The symmetric equation of lines `3x+2y+z-5=0andx+y-2z-3=0`, is

Options

A. `(x-1)/(5)=(y-4)/(7)=(z-0)/(1)`
B. `(x+1)/(5)=(y+4)/(7)=(z-0)/(1)`
C. `(x+1)/(-5)=(y-4)/(7)=(z-0)/(1)`
D. `(x-1)/(-5)=(y-4)/(7)=(z-0)/(1)`

Explanation

To find the symmetric equation of the lines given by the equations \(3x + 2y + z - 5 = 0\) and \(x + y - 2z - 3 = 0\), we will follow these steps:

### Step 1: Rewrite the equations in parametric form
We start with the two equations:
1. \(3x + 2y + z - 5 = 0\) (Equation 1)
2. \(x + y - 2z - 3 = 0\) (Equation 2)

We will express \(z\) in terms of \(x\) and \(y\) from both equations.

From Equation 1:
\[ z = 5 - 3x - 2y \quad \text{(1)} \]

From Equation 2:
\[ z = \frac{x + y - 3}{2} \quad \text{(2)} \]

### Step 2: Set the expressions for \(z\) equal to each other
Now, we set the two expressions for \(z\) equal to each other:
\[ 5 - 3x - 2y = \frac{x + y - 3}{2} \]

### Step 3: Clear the fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 2(5 - 3x - 2y) = x + y - 3 \]
This simplifies to:
\[ 10 - 6x - 4y = x + y - 3 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 10 + 3 = x + y + 6x + 4y \]
\[ 13 = 7x + 5y \]

### Step 5: Express \(y\) in terms of \(x\)
Now, we can express \(y\) in terms of \(x\):
\[ 5y = 13 - 7x \implies y = \frac{13 - 7x}{5} \quad \text{(3)} \]

### Step 6: Substitute \(y\) back into \(z\)
Now, substitute \(y\) from Equation (3) back into either expression for \(z\). We will use Equation (1):
\[ z = 5 - 3x - 2\left(\frac{13 - 7x}{5}\right) \]
This simplifies to:
\[ z = 5 - 3x - \frac{26 - 14x}{5} \]
To combine the terms, convert \(5\) into a fraction:
\[ z = \frac{25 - 15x - 26 + 14x}{5} = \frac{-1 - x}{5} \]

### Step 7: Write the symmetric equations
Now we have:
\[ x = x, \quad y = \frac{13 - 7x}{5}, \quad z = \frac{-1 - x}{5} \]

To write the symmetric form, we express \(x\), \(y\), and \(z\) in terms of a parameter \(t\):
\[ x = t, \quad y = \frac{13 - 7t}{5}, \quad z = \frac{-1 - t}{5} \]

### Final Symmetric Equation
The symmetric equations can be written as:
\[ \frac{x + 1}{-5} = \frac{y - 4}{7} = z - 0 \]