The graph of which of the following equation is parallel to the line with equation `y=-3x-6`?

Options

A. `x-3y=3`
B. `x-(1)/(3)y=2`
C. `x+(1)/(6)y=4`
D. `x+(1)/(3)y=5`

Answer

To determine which equation's graph is parallel to the line given by \( y = -3x - 6 \), we need to find the slope of the given line and then check which of the provided options has the same slope.

### Step-by-Step Solution:

1. **Identify the slope of the given line**:
The equation of the line is given as \( y = -3x - 6 \). In the slope-intercept form \( y = mx + c \), the slope \( m \) is the coefficient of \( x \).
- Here, \( m = -3 \).

2. **Understand the condition for parallel lines**:
For two lines to be parallel, they must have the same slope. Therefore, we are looking for an equation from the options that also has a slope of \( -3 \).

3. **Examine the options**:
We will analyze each option to find its slope.

- **Option 1**: \( x - 3y = 3 \)
- Rearranging gives: \( 3y = x - 3 \) or \( y = \frac{1}{3}x - 1 \).
- Slope \( m = \frac{1}{3} \) (not equal to -3).

- **Option 2**: \( x - \frac{1}{3}y = 2 \)
- Rearranging gives: \( \frac{1}{3}y = x - 2 \) or \( y = 3x - 6 \).
- Slope \( m = 3 \) (not equal to -3).

- **Option 3**: \( x + \frac{1}{6}y = 4 \)
- Rearranging gives: \( \frac{1}{6}y = -x + 4 \) or \( y = -6x + 24 \).
- Slope \( m = -6 \) (not equal to -3).

- **Option 4**: \( x + \frac{1}{3}y = 5 \)
- Rearranging gives: \( \frac{1}{3}y = -x + 5 \) or \( y = -3x + 15 \).
- Slope \( m = -3 \) (this is equal to -3).

4. **Conclusion**:
The fourth option has the same slope as the given line, which means it is parallel to the line \( y = -3x - 6 \).

### Final Answer:
The equation whose graph is parallel to the line \( y = -3x - 6 \) is **Option 4**: \( x + \frac{1}{3}y = 5 \).