a) $$ 130x + 143y = 5957 $$
b) $$ 44x + 19y = 75 $$
the theorem says ax + by = c has solution if and only if d | c
however I work out both question with no solution as
a)
$$ 143 = 130 . 1 + 13 $$
$$ 130 = 13 . 10 + 0 $$
and 5957 is unable to be divided by 13 therefore no solution exists.
b)
$$ 44 = 19 . 2 + 6 $$
$$ 19 = 6 . 3 + 1 $$
$$ 6 = 3 . 2 + 0 $$
for b) I am unsure if there are any solution exists?
Your first problem is correct.
Your second one, however, is not. When doing the Euclidean Algorithm, you mess up between your second and third steps. From $19 = 6*3 + 1$, you should go on the next next step $6 = 1*6 + 0$, indicating that 1 is the gcd of 19 and 44.
Let's also consider for a moment what your solution said: it said that either 2 or 3 divided both 44 and 19 (neither of which are correct). We also know that 19 is prime, so unless 44 is a multiple of 19 (which it isn't), they share no factors.
With that, it turns out that there are infinitely many solutions.