The lines
`2x+3y-4z=0 ,3x-4y+z=7`
`5x-y-3z+12=0, x-7y+5z-6=0``
` are parellel.

Answer

To prove that the given lines are parallel, we need to analyze the direction ratios of the lines represented by the equations. The equations of the lines are:

1. \( 2x + 3y - 4z = 0 \)
2. \( 3x - 4y + z = 7 \)
3. \( 5x - y - 3z + 12 = 0 \)
4. \( x - 7y + 5z - 6 = 0 \)

### Step 1: Convert the equations into vector form

For the first two lines:

1. From \( 2x + 3y - 4z = 0 \), we can express it in vector form as:
\[ \vec{r_1} = \lambda (2\hat{i} + 3\hat{j} - 4\hat{k}) \]
where \( \lambda \) is a parameter.

2. From \( 3x - 4y + z = 7 \), we can express it in vector form as:
\[ \vec{r_2} = \mu (3\hat{i} - 4\hat{j} + \hat{k}) + \vec{a} \]
where \( \vec{a} \) is a point on the line.

### Step 2: Find the direction ratios of the lines

The direction ratios for the first line \( 2x + 3y - 4z = 0 \) are \( (2, 3, -4) \).

The direction ratios for the second line \( 3x - 4y + z = 7 \) are \( (3, -4, 1) \).

### Step 3: Check if the direction ratios are proportional

To check if the lines are parallel, we need to see if the direction ratios are proportional. This means we need to find a constant \( k \) such that:

\[ \frac{2}{3} = \frac{3}{-4} = \frac{-4}{1} \]

Calculating these ratios:

1. \( \frac{2}{3} \) is not equal to \( \frac{3}{-4} \).
2. \( \frac{3}{-4} \) is not equal to \( \frac{-4}{1} \).

Since the ratios are not equal, the first two lines are not parallel.

### Step 4: Repeat for the next two lines

Now, we will analyze the last two lines:

1. From \( 5x - y - 3z + 12 = 0 \), the direction ratios are \( (5, -1, -3) \).
2. From \( x - 7y + 5z - 6 = 0 \), the direction ratios are \( (1, -7, 5) \).

### Step 5: Check if these direction ratios are proportional

To check if the lines are parallel, we need to find a constant \( k \) such that:

\[ \frac{5}{1} = \frac{-1}{-7} = \frac{-3}{5} \]

Calculating these ratios:

1. \( \frac{5}{1} = 5 \)
2. \( \frac{-1}{-7} \approx 0.14 \)
3. \( \frac{-3}{5} = -0.6 \)

Since the ratios are not equal, the last two lines are also not parallel.

### Conclusion

Since neither pair of lines is parallel, we conclude that the given lines are not parallel.