8y - 9x = 31
<=> y = (31 + 9x)/8 (1)
ta có:
\(\dfrac{11}{17}< \dfrac{x}{y}< \dfrac{23}{29}\)
<=> \(\left\{{}\begin{matrix}11y< 17x\\29x< 23y\end{matrix}\right.\) (2)
thay (1) vào (2)
=> \(\left\{{}\begin{matrix}11\left(\dfrac{31+9x}{8}\right)< 17x\\29x< 23\left(\dfrac{31+9x}{8}\right)\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}\dfrac{341+99x}{8}< 17x\\29x< \dfrac{713+207x}{8}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{341-37x}{8}< 0\\\dfrac{25x-713}{8}< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}341-37x< 0\\25x-713< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>\dfrac{341}{37}\\x< 28,52\end{matrix}\right.\)\(\Leftrightarrow\dfrac{341}{7}< x< 28,52\)
=> x ∈ {10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28}
mà x,y nguyên dương => (x,y) = (17,23); (25;32)