Vì (3;5)=1 nên pt có nghiệm nguyên
\(3x-5y=9\\ \Rightarrow y=\frac{3x-9}{5}=\frac{1-2x}{5}+x-2\)
Đặt t=\(\frac{1-2x}{5}\left(t\in Z\right)\)
\(\Rightarrow x=\frac{1-5t}{2}\)\(=\frac{t-1}{2}+1-3t\)
Đặt n=\(\frac{t-1}{2}\left(n\in Z\right)\)\(\Rightarrow t=2n+1\)
\(\Rightarrow\begin{cases}y=t+x-2\\x=n+1-3t\\t=2n+1\end{cases}\Rightarrow\begin{cases}y=-3n-3\\x=-5n-2\end{cases}\left(n\in Z\right)}}\)
\(\Leftrightarrow y=\dfrac{3x-9}{5}=\dfrac{3\left(x-3\right)}{5}\)\(\Rightarrow x-3⋮5\)\(\Rightarrow x=5k+3\left(k\in Z\right)\)\(\Rightarrow y=\dfrac{3.5k}{5}=3k\)
Vậy pt có vô số nghiệm với nghiệm tổng quát (x;y)=(5k+3\(\left(k\in Z\right)\) ;3k).
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