Giải hệ pt và pt sau:
a.\(\left\{{}\begin{matrix}\left(2x-3\right)\cdot\left(2y+4\right)=4x\cdot\left(y-3\right)+54\\\left(x+1\right)\cdot\left(3y-3\right)=3y\left(x+1\right)-12\end{matrix}\right.\)
b.\(\left\{{}\begin{matrix}x+y-1=0\\x^2+xy+3=0\end{matrix}\right.\)
c.\(\left\{{}\begin{matrix}2x-3y=5\\x^2-y^2=40\end{matrix}\right.\)
d.\(\left\{{}\begin{matrix}3x+2y=36\\\left(x-2\right)\left(y-3\right)=18\end{matrix}\right.\)
e.\(\left\{{}\begin{matrix}2x+y=5m-1\\x-2y=2\end{matrix}\right.\) . Tìm m để hệ có nghiệm (x;y) t/m x\(^2\)-2y\(^2\)=1
f. \(\frac{t^2}{t-1}+t=\frac{2t^2+5t}{t+1}\)
g.\(\frac{x^2+2x-3}{x^2-9}+\frac{2x^2-2}{x^2-3x+2}=8\)
a.
\(\Leftrightarrow\left\{{}\begin{matrix}4xy+8x-6y-12=4xy-12x+54\\3xy-3x+3y-3=3xy+3y-12\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}20x-6y=66\\-3x=-9\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=3\\y=-1\end{matrix}\right.\)
b.
\(\Leftrightarrow\left\{{}\begin{matrix}y=1-x\\x^2+xy+3=0\end{matrix}\right.\)
\(\Leftrightarrow x^2+x\left(1-x\right)+3=0\)
\(\Leftrightarrow x+3=0\Rightarrow x=-3\Rightarrow y=4\)
c.
\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{2x-5}{3}\\x^2-y^2=40\end{matrix}\right.\)
\(\Rightarrow x^2-\left(\frac{2x-5}{3}\right)^2-40=0\)
\(\Leftrightarrow9x^2-\left(4x^2-20x+25\right)-360=0\)
\(\Leftrightarrow5x^2+20x-385=0\)
\(\Rightarrow\left[{}\begin{matrix}x=7\Rightarrow y=3\\x=-11\Rightarrow y=-9\end{matrix}\right.\)
d.
\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{36-3x}{2}\\\left(x-2\right)\left(y-3\right)=18\end{matrix}\right.\)
\(\Rightarrow\left(x-2\right)\left(\frac{36-3x}{2}-3\right)=18\)
\(\Leftrightarrow\left(x-2\right)\left(10-x\right)=12\)
\(\Leftrightarrow-x^2+12x-32=0\Rightarrow\left[{}\begin{matrix}x=4\Rightarrow y=12\\x=8\Rightarrow y=6\end{matrix}\right.\)
e.
\(\Leftrightarrow\left\{{}\begin{matrix}4x+2y=10m-2\\x-2y=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}5x=10m\\x-2y=2\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=2m\\y=m-1\end{matrix}\right.\)
\(x^2-2y^2=1\)
\(\Leftrightarrow4m^2-2\left(m-1\right)^2=1\)
\(\Leftrightarrow4m^2-\left(2m^2-4m+2\right)-1=0\)
\(\Leftrightarrow2m^2+4m-3=0\Rightarrow m=\frac{-2\pm\sqrt{10}}{2}\)
f.
ĐKXĐ: \(t\ne\pm1\)
\(\Leftrightarrow\frac{t\left(t-1\right)+t}{t-1}+t=\frac{2t\left(t+1\right)+3t}{t+1}\)
\(\Leftrightarrow t+\frac{t}{t-1}+t=2t+\frac{3t}{t+1}\)
\(\Leftrightarrow\frac{t}{t-1}=\frac{3t}{t+1}\)
\(\Leftrightarrow t\left(t+1\right)=3t\left(t-1\right)\)
\(\Leftrightarrow2t^2-4t=0\Rightarrow\left[{}\begin{matrix}t=0\\t=2\end{matrix}\right.\)
g.
ĐKXĐ: \(x\ne\left\{\pm3;1;2\right\}\)
\(\Leftrightarrow\frac{\left(x-1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}+\frac{2\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x-2\right)}=8\)
\(\Leftrightarrow\frac{x-1}{x-3}+\frac{2\left(x+1\right)}{x-2}=8\)
\(\Leftrightarrow\left(x-1\right)\left(x-2\right)+2\left(x+1\right)\left(x-3\right)=8\left(x-2\right)\left(x-3\right)\)
\(\Leftrightarrow x^2-3x+2+2x^2-4x-6=8x^2-40x+48\)
\(\Leftrightarrow5x^2-33x+52=0\Rightarrow\left[{}\begin{matrix}x=4\\x=\frac{13}{5}\end{matrix}\right.\)
