Now we're getting somewhere. With the research questions in the preceding section complete, we now know all of the solutions to the equation

ax+by=d,where

d= gcd(a,b). Therefore, it remains to determine all of the solutions to the equationax+by=c,where

c=kdandkis an integer. To get a feel for what is going on, let's look at an example. In the preceding section, we looked at the equation 7x+ 2y= 1. Let's change this a bit, say to7 x+ 2y= 5.Using the Euclidean Algorithm, we found that

x= 1,y= –3 is a solution to 7x+ 2y= 1. Thus, it is easy to see thatx= 1 · 5 = 5 andy= –3 · 5 = –15 is a solution to the equation 7x+ 2y= 5. What about the other solutions? The applet from the previous section can be used to search for additional solutions:

## Research Question 5

Suppose that gcd(

a,b) =dand that (x_{0},y_{0}) is a solution toax+by=d. Find the general form of all solutions (x,y) toax+by=kd,giving

xin terms ofx_{0}andyin terms ofy_{0}.

Section 2.1 | Section 2.2 | Section 2.3 | Section 2.4 | Section 2.5

Copyright © 2001 by W. H. Freeman and Company