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To find the equation of a line perpendicular to \( y = \frac{5}{6}x + \frac{5}{6} \) that passes through the point \( (-1, 9) \), we’ll go through these steps:

### Step 1: Identify the Slope of the Given Line
The given line equation is in slope-intercept form \( y = mx + b \), where \( m = \frac{5}{6} \) is the slope.

### Step 2: Find the Slope of the Perpendicular Line
For two lines to be perpendicular, their slopes must be negative reciprocals. Therefore, if the slope of the given line is \( \frac{5}{6} \), the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{6}{5}
\]

### Step 3: Use Point-Slope Form to Find the Equation
The line we are looking for passes through \( (-1, 9) \) and has a slope of \( -\frac{6}{5} \). Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (-1, 9) \) and \( m = -\frac{6}{5} \), we have:
\[
y - 9 = -\frac{6}{5}(x + 1)
\]

### Step 4: Simplify to Slope-Intercept Form (if needed)
Expanding this equation:
\[
y - 9 = -\frac{6}{5}x - \frac{6}{5}
\]
Add 9 to both sides:
\[
y = -\frac{6}{5}x + \frac{39}{5}
\]

### Final Answer
The equation of the line perpendicular to \( y = \frac{5}{6}x + \frac{5}{6} \) and passing through \( (-1, 9) \) is:
\[
y = -\frac{6}{5}x + \frac{39}{5}
\]

Would you like a further breakdown of any steps?

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Here are five related questions you might consider:
1. How do you find the slope of a line from its equation?
2. Why do perpendicular slopes need to be negative reciprocals?
3. What is the point-slope form of a line?
4. How do you convert from point-slope form to slope-intercept form?
5. How can you verify that two lines are perpendicular?

**Tip:** When finding perpendicular slopes, simply flip the fraction and change the sign of the original slope.

Write the equation of the line perpendicular to y = (5/6)x + (5/6) that passes through the point (-1, 9).

Learn how to find the equation of a line perpendicular to y = (5/6)x + (5/6) that passes through the point (-1, 9). This solution involves understanding slope-intercept form, point-slope form, and the concept of perpendicular slopes. Suitable for students in Grades 9-11.

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