The line `x + 2y - z - 3 = 0 = x + 3y - 2 - 4` is parallel to

Options

A. x-y plane
B. y-z plane
C. z-x plane
D. z-axis

Answer

To determine which plane the line given by the equations \( x + 2y - z - 3 = 0 \) and \( x + 3y - 2 - 4 = 0 \) is parallel to, we can follow these steps:

### Step 1: Rewrite the equations of the line
The equations of the line can be rewritten as:
1. \( x + 2y - z = 3 \)
2. \( x + 3y - 6 = 0 \)

### Step 2: Find the direction ratios of the line
To find the direction ratios, we can express the equations in a parametric form. We can set \( z = t \) (a parameter), and then solve for \( x \) and \( y \) in terms of \( t \).

From the first equation:
\[ x + 2y = z + 3 \implies x + 2y = t + 3 \]
From the second equation:
\[ x + 3y = 6 \implies x = 6 - 3y \]

### Step 3: Substitute \( x \) in the first equation
Substituting \( x = 6 - 3y \) into the first equation:
\[ (6 - 3y) + 2y = t + 3 \]
This simplifies to:
\[ 6 - y = t + 3 \implies y = 3 - t \]

### Step 4: Find \( x \) in terms of \( t \)
Now substituting \( y = 3 - t \) back into \( x = 6 - 3y \):
\[ x = 6 - 3(3 - t) = 6 - 9 + 3t = 3t - 3 \]

### Step 5: Parametric equations of the line
Now we have the parametric equations:
\[ x = 3t - 3, \quad y = 3 - t, \quad z = t \]
The direction ratios of the line can be derived from the coefficients of \( t \):
\[ \text{Direction ratios} = (3, -1, 1) \]

### Step 6: Determine the normal vector of the planes
The normal vector of the planes can be derived from their equations:
1. For the XY-plane, the normal vector is \( (0, 0, 1) \).
2. For the YZ-plane, the normal vector is \( (1, 0, 0) \).
3. For the ZX-plane, the normal vector is \( (0, 1, 0) \).

### Step 7: Check for parallelism
A line is parallel to a plane if its direction ratios are perpendicular to the normal vector of the plane.

- For the XY-plane, the dot product of \( (3, -1, 1) \) and \( (0, 0, 1) \) is \( 1 \) (not perpendicular).
- For the YZ-plane, the dot product of \( (3, -1, 1) \) and \( (1, 0, 0) \) is \( 3 \) (not perpendicular).
- For the ZX-plane, the dot product of \( (3, -1, 1) \) and \( (0, 1, 0) \) is \( -1 \) (not perpendicular).

### Conclusion
Since the direction ratios of the line are not perpendicular to the normal vectors of any of the given planes, we conclude that the line is parallel to the ZX-plane.

### Final Answer
The line is parallel to the ZX-plane.