<s>[INST] Given that $1 = 27 \times 11 - 74 \times 4$, solve the following equations in modulo $74$: $ 3x - y = 1$; $2x + 3y = 0$ Given that $1 = 27 \times 11 - 74 \times 4$, solve the following equations in modulo $74$: 
$3x - y = 1$; 
$2x + 3y = 0$. 
Thank you.
 [/INST] In mod $74$, we have 
$$27\times 11\equiv 1\iff 3\times 27\times 11\equiv 3.$$
Then, 
$$\begin{align}y\equiv 3x-1&\Rightarrow 3y\equiv 9x-3\\&\Rightarrow -2x\equiv 9x-3\\&\Rightarrow 11x\equiv 3\\&\Rightarrow 11x\equiv 3\times 27\times 11\\&\Rightarrow x\equiv 3\times 27\equiv 81\equiv \color{red}{7}\end{align}$$
Hence, $$y\equiv 3x-1\equiv 3\times 7-1\equiv \color{red}{20}.$$
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