The equation of a line is $$ y=5 x+2 $$. Find the oquation of a line perpendiculat to it and passing through $$ \left(4,-\frac{3)}{3}\right. $$

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To find the equation of a line that is **perpendicular** to the given line $$ y = 5x + 2 $$ and passes through the point $$ \left(4, -\frac{3}{3}\right) $$, follow these steps:

### 1. Simplify the Given Point
First, simplify the coordinates of the point:
$$
\left(4, -\frac{3}{3}\right) = (4, -1)
$$

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

For $$ y = 5x + 2 $$:
- **Slope ($$ m_1 $$)**: 5

### 3. Find the Slope of the Perpendicular Line
Lines that are perpendicular to each other have slopes that are **negative reciprocals**.

So, if the slope of the given line is $$ m_1 = 5 $$, then the slope of the perpendicular line ($$ m_2 $$) is:
$$
m_2 = -\frac{1}{m_1} = -\frac{1}{5}
$$

### 4. 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:
- $$ m $$ is the slope
- $$ (x_1, y_1) $$ is the point the line passes through

Plugging in the values:
$$
y - (-1) = -\frac{1}{5}(x - 4)
$$
$$
y + 1 = -\frac{1}{5}x + \frac{4}{5}
$$

### 5. Simplify to Slope-Intercept Form
$$
y = -\frac{1}{5}x + \frac{4}{5} - 1
$$
$$
y = -\frac{1}{5}x - \frac{1}{5}
$$

### **Final Equation**
$$
\boxed{y = -\frac{1}{5}x - \frac{1}{5}}
$$

This is the equation of the line perpendicular to $$ y = 5x + 2 $$ that passes through the point $$ (4, -1) $$.
The equation of the perpendicular line is $$ y = -\frac{1}{5}x - \frac{1}{5} $$.

The equation of a line is .
Find the oquation of a line perpendiculat
to it and passing through

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The equation of a line is . Find the oquation of a line perpendiculat to it and passing through