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To find the equation of a line that is perpendicular to the line \( y = 6x - 7 \) and passes through the point \((-2, 9)\), we need to follow these steps:

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

For \( y = 6x - 7 \), the slope \( m = 6 \).

### Step 2: Find the Slope of the Perpendicular Line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. 

So, if the slope of the given line is \( 6 \), the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{6}
\]

### Step 3: Use the Point-Slope Form to Write the Equation
Now, we have a point \((-2, 9)\) and a slope \(-\frac{1}{6}\). We can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \((x_1, y_1) = (-2, 9)\) and \( m = -\frac{1}{6} \):
\[
y - 9 = -\frac{1}{6}(x + 2)
\]

### Step 4: Simplify the Equation
Distribute the slope:
\[
y - 9 = -\frac{1}{6}x - \frac{1}{3}
\]
Add 9 to both sides to solve for \( y \):
\[
y = -\frac{1}{6}x + \frac{26}{3}
\]

### Final Answer
The equation of the line that is perpendicular to \( y = 6x - 7 \) and passes through the point \((-2, 9)\) is:
\[
y = -\frac{1}{6}x + \frac{26}{3}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do we find the equation of a line parallel to a given line?
2. What is the process to convert between slope-intercept form and point-slope form?
3. How can we determine if two lines are perpendicular by looking at their slopes?
4. Can you explain how to find the distance between a point and a line?
5. How do we graph lines in slope-intercept form on the coordinate plane?

### Tip:
When working with perpendicular lines, remember that their slopes are negative reciprocals, meaning the product of their slopes should always be \(-1\).

Find the equation of the line in the xy-plane that is perpendicular to the line y = 6x - 7 and that contains the point (-2, 9).

Learn how to find the equation of a line perpendicular to y = 6x - 7 that passes through the point (-2, 9). This problem covers key algebraic concepts, including slopes of perpendicular lines and using point-slope form. Suitable for students in Grades 8-10.

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