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

Question

UpStudy AI · Accepted Answer

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

Upstudy AI Solution

Answer

Solution

1. Simplify the Given Point

2. Determine the Slope of the Given Line

3. Find the Slope of the Perpendicular Line

4. Use the Point-Slope Form to Find the Equation

5. Simplify to Slope-Intercept Form

Final Equation

Bonus Knowledge

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