Question

Explain me, please. I know how to do it but I want to know the logic on it rather than just apply formula.
Problem, find all solutions of 3x + 4y = 7

Answer 1

gcd (3, 4 ) =1
4 = 3 + 1
1 = 4 - 3 --> 1 = 3 (-1) + 4(1)
multiple all by 7 , we get
\(3 (-7) +4(7) =7\)
Hence \(x_0= -7 \\y_0 = 7\)
\(x = -7 \huge \color{red}{+}\)4t

\(y = 7 \huge \color{blue}{-}\) 3t
I would like to know the logic on those \(\color{red}{+}\) and \(\color{blue}{-}\)

Answer 2

@ganeshie8

Answer 3

You can find the proof for why the solution of a linear diophantine solution has that particular form here: http://math.uh.edu/~minru/5333-07/hj02cs01.pdf