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

Step 1: Determine the slope of the given line

The given equation $5x + 6y = -1$ is in standard form. To find the slope, we first rewrite it in slope-intercept form $y = mx + b$, where $m$ is the slope:

$5x + 6y = -1$ Solve for $y$:

$6y = -5x - 1$

$y = -\frac{5}{6}x - \frac{1}{6}$

The slope of the given line is $m = -\frac{5}{6}$.

Step 2: Use the slope-point form

Lines that are parallel have the same slope, so the new line will also have a slope of $m = -\frac{5}{6}$. Now, use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Where $(x_1, y_1) = (3, 2)$ and $m = -\frac{5}{6}$. Substituting these values:

$y - 2 = -\frac{5}{6}(x - 3)$

Step 3: Simplify the equation

Distribute the slope on the right-hand side:

$y - 2 = -\frac{5}{6}x + \frac{15}{6}$

Simplify:

$y - 2 = -\frac{5}{6}x + \frac{5}{2}$

Now, add 2 to both sides:

$y = -\frac{5}{6}x + \frac{5}{2} + 2$

Convert 2 to a fraction:

$y = -\frac{5}{6}x + \frac{5}{2} + \frac{4}{2}$

Simplify:

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

Final Answer:

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

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

Would you like more details or have any questions?

Related Questions:

1. How do you find the slope of a line from an equation in standard form?
2. What is the slope-point form of a linear equation?
3. How do you find the equation of a perpendicular line?
4. How can you convert a line equation from slope-intercept form to standard form?
5. What is the significance of parallel and perpendicular slopes?

Tip:

Parallel lines always have the same slope, but different y-intercepts!