have changed your equation from 19x=29 mod16 to 19x=16 mod29 as
it is impossible to have 29 mod 16 in modular arithmetic/it makes
no sense.
Anyway, any modular equation a=b mod m can be rewritten as
a-b=km where k is an integer. or in other words, m divides a-b
without remainder. Applying this to the equation above we get
(19x-16)/29=k. Now just stick it into a calculator and replace x
with integers until the equation gives out an integer, in this case
x turns out to be 10.