question

x + 2y = 1
x - y  = 4

how to solve

1  2 | 1   : a
1 -1 | 4   : b

1  2 | 1   : a <- a
0 -3 | 3   : b <- b - a

1  2 | 1   : a <- a
0 -1 | 1   : b <- b / 3

1  2 |  1  : a <- a
0  1 | -1  : b <- b * -1

1  0 |  3  : a <- a - 2 * b
0  1 | -1  : b <- b

answer

x = 3, y = -1

check

a : x + 2y = 1
3 + 2 (-1) = 3 - 2 = 1

b : x - y = 4
3 - (-1)   = 3 + 1 = 4