I am the prime next to Optimus

As the numbers are given to be positive $\color{#3D99F6}{\text{integers}}$ , we can see that $y$ can take values from $1,2,3,...,16,17$ . (If $y\geq 18$ , then $x \leq \frac{-1}{3}$ , but $x \in \mathbb{Z}^+$ )

From the given equation, for any values of $x$ and $y$ , we can say

$3x +8y \equiv 143 \pmod{3}$

Thus $8y \equiv 143 \pmod{3}$

$2y \equiv 2 \pmod{3} \implies \boxed{y\equiv 1 \pmod{3}}$

Thus out of the possible values of $y$ , we just have to consider the ones of the form $3k+1$ .

These values are $1,4,7,10,13,16$ . But we want $\text{a,b,c}$ to be prime numbers hence we only have to check for $y =7$ and for $y=13$ .

For $y=13, x=13$ , and for $y=7, x =29$ .

Hence the *prime * co-ordinate solutions are $(13,13)$ and $(29,7)$

Hence the answer is $7+29+13 = \boxed{49}$