Solve the given linear diophantine equation. a) 12x+9y=15 b) 8x+10y=17

  A) (x, y) = (2-3n, 4n-1) . . . . for any integer n

  B) no solution

Step-by-step explanation:

A) All coefficients have a common factor of 3, so any solution of the reduced equation will be a solution of the given equation. The reduced equation is ...

  4x +3y = 5

A graph of the original shows (x, y) = (2, -1) is a solution. Then other solutions will be those values with a multiple of 4 added to y and the same multiple of 3 subtracted from x:

  (x, y) = (2 -3n, 4n -1)

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B) The left side of the equation is an even number for any integer values of x and y. The right side is an odd number. There can be no solution.