linear diophantine equations show that there are no integral

linear diophantine equations. show that there are no integral solutions or find all solutions

a. 6x + 15 y =21

b. 15x+ 11 y =14

c. 30x + 18 y =96 *list all positive integral solutions*

Solution

a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns.

The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem:

This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution, then the other solutions have the form (x + kv, y ku), where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b.

solve the two equation by multoplying both (a) &(b) with15 and 6 inorder to get x and y values

15(6x + 15 y =21)---(1)

6(15x+ 11 y =14)----(2)   multiply and subtracing both (1)*(2)

we get x=1.45 and y=0.125

Note:

we get Y value by substitute in equation (a) and here equation (1),(2) are multiply opposite value decimals to get values