In my dec 16<\/a> 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.<\/p>\n An equation of form Ax+By=C can be referred to as a linear diophantine<\/em> 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.<\/p>\n Example 1: Determine if 2x-3y=11 has integer solutions; if so, find one.<\/strong><\/p>\n Solution:<\/p>\n The greatest common factor of 2 and -3 is 1, which is also a factor of 11. Therefore, Example 2: Determine if 4x-8y=6 has integer solutions; if so, find one.<\/strong><\/p>\n Solution:<\/p>\n The greatest common factor of 4 and -8 is 4, which is not a factor 6. This equation has no integer solution.<\/p>\n 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.<\/p>\n I’ll be further discussing linear diophantine equations in a future post:)<\/p>\n Source:<\/p>\n Dudley, Underwood. Elementary Number Theory<\/u>. New York: W H Freeman and
2x-3y=11 does indeed have integer solutions, one being (4,-1): 2(4)-3(-1)=8+3=11.<\/p>\n
Company, 1978.<\/p>\n