Solve the linear congruence x+2y -= 1(mod 5) 2x+y -= 1(mod 5)

Question

Solve the linear congruence

x + 2 y \equiv 1 (mod 5)

2 x + y \equiv 1 (mod 5)

Answer 1

Step 1
Given a system of linear congruences

x + 2 y \equiv 1 (mod 5)

2 x + y \equiv 1 (mod 5)

Solve it.
Step 2
Multiply the first congruence by 2.

2 x + 4 y \equiv 2 (mod 5)

Add the preceding one with the second congruence in the system.

4 x + 5 y \equiv 3 (mod 5)

\Rightarrow 4 x \equiv 3 (mod 5)

Step 3
Multiply the congruence by

4^{- 1} (mod 5) = 4

to get

16 x \equiv 12 (mod 5)

\Rightarrow x \equiv 2 (mod 5)

Step 4
Plugging back in x for the second equation.

4 + y \equiv 1 (mod 5)

\Rightarrow y \equiv - 3 (mod 5) or y \equiv 2 (mod 5)

Thus the solution of the system of congruences is

x \equiv 2 (mod 5), y \equiv 2 (mod 5)

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