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 \( (3,2) \), 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!

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

Learn how to find the equation of a line parallel to 5x + 6y = -1 that passes through the point (3,2). This step-by-step solution explains the use of slope-intercept form, point-slope form, and the concept of parallel lines. Suitable for students in Grades 9-11.

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