I'm trying to calculate euclidean algorithm for my question $4x+6y=46$ Now I figure out that my GCD is $(6,4)=2$ and my lcm is $12$ now I don't know how to figure out the other solution by using linear combination for $K$ by using lcm.
A website calculator gives the answer is
$x = -23 + 3k$
and
$y = 23 - 2k$
but how did they came up with this solution ?
We want to solve $$4x + 6y = 46$$ divide all terms by $2$ $$ 2x + 3y = 23$$
But we know that $2(2) + 3(-1) = 1 $ multiply with $23$ we get $$2(46) + 3(-23) = 23$$ So $(46, -23)$ is a solution in general $x = 46 + 3k , y= -23 - 2k\;$ for any integer $k$
so see why this is true $$2(46 + 3k) + 3(-23 - 2k ) = 2(46) + \color{red}{2(3k)} + 3(-23) + \color{red}{3(-2k)} = 2(46) + 3(-23) = 23$$ red terms cancel To get the answer you have sub $k=-3$
