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

Question
Congruence
Solve the linear congruence
$$\displaystyle{x}+{2}{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{2}{x}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$

2021-02-06
Step 1
Given a system of linear congruences
$$\displaystyle{x}+{2}{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{2}{x}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
Solve it.
Step 2
Multiply the first congruence by 2.
$$\displaystyle{2}{x}+{4}{y}\equiv{2}{\left(\text{mod}{5}\right)}$$
Add the preceding one with the second congruence in the system.
$$\displaystyle{4}{x}+{5}{y}\equiv{3}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{4}{x}\equiv{3}{\left(\text{mod}{5}\right)}$$
Step 3
Multiply the congruence by $$\displaystyle{4}^{{-{1}}}{\left(\text{mod}{5}\right)}={4}$$ to get
$$\displaystyle{16}{x}\equiv{12}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{x}\equiv{2}{\left(\text{mod}{5}\right)}$$
Step 4
Plugging back in x for the second equation.
$$\displaystyle{4}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{y}\equiv-{3}{\left(\text{mod}{5}\right)}{\quad\text{or}\quad}{y}\equiv{2}{\left(\text{mod}{5}\right)}$$
Thus the solution of the system of congruences is
$$\displaystyle{x}\equiv{2}{\left(\text{mod}{5}\right)},{y}\equiv{2}{\left(\text{mod}{5}\right)}$$

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