= Solution
Use the <Euclidean algorithm>: $23=18+5$, $18=3\cdot5+3$, $5=3+2$, and $3=2+1$. Back-substitution gives the <Bezout identity> $1=9\cdot18-7\cdot23$. Multiplying by $101$ gives one solution of the <linear Diophantine equation>, and shifting the coefficients by multiples of $(23,-18)$ reduces it to
$$
\boxed{x=12,\qquad y=-5.}
$$
Indeed, $18\cdot12+23(-5)=216-115=101$. More generally all solutions are $x=12+23k$, $y=-5-18k$, $k\in\mathbb Z$: subtracting this particular solution gives $18(x-12)=-23(y+5)$, and the <coprime integers> $18,23$ <force> $23\mid(x-12)$.
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