The tutor delves deeper into when integer solutions can be expected for Ax+By=C.

In my dec 16 post I discussed finding integer coordinates for a linear equation in the form Ax+By=C. I pointed out that not always can integer solutions be found.

An equation of form Ax+By=C can be referred to as a linear diophantine equation. Furthermore, there is an easy way to tell if it has integer solutions. Specifically, if the greatest common factor of A and B is also a factor of C, the equation does have integer solutions.

Example 1: Determine if 2x-3y=11 has integer solutions; if so, find one.

Solution:

The greatest common factor of 2 and -3 is 1, which is also a factor of 11. Therefore,
2x-3y=11 does indeed have integer solutions, one being (4,-1): 2(4)-3(-1)=8+3=11.

Example 2: Determine if 4x-8y=6 has integer solutions; if so, find one.

Solution:

The greatest common factor of 4 and -8 is 4, which is not a factor 6. This equation has no integer solution.

Note that, for graphing purposes, convenient coordinates can still be found for 4x-8y=6. Despite its not having integer solutions, it does have (-2.5,-2) and (1.5,0), which are quite convenient for graphing.

I’ll be further discussing linear diophantine equations in a future post:)

Source:

Dudley, Underwood. Elementary Number Theory. New York: W H Freeman and
   Company, 1978.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.