Given the first order diophantine equation...

$\displaystyle 35 \cdot y \equiv 13\ \text{mod}\ 97$ (1)

... a standard way to solve it is to multiply both members by the multiplicative inverse of 35, if that multiplicative inverse do exists, obtaining...

$\displaystyle y \equiv 35^{-1} \cdot 13\ \text{mod}\ 97$ (2)

If n is prime, then any number mod n different from zero has one and only one multiplicative inverse. In Your case 97 is prime and the multiplicative inverse of 35 is 61 because $\displaystyle 35 \cdot 61 = 2135 = 97 \cdot 22 + 1$, so that the solution is...

$\displaystyle y = 61 \cdot 13 = 793 \equiv 17\ \text{mod}\ 97$ (3)

Kind regards

$\chi$ $\sigma$