How to check for solution of system of linear equation

Question

I have seen many ques related to linear equation but can't find any example from where I can understand how to solve this ques

The following system of linear equations have :

$$ x+2y+z-3w = 1$$

$$ 2x+4y+3z+w = 3$$

$$ 3x+6y+4z-2w = 5$$

(a.) No solution

(b.) Unique solution

(a.) Infinite solution

(a.) Finite but more than one solution

Answer 1

Let $x + 2y = a$.

Then ,

$$x+2y+z-3w = 1 \implies a = 1+3w-z \space \space \text{ (1) } $$

$$2a + 3z+w = 3 \implies 2+6w -2z +3z+w = 3 \implies7w +z =1 \space \space \text{ (2) }$$

And

$$3a + 4z -2w = 5 \implies3+9w-3z +4z-2w = 5 \implies 7w +z = 2\space \space \text{ (3) }$$

Which is contradiction.

Hence there is no solution.