Find the equation of the line that is perpendicular to 
Ax + By = C
 and passes through the point 
(−7, 8)
.
Assume A, B, and C are all positive constants, not equal to each other.

To find the equation of a line that is perpendicular to the given line \( Ax + By = C \) and passes through the point \((-7, 8)\), follow these steps:

### 1. **Determine the slope of the given line**:
   The equation \( Ax + By = C \) can be rewritten in slope-intercept form \( y = mx + b \) to identify its slope. 

   \[
   By = -Ax + C
   \]
   \[
   y = -\frac{A}{B}x + \frac{C}{B}
   \]

   The slope of the given line is \( m_1 = -\frac{A}{B} \).

### 2. **Find the slope of the perpendicular line**:
   The slope of the line perpendicular to the given line is the negative reciprocal of \( m_1 \). 

   \[
   m_2 = \frac{B}{A}
   \]

### 3. **Use the point-slope form to find the equation of the perpendicular line**:
   The point-slope form of the equation of a line is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( (x_1, y_1) = (-7, 8) \) is the point the line passes through and \( m_2 = \frac{B}{A} \) is the slope.

   Substitute the values:
   \[
   y - 8 = \frac{B}{A}(x + 7)
   \]

### 4. **Simplify the equation**:
   Distribute \( \frac{B}{A} \) on the right side:
   \[
   y - 8 = \frac{B}{A}x + \frac{7B}{A}
   \]
   Add 8 to both sides to get the equation in slope-intercept form:
   \[
   y = \frac{B}{A}x + \frac{7B}{A} + 8
   \]

This is the equation of the line that is perpendicular to \( Ax + By = C \) and passes through the point \( (-7, 8) \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the slope of a line given two points?
2. What is the negative reciprocal, and why is it used for perpendicular lines?
3. How can you convert an equation from point-slope form to standard form?
4. What is the relationship between parallel and perpendicular lines in terms of slope?
5. How do you find the distance between a point and a line?

### Tip:
Always remember that the product of the slopes of two perpendicular lines is \(-1\). This property is crucial for solving problems involving perpendicularity.

Learn how to find the equation of a line that is perpendicular to the line Ax + By = C and passes through the point (-7, 8). This problem involves geometry concepts, including the point-slope form of a line equation and the theorem on perpendicular lines. Suitable for students in Grades 9-12.

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