Note that 11×1+1=12=3×4, so 3 and 4 are multiplicative inverses of each other mod 11.
Similarly, we have 11×4+1=45=5×9 so 5 and 9 are multiplicative inverses of each other mod 11.
Also, 11×5+1=56=7×8 so 7 and 8 are multiplicative inverses of each other mod 11.
Therefore, a≡(3−1+5−1+7−1)−1≡(4+9+8)−1≡21−1(mod11)≡−1−1≡10(mod11).