b. \(\left\{{}\begin{matrix}2x+3y=1\\x+\dfrac{3}{2}-y=\dfrac{1}{2}\end{matrix}\right.\)
⇔ \(\left\{{}\begin{matrix}2x+3y=1\\x=y-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2\left(y-1\right)+3y=1\\x=y-1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}5y=3\\x=y-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{3}{5}\\x=-\dfrac{2}{5}\end{matrix}\right.\)
Vậy hệ phương trình đã cho có nghiệm duy nhất \(\left(x;y\right)=\left(-\dfrac{2}{5};\dfrac{3}{5}\right)\)
c. \(\left\{{}\begin{matrix}\dfrac{1}{2}x-y+2=0\\x-2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2}x-y=-2\\x=2y+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2}\left(2y+1\right)-y=-2\\x=2y+1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}0=-2\\x=2y+1\end{matrix}\right.\)
Vậy hệ phương trình đã cho vô nghiệm.
d. \(\left\{{}\begin{matrix}\dfrac{5x}{3}-\dfrac{2y}{5}=19\\4x+\dfrac{3y}{2}=21\end{matrix}\right.\)
⇔ \(\left\{{}\begin{matrix}\dfrac{5x}{3}-\dfrac{2y}{5}=19\\4x=21-\dfrac{3y}{2}\end{matrix}\right.\)
⇔ \(\left\{{}\begin{matrix}\dfrac{5x}{3}-\dfrac{2y}{5}=19\\x=\dfrac{21}{4}-\dfrac{3y}{8}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{21}{4}-\dfrac{3y}{8}\\\dfrac{5}{3}\left(\dfrac{21}{4}-\dfrac{3y}{8}\right)-\dfrac{2y}{5}=19\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=-10\\x=9\end{matrix}\right.\)
Vậy hệ phương trình đã cho có nghiệm duy nhất \(\left(x;y\right)=\left(9;-10\right)\)
