \quad \Rightarrow \quad y = \frac{5}{4}x - 1$ The slope $m$ of the line is $\frac{5}{4}$.

2. Determine the equation of the parallel line: A line parallel to this one will have the same slope, $\frac{5}{4}$. Using the point-slope form: $y - y_1 = m(x - x_1)$ where $m = \frac{5}{4}$, $(x_1, y_1) = (-9, -6)$, the equation becomes: $y - (-6) = \frac{5}{4}(x - (-9)) \quad \Rightarrow \quad y + 6 = \frac{5}{4}(x + 9)$

3. Simplify the equation: Distribute $\frac{5}{4}$: $y + 6 = \frac{5}{4}x + \frac{45}{4}$ Subtract 6 from both sides: $y = \frac{5}{4}x + \frac{45}{4} - 6 \quad \Rightarrow \quad y = \frac{5}{4}x - \frac{24}{4} \quad \Rightarrow \quad y = \frac{5}{4}x - 6$

4. Convert to standard form: Multiply through by 4 to eliminate the fraction: $4y = 5x - 24$ Rearrange into standard form: $-5x + 4y = -24 \quad \Rightarrow \quad 5x - 4y = 24$

Final Answer:

The equation of the required line in standard form is: $5x - 4y = 24$

Let me know if you'd like further clarification or have any additional questions!

Expanding Questions:

1. What changes if the line is perpendicular instead of parallel?
2. How do you verify that a point lies on the given line?
3. What is the interpretation of the slope $\frac{5}{4}$ in real-world terms?
4. Can this problem be solved using a matrix approach?
5. How can this method generalize to 3D planes instead of lines?

Tip:

Always convert equations to slope-intercept form first to easily identify the slope for parallel and perpendicular problems.