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 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 is expressed in the symmetric form:
\[ \frac{x-5}{3} = \frac{y+1}{-2} = \frac{z}{1} \]
From this equation, we can identify the direction ratios (or direction vector) of the line as \( \langle 3, -2, 1 \rangle \).

### Step 2: Use the point through which the new line passes
The line we want to find must pass through the point \( (0, -1, 2) \). We will denote this point as \( (x_1, y_1, z_1) = (0, -1, 2) \).

### Step 3: Write the equation of the new line
The equation of a line in space can be expressed in symmetric form as:
\[ \frac{x - x_1}{\alpha} = \frac{y - y_1}{\beta} = \frac{z - z_1}{\gamma} \]
where \( \langle \alpha, \beta, \gamma \rangle \) are the direction ratios of the line.

Since the new line is parallel to the given line, we will use the same direction ratios \( \langle 3, -2, 1 \rangle \).

### Step 4: Substitute the values into the equation
Substituting \( (x_1, y_1, z_1) = (0, -1, 2) \) and \( \langle \alpha, \beta, \gamma \rangle = \langle 3, -2, 1 \rangle \) into the equation gives us:
\[ \frac{x - 0}{3} = \frac{y + 1}{-2} = \frac{z - 2}{1} \]

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

### Final Equation
Thus, the equation of the line parallel to the given line and passing through the point \( (0, -1, 2) \) is:
\[ \frac{x}{3} = \frac{y + 1}{-2} = \frac{z - 2}{1} \]