parallel to 5x+6y=-1 and passes through (3,2)

To find the equation of the line parallel to \( 5x + 6y = -1 \) and passing through the point \( (3, 2) \), we need to follow these steps:

1. **Find the slope of the given line**:  
   The equation of the line is in standard form: \( 5x + 6y = -1 \).  
   To find the slope, we first rewrite the equation in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

   \[
   6y = -5x - 1
   \]
   \[
   y = -\frac{5}{6}x - \frac{1}{6}
   \]
   The slope of this line is \( m = -\frac{5}{6} \).

2. **Use the point-slope form**:  
   Since parallel lines have the same slope, the slope of the new line is also \( m = -\frac{5}{6} \).  
   Now, use the point-slope form of a line equation:
   
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( (x_1, y_1) = (3, 2) \) is the given point and \( m = -\frac{5}{6} \).

   Substituting the values:
   \[
   y - 2 = -\frac{5}{6}(x - 3)
   \]
   
3. **Simplify the equation**:
   Distribute the slope on the right-hand side:
   \[
   y - 2 = -\frac{5}{6}x + \frac{15}{6}
   \]
   \[
   y - 2 = -\frac{5}{6}x + \frac{5}{2}
   \]

   Now, add 2 to both sides:
   \[
   y = -\frac{5}{6}x + \frac{5}{2} + 2
   \]
   \[
   y = -\frac{5}{6}x + \frac{9}{2}
   \]

Thus, the equation of the line parallel to \( 5x + 6y = -1 \) and passing through \( (3, 2) \) is:

\[
y = -\frac{5}{6}x + \frac{9}{2}
\]

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Would you like more details on any step, or do you have any questions? 

Here are 5 related questions to explore this concept further:

1. How do you convert from point-slope form to standard form?
2. What is the difference between parallel and perpendicular lines in terms of slope?
3. Can you find the equation of a line perpendicular to \( 5x + 6y = -1 \)?
4. What happens if a line passes through the origin?
5. How do you find the distance between two parallel lines?

**Tip:** Parallel lines always have the same slope but different intercepts!

Learn how to find the equation of a line parallel to 5x + 6y = -1 and passing through the point (3,2). This solution covers algebraic concepts such as slope-intercept form, point-slope form, and the properties of parallel lines. Suitable for students in Grades 9-11.

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