Find the equation of a line parallel to the line `(x-5)/(3) = (y+1)/(-2) = (z)/(1)` and passes through the point `(0,-1,2)`.

Answer

To find the equation of a line that is parallel to the given line \(\frac{x-5}{3} = \frac{y+1}{-2} = \frac{z}{1}\) and passes through the point \((0, -1, 2)\), we can follow these steps:

### Step 1: Identify the direction ratios of the given line
The given line can be expressed in the symmetric form:
\[ \frac{x-5}{3} = \frac{y+1}{-2} = \frac{z}{1} \]
From this, we can identify the direction ratios (or direction vector) of the line:
\[ \mathbf{b} = (3, -2, 1) \]

### Step 2: Write the direction vector of the required line
Since the required line is parallel to the given line, it will have the same direction vector:
\[ \mathbf{b}_{\text{required}} = (3, -2, 1) \]

### Step 3: Use the point through which the line passes
The required line passes through the point \((0, -1, 2)\). We denote this point as:
\[ (x_1, y_1, z_1) = (0, -1, 2) \]

### Step 4: Write the equation of the line in symmetric form
The equation of a line in symmetric form that passes through a point \((x_1, y_1, z_1)\) with direction ratios \((b_1, b_2, b_3)\) is given by:
\[ \frac{x - x_1}{b_1} = \frac{y - y_1}{b_2} = \frac{z - z_1}{b_3} \]
Substituting the values we have:
- \(x_1 = 0\), \(y_1 = -1\), \(z_1 = 2\)
- \(b_1 = 3\), \(b_2 = -2\), \(b_3 = 1\)

The equation becomes:
\[ \frac{x - 0}{3} = \frac{y - (-1)}{-2} = \frac{z - 2}{1} \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{x}{3} = \frac{y + 1}{-2} = \frac{z - 2}{1} \]

Thus, the final equation of the required line is:
\[ \frac{x}{3} = \frac{y + 1}{-2} = \frac{z - 2}{1} \]