# Linear Diophantine Equation/Examples/5x + 6y = 1

## Example of Linear Diophantine Equation

$5 x + 6 y = 1$

has the general solution:

$x = -1 + 6 t, y = 1 - 5 t$

## Proof

We have that:

$\gcd \set {5, 6} = 1$

which is (trivially) a divisor of $1$.

So, from Solution of Linear Diophantine Equation, a solution exists.

First we find a single solution to $5 x + 6 y = 1$:

$1 = 1 \times 6 - 1 \times 5$

So $y_0 = 1, x_0 = -1$ is a solution.

From Solution of Linear Diophantine Equation, the general solution is then:

$\forall t \in \Z: x = x_0 + \dfrac b d t, y = y_0 - \dfrac a d t$

where $d = \gcd \set {a, b}$.

In this case:

 $\ds x_0$ $=$ $\ds -1$ $\ds y_0$ $=$ $\ds 1$ $\ds a$ $=$ $\ds 5$ $\ds b$ $=$ $\ds 6$ $\ds d$ $=$ $\ds 1$

giving:

 $\ds x$ $=$ $\ds -1 + 6 t$ $\ds y$ $=$ $\ds 1 - 5 t$

$\blacksquare$