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Let's solve the problem step by step.

### Problem Recap:
We are tasked with finding the equation of the line that:
1. Passes through the point \( (4, -6) \).
2. Is **perpendicular** to the line \( y = \frac{4}{5}x - 3 \).

### Step 1: Find the slope of the given line
The equation \( y = \frac{4}{5}x - 3 \) is in **slope-intercept form** \( y = mx + b \), where \( m \) is the slope. 

For this line:
- The slope is \( m = \frac{4}{5} \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the **negative reciprocal** of the original line's slope.

So, the perpendicular slope \( m_{\perp} \) is:
\[
m_{\perp} = -\frac{5}{4}
\]

### Step 3: Use the point-slope form to find the equation of the perpendicular line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point on the line, and \( m \) is the slope.

We know:
- The slope \( m_{\perp} = -\frac{5}{4} \).
- The point \( (4, -6) \).

Substituting these values into the point-slope form:
\[
y - (-6) = -\frac{5}{4}(x - 4)
\]
This simplifies to:
\[
y + 6 = -\frac{5}{4}(x - 4)
\]

### Step 4: Convert to slope-intercept form
Now we simplify the equation and put it into slope-intercept form \( y = mx + b \).

Distribute the slope:
\[
y + 6 = -\frac{5}{4}x + 5
\]

Now subtract 6 from both sides to solve for \( y \):
\[
y = -\frac{5}{4}x + 5 - 6
\]
\[
y = -\frac{5}{4}x - 1
\]

### Final Answer:
The equation of the perpendicular line in slope-intercept form is:
\[
y = -\frac{5}{4}x - 1
\]

Would you like further details or clarifications?

---

### 5 Follow-Up Questions:
1. What is the point-slope form, and how is it different from slope-intercept form?
2. How do you find the slope of a line perpendicular to another?
3. What happens if two lines have the same slope?
4. How do you interpret the y-intercept in a slope-intercept equation?
5. How would you find the equation of a line parallel to \( y = \frac{4}{5}x - 3 \)?

### Tip:
Always remember that perpendicular lines have slopes that are negative reciprocals of each other!

Find the equation of the line that contains the given point (4, -6) and is perpendicular to the line y = 4/5x - 3.

Learn how to find the equation of a line passing through the point (4, -6) that is perpendicular to the line y = 4/5x - 3. This problem uses concepts like slopes of perpendicular lines, point-slope form, and slope-intercept form. Suitable for students in Grades 9-12.

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