The particular solution of 2.a is and the complete solution is .
The particular solution of 2.b is and the complete solution is
Step-by-step explanation:
The Diophantine equation ax+by=n has solutions if and only if gcd(a,b) | n. If this is true, it has infinitely many solutions, and any solution can be used to generate a complete solution.
This are the steps that you need to follow:
Use the Euclidean algorithm to compute gcd(a,b)=d
So for the equation ,
When the remainder r = 0, the gcd is the divisor, 3, in the last equation so gcd(21,15) = 3
Then, observe that 3|6 (that means when we divide 6 by 3, the remainder is 0.) is true therefore there are integer solutions to the equation.
The same for the equation ,
When the remainder r = 0, the gcd is the divisor, 1, in the last equation so gcd(22,15) = 1 and 1|6 is true therefore there are integer solutions to this equation.
The next step is reformat the equations from the Euclidean algorithm as follows:
For the equation
For the equation
Using substitution, go through the steps of the Euclidean algorithm to find a solution to the equation ax_{i}+by_{i}=d
For the equation
This gives x_i=-2 and y_i=3 as a solution to the equation
For the equation
This gives u_i=-2 and v_i=3 as a solution to the equation
The initial solution to the equation ax+by=n is the ordered pair
Then an initial solution to the equation is
For the equation
Find the complete solutions of the equation ax+by=n
For this you can use this Theorem:
if is an integer solution of the Diophantine equation ax+by=n, then all integer solutions to the equation are of the form
for some integer m.
The complete solutions for the equation are:
The complete solutions for the equation are: