What is the equation of the line parallel to 3x + 5y = 11 that passes through the point (15, 4)?
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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.

What is the equation of the line parallel to 3x + 5y = 11 that passes through the point (15, 4)?

Learn how to find the equation of a line parallel to 3x + 5y = 11 that passes through the point (15, 4). This problem involves key algebraic concepts like slope-intercept and point-slope forms, making it ideal for students in Grades 8-10.

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