Linear Diophantine Equation/Examples/10x - 8y = 42

Example of Linear Diophantine Equation
The linear diophantine equation:
 * $10 x - 8 y = 42$

has the general solution:
 * $\tuple {x, y} = \tuple {21 - 4 t, 21 - 5 t}$

Proof
Using the Euclidean Algorithm:

Thus we have that:
 * $\gcd \set {10, -8} = 2$

which is a divisor of $42$:
 * $42 = 21 \times 2$

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

Next we find a single solution to $10 x - 8 y = 42$.

Again with the Euclidean Algorithm:

and so:

is a solution.

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


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

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

In this case:

giving: