\(2xy+y=10x+17\Leftrightarrow y\left(2x+1\right)=10x+17\Leftrightarrow y=\frac{10x+17}{2x+1}.Vì,y,nguyên,nên:10x+17⋮2x+1\Leftrightarrow10x+5+12⋮2x+1\Leftrightarrow5\left(2x+1\right)+12⋮2x+1\Leftrightarrow12⋮2x+1.Mà,2x+1,lẻ,nên:2x+1\in\left\{-1;1;3;-3\right\}\Leftrightarrow2x\in\left\{-2;0;2;-4\right\}\Leftrightarrow x\in\left\{-1;0;1;-2\right\}\)
\(+,x=-1\Rightarrow-2y+y=7\Leftrightarrow-y=7\Leftrightarrow y=-7\Rightarrow\left(x,y\right)=\left(-1;-7\right)\left(thoaman\right)\)
\(+,x=0\Rightarrow y=17\left(thoaman\right)\)
\(+,x=1\Rightarrow3y=27\Rightarrow y=9\left(thoaman\right)\)
\(+,x=-2\Rightarrow-3y=-3\Leftrightarrow3y=3\Leftrightarrow y=1\left(thoaman\right)\)
\(Vậy:\left(x,y\right)\in\left\{\left(-1;-7\right);\left(0;17\right);\left(1;9\right);\left(-2;1\right)\right\}\)
\(2xy-10x+y=17\)
\(\Leftrightarrow2x\left(y-5\right)+\left(y-5\right)=12\)
\(\Leftrightarrow\left(2x+1\right)\left(y-5\right)=12\)
Vì x \(\in\) N nên 2x + 1 là ước lẻ của 12 \(\Rightarrow2x+1\in\left\{1;3\right\}\)
Ta có bảng sau:
| 2x + 1 | 1 | 3 |
| y - 5 | 12 | 4 |
| x | 0 | 1 |
| y | 17 | 9 |
