Method of successive substitution

The method of successive substitution, in mathematics, is a method of solving problems of simultaneous congruences by using the definition of the congruence equation.

For example, consider the simple set of simultaneous congruences

x ≡ 3 (mod 4)

x ≡ 11 (mod 12)

Now, for x ≡ 3 (mod 4) to be true, x=3+4j for some integer j. Substitute this in the second equation

3+4j ≡ 11 (mod 12)

since we are looking for a solution to both equations.

Subtract 3 from both sides (this is permitted in modular arithmetic)

4j ≡ 11 (mod 12)

We need to find the multiplicative inverse of 4 mod 12, which we can find as 4. Multiply throughout to get

j ≡ 44 (mod 12)

j ≡ 8 (mod 12)

For the above to be true, j=8+12k for some integer k. Now substitute back into 3+4j and we obtain

x=3+4(8+12k)

Expand out

x=35+48k

to obtain the solution

x≡ 35 (mod 48)

In general:

• write the first equation in its equivalent form
• substitute it into the next
  • simplify, use the multiplicative inverse if necessary
• continue until the last equation
• back substitute, then simplify
• rewrite back in the congruence form