Linear Diophantine equations are Diophantine equations of the form A₁x₁ + A₂x₂ + ... + A_nx_n= B, where all A_j and B are integers and each x_j is an unknown integer. They are used discrete mathematics and in integer programming in particular, such as the following system of equations:

A linear Diophantine equation has no solutions if gcd(A₁, A₂, ..., A_n) does not divide B. Otherwise, the solution proceeds as follows:

(Assuming you've done some algebra to narrow the system of equations down to a single Ax + By = C.)
1. Divide through by gcd(A,B) to get a reduced equation Dx + Ey = F. [gcd(D,E)=1]
2. Solve either of these guys for integers:
   

   a) x = (F - Ey) / D     where y is in {0, 1, 2, ..., (D-1)}
   
   b) y = (F - Dx) / E     where x is in {0, 1, 2, ..., (E-1)}
   

3. The resulting (x,y) is a particular solution of the equation.

To illustrate how to solve for a general solution of the equation, the solution to 5x + 7y = 10 is given:

1. x = (10 - 7y) / 5
2. For y = 0, x = 2; therefore (2,0) is a particular solution.

3.      5x     + 7y     = 10
   -    5(2)   + 7(0)   = 10      We plug in our particular solution here.
   ----------------------
        5(x-2) + 7(y-0) = 0   =====>5(x-2) = -7(y-0)
   

4. Then, (x-2) is divisible by -7, and (y-0) = y is divisible by 5. This means y = 5q and x = -7q + 2 for any integer q.
5. To find other solutions besides (2,0) just plug in different integers for q.