How to solve congruence system What is the step how can I solve following system of congruences (that is one system):


  
  $7x-8y≡5 \pmod {11}$
  $2x+5y≡9 \pmod {11}$
Adding the first equation with twice the second one we obtain

$$7x-8y+2(2x+5y)≡5+2(9)  \implies 11x+2y\equiv23 \implies2y\equiv 1 \mod 11$$

$$\implies 6\cdot 2y\equiv 6\cdot 1 \implies 12y\equiv6\implies y\equiv 6 \mod 11$$

Then from the second equation

$$2x+5y≡9  \implies 6\cdot 2x+6\cdot 5y \equiv 6\cdot 9 \implies12x+4\equiv 10 \implies x\equiv 6 \mod 11$$
Hint:

$5\cdot(1)+8\cdot(2)$

$$(35+16)x\equiv5\cdot5+9\cdot8\pmod{11}$$

$$-4x\equiv3-5\pmod{11}\iff2x\equiv1\iff x\equiv2^{-1}\equiv6$$