document.write( "Question 618734: Find all solutions (x,y) in positive integers to:
\n" ); document.write( "13x + 14y = 2008\r
\n" ); document.write( "\n" ); document.write( "I know we are supposed to use the Diophantine equation but I'm not
\n" ); document.write( "comletely sure how to yet.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #389124 by fcabanski(1391)\"\" \"About 
You can put this solution on YOUR website!
Step one - use trial and error to find one solution for x,y.


\n" ); document.write( "In this case, use x=162 and y=-7: (13*162+14(-7) = 2106 - 98 = 2008)


\n" ); document.write( "The general solution is x=x(0) + pt and y = y(0) - qt where x(0) and y(0) are the values found from trial and error, p = b (coefficient of y) / GCD (a,b) and q = a (coefficient of x) / GCD(a,b) and t is an integer.


\n" ); document.write( "The GCD of a and b is 1. So p = 14/1 = 14 and q = 13/1 = 13.


\n" ); document.write( "So the solution in this case is x = 162+14t and y = -7 - 13t
\n" ); document.write( "

Hope the solution helped. Sometimes you need more than a solution. Contact fcabanski@hotmail.com for online, private tutoring, or personalized problem solving (quick for groups of problems.)
\n" ); document.write( " \n" ); document.write( "

\n" );