Vtyjkyuk

How to determine the coefficient signs for the solution of a linear diophantine equation?

Take $24x + 69y = 33$ for example.

I know the solution is $x = 33 − 23k , y = −11 + 8k$, and I understand fully how to arrive at the values, but how do I determine the sign on the coefficients?

In other words, why is $23k$ negative, and why is $8k$ positive?

Through all the examples I can find, there seems to be no pattern.

You want to parameterize the solutions to the equation$$24x + 69y = 33$$

You know that $(x_0, y_0) = (33, -11)$ is a solution.

beginalign
24(phantom3x) + 69(phantom-1y) &= 33 \
24(33) + 69(-11) &= 33 &text(subtract) \
hline
24(x-33) + 69(y+11) &= 0
endalign

Since $24$ divides $24(x-33)$, then it must also divide $69(y+11)$.

Since $24$ and $69$ are relatively prime to each other, then we must have $24$ divides
$y+11$. So, for some integer $t, y+11 = 24t$. Hence $y=24t-11$. We can now solve for $x$.

beginalign
24(x-33) + 69(y+11) &= 0 \
24(x-33) + 69(24t-11+11) &= 0 \
24(x-33) + 69(24t) &= 0 \
24(x-33) &= -69(24t) &textNote the change in sign. \
x-33 &= -69t \
x &= -69t + 33
endalign

And you get
beginalign
x &= -69t + 33 \
y &= 24t - 11
endalign

If you let $t=-u$, you get
beginalign
x &= 69u + 33 \
y &= -24u - 11
endalign