## 5.1   Techniques Used with Rationals and Integers

### 5.1.1  Finding Rational Solutions for Linear Equations

If a and b are rational numbers and a0, the equation ax = b has exactly one rational solution, namely b/a. We can find this by dividing by a, or equivalently, by multiplying both sides of the equation by 1/a. So, solving the equation is easy if a has a multiplicative inverse (a number which when multiplied by a gives 1).

The remaining case is when a = 0. Here our equation takes the form 0x = b. If b0, then there are no solutions to the equation, and if b = 0, then every rational number x is a solution to the equation.

### 5.1.2  Finding Integer Solutions for Linear Equations

Determining integer solutions to ax = b, when a and b are integers, is similar to the rational case, but a bit more complicated. The degenerate cases are the same: if a = b = 0, then every integer x is a solution; if a = 0 and b0, then there are no integer solutions to ax = b.

Now suppose a0. It follows directly from the definition of divisibility that ax = b has a solution if and only if a | b. Moreover, if a | b, then there is a unique solution, namely x = b/a.

Section 5.1 | Section 5.2 | Section 5.3 | Section 5.4 | Section 5.5 | Section 5.6

Chapter 5 | DNT Table of Contents

Copyright © 2001 by W. H. Freeman and Company