To find the equation of the line parallel to $3x + 5y = 11$ that passes through the point $(15, 4)$, we need to follow these steps:

Step 1: Find the Slope of the Given Line

First, let's determine the slope of the line given by the equation $3x + 5y = 11$.

1. Rewrite the equation in slope-intercept form $(y = mx + b)$:

   Starting with: $3x + 5y = 11$

   Solving for $y$: $5y = -3x + 11$ $y = -\frac{3}{5}x + \frac{11}{5}$

   The slope $m$ of the line is $-\frac{3}{5}$.

Step 2: Use the Point-Slope Form to Find the New Line

Since parallel lines have the same slope, the slope of the new line is also $-\frac{3}{5}$. We use the point-slope form of the line equation to find the equation of the line that passes through the point $(15, 4)$.

The point-slope form is: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Plugging in the point $(15, 4)$ and the slope $-\frac{3}{5}$: $y - 4 = -\frac{3}{5}(x - 15)$

Step 3: Simplify the Equation

Distribute the slope and simplify: $y - 4 = -\frac{3}{5}x + \frac{3}{5} \times 15$ $y - 4 = -\frac{3}{5}x + 9$

Now, add 4 to both sides to isolate $y$: $y = -\frac{3}{5}x + 9 + 4$ $y = -\frac{3}{5}x + 13$

Final Equation

Thus, the equation of the line parallel to $3x + 5y = 11$ and passing through the point $(15, 4)$ is: $\boxed{y = -\frac{3}{5}x + 13}$

Additional Questions to Explore

1. What would be the equation of a line perpendicular to $3x + 5y = 11$ that passes through the point $(15, 4)$?
2. How would the equation change if the line had to pass through a different point, say $(7, -2)$?
3. What are the x-intercept and y-intercept of the line $y = -\frac{3}{5}x + 13$?
4. How do you find the distance between two parallel lines?
5. Can you find the midpoint of a segment connecting two points on this line?

Tip: When working with equations of lines, always make sure to clearly identify the slope and use appropriate forms (slope-intercept, point-slope) based on the information given.