# Linear Diophantine Equation/Examples

## Examples of Linear Diophantine Equations

### Example: $15 x + 27 y = 1$

$15 x + 27 y = 1$

has no solutions for $x$ and $y$ integers.

### Example: $5 x + 6 y = 1$

$5 x + 6 y = 1$

has the general solution:

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

### Example: $2 x + 3 y = 4$

$2 x + 3 y = 4$

has the general solution:

$x = -4 + 3 t, y = 4 - 2 t$

### Example: $17 x + 19 y = 23$

$17 x + 19 y = 23$

has the general solution:

$\tuple {x, y} = \tuple {207 + 19 t, -184 - 17 t}$

### Example: $15 x + 51 y = 41$

$15 x + 51 y = 41$

has no solutions for $x$ and $y$ integers.

### Example: $23 x + 29 y = 25$

$23 x + 29 y = 25$

has the general solution:

$\tuple {x, y} = \tuple {-125 + 29 t, 100 - 23 t}$

### Example: $10 x - 8 y = 42$

$10 x - 8 y = 42$

has the general solution:

$\tuple {x, y} = \tuple {21 - 4 t, 21 - 5 t}$

### Example: $121 x - 88 y = 572$

$121 x - 88 y = 572$

has the general solution:

$\tuple {x, y} = \tuple {156 - 8 t, 208 - 11 t}$

### Example: $17 x + 15 y = 143$

$17 x + 15 y = 143$

has the general solution in positive integers:

$\tuple {x, y} = \tuple {4, 5}$

### Example: $35 x - 256 y = 48$

$35 x - 256 y = 48$

has the general solution:

$\tuple {x, y} = \tuple {16 + 256 t, 2 + 35 t}$