Linear Diophantine Equation/Examples
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Examples of Linear Diophantine Equations
Example: $15 x + 27 y = 1$
The linear diophantine equation:
- $15 x + 27 y = 1$
has no solutions for $x$ and $y$ integers.
Example: $5 x + 6 y = 1$
The linear diophantine equation:
- $5 x + 6 y = 1$
has the general solution:
- $x = -1 + 6 t, y = 1 - 5 t$
Example: $2 x + 3 y = 4$
The linear diophantine equation:
- $2 x + 3 y = 4$
has the general solution:
- $x = -4 + 3 t, y = 4 - 2 t$
Example: $17 x + 19 y = 23$
The linear diophantine equation:
- $17 x + 19 y = 23$
has the general solution:
- $\tuple {x, y} = \tuple {207 + 19 t, -184 - 17 t}$
Example: $15 x + 51 y = 41$
The linear diophantine equation:
- $15 x + 51 y = 41$
has no solutions for $x$ and $y$ integers.
Example: $23 x + 29 y = 25$
The linear diophantine equation:
- $23 x + 29 y = 25$
has the general solution:
- $\tuple {x, y} = \tuple {-125 + 29 t, 100 - 23 t}$
Example: $10 x - 8 y = 42$
The linear diophantine equation:
- $10 x - 8 y = 42$
has the general solution:
- $\tuple {x, y} = \tuple {21 - 4 t, 21 - 5 t}$
Example: $121 x - 88 y = 572$
The linear diophantine equation:
- $121 x - 88 y = 572$
has the general solution:
- $\tuple {x, y} = \tuple {156 - 8 t, 208 - 11 t}$
Example: $17 x + 15 y = 143$
The linear diophantine equation:
- $17 x + 15 y = 143$
has the general solution in positive integers:
- $\tuple {x, y} = \tuple {4, 5}$
Example: $35 x - 256 y = 48$
The linear diophantine equation:
- $35 x - 256 y = 48$
has the general solution:
- $\tuple {x, y} = \tuple {16 + 256 t, 2 + 35 t}$