Number Theory : Solve the system of congruence
(1) $28x+17y\equiv 18 \pmod{41}$
(2) $31x+11y\equiv 35\pmod{41}$
Attempt :
we know that equation (1) and (2) are in the same$\pmod{41}$. so we can use Modular arithmetic
lets multiply the first equation (1) by $31$ and the second equation(2) by $28$.
(1) $31\cdot(28x+17y)\equiv 31\cdot18 \pmod{41}$
(1) $868x+527y\equiv 558\pmod{41}$
(2) $28\cdot(31x+11y)\equiv 28\cdot35\pmod{41}$
(2) $868x+308y\equiv 980\pmod{41}$
So finally we can subtract equation (1) from (2) we get:
$219y\equiv -422\pmod{41}$
lets check the $\gcd(219,41)$ by Euclidian algorithm :
$219 = 41\cdot 5 + 14$
$41= 14\cdot 2 + 13$
$14= 13\cdot 1 + 1$
$\gcd(219,41)=1$
Hence, because the $\gcd$ is equal to $1$ we can find the Inverse and multiply the equation by the Inverse to find $y$.
$219y\equiv 1\pmod{41}$
$219a = 1+41k$ , the $41k$ must end with the digit of $8$ for $1+$digit $8$ will be $9$ so $k$ must be multiply of number with end digit of $8$.
I don't know how to continue from here .
Best Answer
Starting from your result of
$$219y \equiv -422 \pmod{41} \tag{1}\label{eq1A}$$
the first thing I recommend doing is reducing the coefficients to smaller values since these are generally easier to deal with, plus sometimes you may be able to find simplifications you can use, such as common factors relatively prime to $41$ so they can be "removed" using their multiplicative inverses. Doing this, and using that $\gcd(2, 41) = \gcd(7, 41) = 1$, gives
$$\begin{equation}\begin{aligned} 14y & \equiv -12 \pmod{41} \\ 7y & \equiv -6 \pmod{41} \\ 7y & \equiv 35 \pmod{41} \\ y & \equiv 5 \pmod{41} \end{aligned}\end{equation}\tag{2}\label{eq2A}$$
Using this result in your first congruence equation, along with $\gcd(28, 41) = 1$, gives
$$\begin{equation}\begin{aligned} 28x + 85 & \equiv 18 \pmod{41} \\ 28x + 3 & \equiv 18 \pmod{41} \\ 28x & \equiv 15 \pmod{41} \\ 28x & \equiv 56 \pmod{41} \\ x & \equiv 2 \pmod{41} \end{aligned}\end{equation}\tag{3}\label{eq3A}$$
In general, though, if you can't fairly easily simplify the results as I did above, you can instead use something like either the constructive proof or direct proof methods described in Wikipedia's "Chinese remainder theorem" article.