# Using Diophantine Equation to find the solutions of another equation

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 $$19x+57y$$

• Just plug this in to 19x+57y – Aqua May 18 at 9:42

Hint:

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

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

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

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 $$95$$.

• you mean substitute $5$ and $0$ for x and y into $19x+57y$? – user8358234 May 18 at 17:22
• also am i correct in my general solution for $17x+51y=85$? – user8358234 May 18 at 17:23
• Yes, your general solution is correct and you can just substitute $5$ and $0$. – Toby Mak May 19 at 0:48

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

• thanks for this, I knew how to do it that way but I just wanted to use the Diophantine equation because I'm just a little confused on how to apply the solution I have for $17x+51y=85$ to get the solution for $19x+57y$ – user8358234 May 18 at 17:22
• @user8358234 The above equivalences shows that they have the same set of solutions, namely $\, (x,y) = (5,0) + (3,-1)n\$ (note the correction to your solution). – Bill Dubuque May 18 at 17:34