How to solve the equation $15x- 16y= 10$

Question

I am trying to find an $x$ and $y$ that solve the equation $15x - 16y = 10$, usually in this type of question I would use Euclidean Algorithm to find an $x$ and $y$ but it doesn't seem to work for this one. Computing the GCD just gives me $16 = 15 + 1$ and then $1 = 16 - 15$ which doesn't really help me. I can do this question with trial and error but was wondering if there was a method to it.

Thank you

Answer 1

Note that by Bezout's identity since $\gcd(15,16)=1$ we have

$$15\cdot (-1+k\cdot 16)+16 \cdot (1-k\cdot 15)=1 \quad k\in\mathbb{Z}$$

are all the solution for $15a+16b=1$ and from here just multiply by $10$.

Answer 2

In this case you don't really need the full power of the Euclidean algorithm. Since you know $$ 16 - 15 = 1 $$ you can just multiply by $10$ to conclude that $$ 16 \times 10 + 15 \times(-10) = 10. $$ Now you have your $y$ and $x$.

Answer 3

You have $16-15=1$

What about $$ x=-10+16k, y= -10+15k ?$$

That implies

$$ 15 x-16y=10$$ Which is a solution