through: $$ (5,2) $$, perp. to $$ y=-\frac{5}{3} x-2 $$

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To find the equation of a line that is **perpendicular** to the given line $$ y = -\frac{5}{3}x - 2 $$ and **passes through the point** $$ (5, 2) $$, follow these steps:

### 1. Determine the Slope of the Given Line
The given line is in the slope-intercept form $$ y = mx + b $$, where $$ m $$ is the slope.

For $$ y = -\frac{5}{3}x - 2 $$:
- **Slope ($$ m $$)**: $$ -\frac{5}{3} $$

### 2. Find the Slope of the Perpendicular Line
Perpendicular lines have slopes that are **negative reciprocals** of each other.

- **Slope of perpendicular line ($$ m_{\perp} $$)**:
  $$
  m_{\perp} = -\frac{1}{m} = -\frac{1}{-\frac{5}{3}} = \frac{3}{5}
  $$

### 3. Use the Point-Slope Form to Find the Equation
The **point-slope form** of a line is:
$$
y - y_1 = m(x - x_1)
$$
where $$ (x_1, y_1) $$ is a point on the line and $$ m $$ is the slope.

Given the point $$ (5, 2) $$ and slope $$ \frac{3}{5} $$:
$$
y - 2 = \frac{3}{5}(x - 5)
$$

### 4. (Optional) Convert to Slope-Intercept Form
To express the equation in the form $$ y = mx + b $$:
$$
y - 2 = \frac{3}{5}x - 3 \
y = \frac{3}{5}x - 3 + 2 \
y = \frac{3}{5}x - 1
$$

### **Final Equation**
**Using point-slope form:**
$$
y - 2 = \frac{3}{5}(x - 5)
$$

**Or, in slope-intercept form:**
$$
y = \frac{3}{5}x - 1
$$
The equation of the line perpendicular to $$ y = -\frac{5}{3}x - 2 $$ and passing through $$ (5, 2) $$ is: $$ y = \frac{3}{5}x - 1 $$

through: \( (5,2) \), perp. to \( y=-\frac{5}{3} x-2 \)

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