5.1.1 Finding Rational Solutions for Linear Equations
If a and b are rational numbers and
a
0, 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.