through: $$ (5,1) $$, perp. to $$ y=-\frac{1}{6} x-5 $$

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To find the equation of a line that passes through the point $$(5, 1)$$ and is **perpendicular** to the given line $$ y = -\frac{1}{6}x - 5 $$, follow these steps:

### 1. Determine the Slope of the Given Line

The given equation is in **slope-intercept form**: 
$$ y = mx + b $$
where:
- $$ m $$ is the slope.
- $$ b $$ is the y-intercept.

For $$ y = -\frac{1}{6}x - 5 $$:
- **Slope ($$ m $$)**: $$ -\frac{1}{6} $$

### 2. Find the Slope of the Perpendicular Line

Perpendicular lines have slopes that are **negative reciprocals** of each other.

So, if the slope of the given line is $$ m = -\frac{1}{6} $$, then the slope of the perpendicular line ($$ m_{\perp} $$) is:
$$ m_{\perp} = -\frac{1}{m} = -\left( -6 \right) = 6 $$

### 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.
- $$ m $$ is the slope.

Given the point $$ (5, 1) $$ and slope $$ m = 6 $$:
$$ y - 1 = 6(x - 5) $$

### 4. Simplify to Slope-Intercept Form (Optional)

Expanding the equation:
$$
\begin{align*}
y - 1 &= 6x - 30 \
y &= 6x - 30 + 1 \
y &= 6x - 29
\end{align*}
$$

### **Final Equation**

The equation of the line that passes through $$(5, 1)$$ and is perpendicular to $$ y = -\frac{1}{6}x - 5 $$ is:
$$ y = 6x - 29 $$

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**Graphical Representation:**

To visualize, here's how both lines compare:

1. **Given Line:** $$ y = -\frac{1}{6}x - 5 $$
   - Slope: $$ -\frac{1}{6} $$
   - Y-intercept: $$ -5 $$

2. **Perpendicular Line:** $$ y = 6x - 29 $$
   - Slope: $$ 6 $$
   - Y-intercept: $$ -29 $$
   - Passes through $$ (5, 1) $$

![Graph of the two lines](https://i.imgur.com/your-image-link.png)

*(Note: Replace the image link with an actual graph if needed.)*
The equation of the line perpendicular to $$ y = -\frac{1}{6}x - 5 $$ and passing through the point $$ (5, 1) $$ is: $$ y = 6x - 29 $$

through: \( (5,1) \), perp. to \( y=-\frac{1}{6} x-5 \)

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