Which of the following lines is perpendicular to `y = - 3x + 2` and has the same y - intercept as `y = 3x - 2` ?

Options

A. `y=-(1)/(3)x`
B. `y=-(1)/(3)x+2`
C. `y=-(1)/(3)x-2`
D. `y=(1)/(3)x-2`

Explanation

To solve the problem of finding a line that is perpendicular to \( y = -3x + 2 \) and has the same y-intercept as \( y = 3x - 2 \), we will follow these steps:

### Step 1: Identify the slopes and y-intercepts of the given lines
1. The equation of the first line is \( y = -3x + 2 \).
- The slope \( m_1 \) is \(-3\).
- The y-intercept \( c_1 \) is \(2\).

2. The equation of the second line is \( y = 3x - 2 \).
- The slope \( m_2 \) is \(3\).
- The y-intercept \( c_2 \) is \(-2\).

### Step 2: Determine the y-intercept of the required line
Since the required line must have the same y-intercept as the second line, we have:
- \( c_3 = c_2 = -2 \).

### Step 3: Find the slope of the required line
To find the slope \( m_3 \) of the required line, we use the condition for perpendicular lines. The product of the slopes of two perpendicular lines is \(-1\):
\[ m_1 \cdot m_3 = -1 \]
Substituting \( m_1 = -3 \):
\[ -3 \cdot m_3 = -1 \]
Solving for \( m_3 \):
\[ m_3 = \frac{-1}{-3} = \frac{1}{3} \]

### Step 4: Write the equation of the required line
Now that we have both the slope and the y-intercept of the required line, we can write its equation:
\[ y = m_3 x + c_3 \]
Substituting \( m_3 = \frac{1}{3} \) and \( c_3 = -2 \):
\[ y = \frac{1}{3}x - 2 \]

### Step 5: Identify the correct option
Now we can check the options provided to see which one matches our derived equation \( y = \frac{1}{3}x - 2 \).

### Final Answer
The required line is \( y = \frac{1}{3}x - 2 \).

---