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?