2016/038) Solve the equation $3x-7y \equiv 11 \pmod {13}$

to solve this we solve 2 equations below and combine the two
$3x - 7y = 11\cdots(1)$
$3x - 7y = 13\cdots(2)$
let us solve $3x - 7y = 11$ first
we need to find solution of $3x-7y =1$. this can be done by extended euler algorithm but
we see that $(-2,-1)$ satisfy $3x-7y = 1$ so (-22,-11) satisfies $3x-7y = 11$
$(-22 + 7t, -11 + 3t)$  is a solution. choose t to be 4 to get (6,1) to be a soution.
for solving $3x - 7y = 13$ we have a solution $(-26,-13)$ so also $( -26 + 7t, - 13+ 3t)$ and
 putting t = 4 we get $(2,-1)$ a solution.
So solution for $3x-7y \equiv 11 \pmod {13}$ is $(6 + 2t, 1 - t)$. because the value need to be computed in mod 13 so the ans is $(6 + 2t \pmod {13} , (1 - t) \pmod {13})$.