through: $$ (-1,0) $$, perp. to $$ y=\frac{1}{3} x+2 $$

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To find the equation of a line that passes through the point $$(-1, 0)$$ and is perpendicular to the line $$ y = \frac{1}{3}x + 2 $$, 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
- $$ b $$ is the y-intercept

For $$ y = \frac{1}{3}x + 2 $$:
- Slope ($$ m_1 $$) = $$ \frac{1}{3} $$

### 2. Find the Slope of the Perpendicular Line
Perpendicular lines have slopes that are **negative reciprocals** of each other. Therefore, if the slope of one line is $$ m $$, the slope of the perpendicular line ($$ m_2 $$) is:
$$
m_2 = -\frac{1}{m_1}
$$

Calculating:
$$
m_2 = -\frac{1}{\frac{1}{3}} = -3
$$

### 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 point $$ (-1, 0) $$ and slope $$ m_2 = -3 $$:
$$
y - 0 = -3(x - (-1))
$$
$$
y = -3(x + 1)
$$
$$
y = -3x - 3
$$

### 4. Final Equation
The equation of the line that passes through $$(-1, 0)$$ and is perpendicular to $$ y = \frac{1}{3}x + 2 $$ is:
$$
\boxed{y = -3x - 3}
$$

### Graphical Representation

To visualize, here's a simple sketch of both lines:

1. **Given Line:** $$ y = \frac{1}{3}x + 2 $$ (Light Blue)
2. **Perpendicular Line:** $$ y = -3x - 3 $$ (Orange)

![Graph of the two lines](https://i.imgur.com/yourimagelink.png)

*Note: Replace the image link with an actual graph if needed.*

### Verification

To ensure correctness, let's verify that the two lines are perpendicular:

- Slope of the given line ($$ m_1 $$): $$ \frac{1}{3} $$
- Slope of the perpendicular line ($$ m_2 $$): $$ -3 $$

Check the product:
$$
m_1 \times m_2 = \frac{1}{3} \times (-3) = -1
$$
Since the product is $$-1$$, the lines are indeed perpendicular.
The equation of the line that passes through $$(-1, 0)$$ and is perpendicular to $$ y = \frac{1}{3}x + 2 $$ is: $$ y = -3x - 3 $$

through: , perp. to

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