Bài 1:
a) \(\left\{{}\begin{matrix}x+2y=3\\3x-2y=5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3-2y\\3\left(3-2y\right)-2y=5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3-2y\\9-6y-2y=5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3-2y\\8y=9-5\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3-2\cdot\dfrac{1}{2}\\y=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=\dfrac{1}{2}\end{matrix}\right.\)
b) \(\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}=1\\\dfrac{2}{x}\cdot\dfrac{3}{y}=1\end{matrix}\right.\left(x;y\ne0\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{y}{xy}+\dfrac{x}{xy}=1\\\dfrac{6}{xy}=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x+y}{xy}=1\\xy=6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=xy=6\\xy=6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=6\\xy=6\end{matrix}\right.\)
Khi đó x và y là nghiệm cùa pt:
\(t^2-6t+6=0\)
\(\Delta=\left(-6\right)^2-4\cdot1\cdot6=12>0\)
\(t_1=\dfrac{6+\sqrt{12}}{2}=3+\sqrt{3}\)
\(t_2=\dfrac{6-\sqrt{12}}{2}=3-\sqrt{3}\)
Vậy: \(\left(x;y\right)=\left\{\left(3+\sqrt{3};3-\sqrt{3}\right);\left(3-\sqrt{3};3+\sqrt{3}\right)\right\}\)
Bài 2:
a) Hàm số: `y=ax+b` đi qua 2 điểm `A(2;1)` và `B(1;2)` ta lần lượt thay tọa độ của chúng vào ta có hpt"
\(\left\{{}\begin{matrix}a\cdot2+b=1\\a\cdot1+b=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2a+b=1\\2a+2b=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}b=3\\2a+3=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}b=3\\a=-1\end{matrix}\right.\)
Vậy: \(y=-x+3\)
b) \(\left\{{}\begin{matrix}4x-y=3\\mx+3y=5\end{matrix}\right.\)
Hệ vô nghiệm khi: \(\dfrac{4}{m}=\dfrac{-1}{3}\ne\dfrac{3}{5}\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{4}{m}=\dfrac{-1}{3}\\\dfrac{4}{m}\ne\dfrac{3}{5}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m=\dfrac{4\cdot3}{-1}=12\\m\ne\dfrac{20}{3}\end{matrix}\right.\left(tm\right)\)
Vậy khi m = 12 thì hpt vô nghiệm
