## 5.1.1 Finding Rational Solutions for Linear Equations

If

aandbare rational numbers anda_{}0, the equationax=bhas exactly one rational solution, namelyb/a. We can find this by dividing bya, or equivalently, by multiplying both sides of the equation by 1/a. So, solving the equation is easy ifahas amultiplicative inverse(a number which when multiplied byagives 1).The remaining case is when

a= 0. Here our equation takes the form 0x=b. Ifb_{}0, then there are no solutions to the equation, and ifb= 0, theneveryrational numberxis a solution to the equation.## 5.1.2 Finding Integer Solutions for Linear Equations

Determining integer solutions to

ax=b, whenaandbare integers, is similar to the rational case, but a bit more complicated. The degenerate cases are the same: ifa=b= 0, then every integerxis a solution; ifa= 0 andb_{}0, then there are no integer solutions toax=b.Now suppose

a_{}0. It follows directly from the definition of divisibility thatax=bhas a solution if and only ifa|b. Moreover, ifa|b, then there is a unique solution, namelyx=b/a.

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

Copyright © 2001 by W. H. Freeman and Company