How to solve linear equations using the Gauss method?

Question

Mathematics

Yes, the moderators will forgive me for a slightly general question, but I would like to get an answer from professionals, I can easily solve ordinary linear equations of size 2x2, 3x3, but when solving equations with 4 unknowns, I get a stupor:
1. The teacher explained to us that we need some or multiply the string by the number n and add to another string to get zero, but the question arose, is it possible to add any strings, that is, from anywhere in the equation? For example, add 2 lines to 4 or add 4 to 3
2. Do you just need to make sure that 1 line has zeros and a positive / negative number to immediately find the root?
3. Is it possible to solve a linear equation simply by multiplying the equations by some numbers so that 0 remains in the equations?

Answer 1

1) Yes, you can add any string multiplied by any factor to any other string.
2) It is necessary to make sure that in all cells (a, b), where a > b, there are zeros. Then the matrix will take a triangular form.
For example, from the system

x + y + z = 1
2*x + 3*y - z = 2
  -x + y + 2*z = 0

succeed

x + y + z = 1
    y - 3*z = 0
        9*z = 1

After that, find sequentially z, y, x
3) You can, but you won't like it. It is necessary to count minors (determinants of matrices (N-1)*(N-1)). But this will no longer be the Gaussian method.