$\frac{x-144}{10}+\frac{x-130}{12}+\frac{x-112}{14}+\frac{x-90}{16}+\frac{x-69}{17}=15$
$⇔(\frac{x-144}{10}-1)+(\frac{x-130}{12}-2)+(\frac{x-112}{14}-3)+(\frac{x-90}{16}-4)+(\frac{x-69}{17}-5)=0$
$⇔\frac{x-154}{10}+\frac{x-154}{12}+\frac{x-154}{14}+\frac{x-154}{16}+\frac{x-154}{17}=15$
$⇔(x-154)(\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}+\frac{1}{17})=0$
$⇔x-154=0$ (vì $\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}+\frac{1}{17}\neq0$)
$⇔x=154$
Vậy $S=\{154\}$.
