In Exercise , Solve the systems of equations in .
The solution of the system of equation in
1) Multiplicative inverses in :
An element of has a multiplicative inverse in , if and only if and are relatively prime.
2) is the unique solution to the equation in .
The set of congruence classes
Consider the system of linear equations
Need to eliminate one of the element and .
can be eliminate by multiplying the top equation by .
The top equation becomes .
It is same as,
, since in .
Therefore, system becomes
Now subtract top equation from the bottom one,
This simplifies to
Since and are relatively prime, has a multiplicative inverse in .
Using Euclidean Algorithm to find the multiplicative inverse of in .
Substituting the remainders in turn,
, as in
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