B Integer solutions for equations

[69911e]

I am trying to understand how to find solutions for a problem when parameters are limited to positive integers.
Example:
30x+19= 7y+1 =a ; where x,y,a are positive integers

Wolframalpha outputs:
a = 210 n + 169, x = 7 n + 5, y = 30 n + 24, n element Z(integers)

30*7= 210 (obviously)
How do I calculate 169? I can do it by hand, but am likely missing something obvious generating a universal equation.

I am trying to find a formula to solve for C5 below:
(C1 * X) +C2 = (C3*Y) + C4 = (C1*C3*n) +C5 ; where C1,C2,C3,C4 are constants and X,Y,n are the set of positive integers. Looking for a formula for smallest integer solution C5

Any suggestions?

 

[mfb]

The existence is a result of the Chinese remainder theorem, and with the formulas there you can also construct the smallest solution.

$30x+19=a$ can be interpreted as $a=19 \pmod{30}$.

 

[69911e]

MFB: Thanks for the link.
I forgot to specify, coefficients (C1&C3) of x and y are also defined as co-prime.

This is a small part of a (non-school) prime number related summer project I am working on with my 13&16 year old and we trying no to get bogged down on this equation. If it does not a have a formula solution for C5 (above), we may need to move onto a different path.

 

[mfb]

There is an algorithm to find it. It is not directly a nice closed formula.

 

[69911e]

MFB: It seems not too complex of an algorithm if limited to 2 equations. We may be able to use it and define the bounds needed.
Thanks