a) Thay m=1 vào hệ phương trình, ta được:
\(\left\{{}\begin{matrix}x+2y=4\\2x-3y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+4y=8\\2x-3y=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}7y=7\\x+2y=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=4-2y=4-2=2\end{matrix}\right.\)
Vậy: Khi m=1 thì hệ phương trình có nghiệm duy nhất là (x,y)=(2;1)
b) Ta có: \(\left\{{}\begin{matrix}x+2y=m+3\\2x-3y=m\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2y\\2\left(m+3-2y\right)-3y=m\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2y\\2m+6-4y-3y-m=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2y\\-7y+m+6=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2y\\-7y=-m-6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2y\\y=\dfrac{m+6}{7}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=m+3-2\cdot\dfrac{m+6}{7}\\y=\dfrac{m+6}{7}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=m+3-\dfrac{2m+12}{7}\\y=\dfrac{m+6}{7}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{7m+21-2m-12}{7}=\dfrac{5m+9}{7}\\y=\dfrac{m+6}{7}\end{matrix}\right.\)
Để hệ phương trình có nghiệm duy nhất thỏa mãn x+y=3 thì \(\dfrac{5m+9}{7}+\dfrac{m+6}{7}=3\)
\(\Leftrightarrow6m+15=21\)
\(\Leftrightarrow6m=6\)
hay m=1
Vậy: Khi m=1 thì hệ phương trình có nghiệm duy nhất thỏa mãn x+y=3
a/ Thay \(m=1\) vào hpt ta có :
\(\left\{{}\begin{matrix}x+2y=4\\2x-3y=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)
Vậy...
b/ Ta có :
\(\left\{{}\begin{matrix}x+2y=m+3\\2x-3y=m\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{m+3}{2y}\\\dfrac{2\left(m+3\right)}{2y}-3y=m\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{m+3}{2y}\\\dfrac{m+3}{y}-3y=m\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{m+3}{2y}\\m-3y^2+3=my\end{matrix}\right.\)