Using Diophantine Equation to find the solutions of another equation

Question

If $17x+51y=85$, find the value of $19x+57y$

I know I could use substitution and figure this out but i wanted to use Diophantine equation. I'm just a little confused because I know that $\gcd(17,51)=17$ and $17|85$

I could use the extended EA to get that a particular solution is $x_0 =5$ and $y_0=0$

The complete solution is $x=5+51n$ and $y=0-17n$

Not sure where to go from here.

Also I know how to do the EA, I just didn't want to write out all the steps so I just put it into a EA calculator. I'm just a little confused on how to get the solutions for $19x+57y$

Answer 1

Hint:

You don't need the Euclidean algorithm to solve the problem. Note that

$19x+57y=(17x+51y)+2x+6y$.

On the other hand, $17x+51y=85\iff x+3y=5$.

Answer 2

Expanding on Maria Mazur's comment:

Since you found one possible solution is $x_0 = 5$ and $y_0 = 0$, you can just substitute to get $17(5) + 51(0) = 85$, so it satisfies the original expression.

Now you can substitute this into $19x + 57y$ this to get $95$.

Answer 3

$17x+51y = 85\!\!\overset{\large (\ \ )/17_{\phantom{1_{1_1}}}\!\!\!\!\!\!\!\!\!}\iff x+3y= 5\!\!\overset{\large 19\,(\ \ )_{\phantom{1_{1}}}\!\!\!\!\!\!\!}\iff 19x+57y = 95$