đúng rùi đó

a) Ta có: \(\left\{{}\begin{matrix}\dfrac{5}{x-1}+\dfrac{1}{y-1}=10\\\dfrac{1}{x-1}-\dfrac{3}{y-1}=18\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{5}{x-1}+\dfrac{1}{y-1}=10\\\dfrac{5}{x-1}-\dfrac{15}{y-1}=90\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{16}{y-1}=-80\\\dfrac{1}{x-1}-\dfrac{3}{y-1}=18\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}y-1=\dfrac{-1}{5}\\\dfrac{1}{x-1}=18+\dfrac{3}{y-1}=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{4}{5}\\x-1=\dfrac{1}{3}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{4}{3}\\y=\dfrac{4}{5}\end{matrix}\right.\)

\(\left(\dfrac{x}{x+1}+\dfrac{x-1}{x}\right):\left(\dfrac{x}{x+1}-\dfrac{x-1}{x}\right)\) \(\left(đk:x\ne0;-1\right)\)

\(=\dfrac{x^2+\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}:\left(\dfrac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}\right)\)

\(=\dfrac{x^2+x^2-1}{x\left(x+1\right)}.\dfrac{x\left(x+1\right)}{x^2-x^2+1}\)

\(=\dfrac{\left(2x^2-1\right)x\left(x+1\right)}{x\left(x+1\right)}=2x^2-1\)

a: \(\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{x+y}\cdot\left(\frac{1}{x}+\frac{1}{y}\right)\)

\(=\frac{x^2+y^2}{x^2y^2}+\frac{2}{x+y}\cdot\frac{x+y}{xy}=\frac{x^2+y^2}{x^2y^2}+\frac{2}{xy}=\frac{x^2+y^2+2xy}{x^2y^2}=\frac{\left(x+y\right)^2}{x^2y^2}\)

\(\left\lbrack\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{x+y}\cdot\left(\frac{1}{x}+\frac{1}{y}\right)\right\rbrack:\frac{x^3+y^3}{x^2y^2}\)

\(=\frac{\left(x+y\right)^2}{x^2y^2}\cdot\frac{x^2y^2}{\left(x+y\right)\left(x^2-xy+y^2\right)}=\frac{x+y}{x^2-xy+y^2}\)

b: \(\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\)

\(=\frac{\left(2x+y\right)^2+2\left(2x-y\right)\left(2x+y\right)+\left(2x-y\right)^2}{\left(2x-y\right)^2\cdot\left(2x+y\right)^2}\)

\(=\frac{\left(2x+y+2x-y\right)^2}{\left(2x+y\right)^2\cdot\left(2x-y\right)^2}=\frac{16x^2}{\left(2x+y\right)^2\cdot\left(2x-y\right)^2}\)

\(\left\lbrack\frac{1}{\left(2x-y\right)^2}+\frac{2}{4x^2-y^2}+\frac{1}{\left(2x+y\right)^2}\right\rbrack\cdot\frac{4x^2+4xy+y^2}{16x}\)

\(=\frac{16x^2}{\left(2x+y\right)^2\cdot\left(2x-y\right)^2}\cdot\frac{\left(2x+y\right)^2}{16x}=\frac{x}{\left(2x-y\right)^2}\)

a: \(A=\dfrac{x^2+2xy+y^2-x^2+xy+2y^2}{\left(x-y\right)\left(x+y\right)}\)

\(=\dfrac{3y^2+3xy}{\left(x-y\right)\left(x+y\right)}=\dfrac{3y}{x-y}\)

a) Ta có: \(\left(x-\dfrac{1}{1-x}\right):\dfrac{x^2-x+1}{x^2-2x+1}\)

\(=\left(x+\dfrac{1}{x-1}\right):\dfrac{x^2-x+1}{\left(x-1\right)^2}\)

\(=\dfrac{x^2-x+1}{x-1}\cdot\dfrac{\left(x-1\right)^2}{x^2-x+1}\)

\(=x-1\)

b) Ta có: \(\left(1+\dfrac{x}{y}+\dfrac{x^2}{y^2}\right)\left(1-\dfrac{x}{y}\right)\cdot\dfrac{y^2}{x^3-y^3}\)

\(=\left(\dfrac{y^2}{y^2}+\dfrac{xy}{y^2}+\dfrac{x^2}{y^2}\right)\cdot\left(\dfrac{y-x}{y}\right)\cdot\dfrac{y^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)

\(=\dfrac{x^2+xy+y^2}{y^2}\cdot\dfrac{-\left(x-y\right)}{y}\cdot\dfrac{y^2}{\left(x-y\right)\left(x^2+xy+y^2\right)}\)

\(=\dfrac{-1}{y}\)

a: ĐKXĐ: x<>0; y<>0

\(\begin{cases}\frac{1}{x}+\frac{1}{y}=\frac12\\ \frac{3}{x}-\frac{4}{y}=-1\end{cases}\Rightarrow\begin{cases}\frac{1}{y}=\frac12-\frac{1}{x}\\ \frac{3}{x}-4\left(\frac12-\frac{1}{x}\right)=-1\end{cases}\)

=>\(\begin{cases}\frac{1}{y}=\frac12-\frac{1}{x}\\ \frac{3}{x}-2+\frac{4}{x}=-1\end{cases}\Rightarrow\begin{cases}\frac{7}{x}=1\\ \frac{1}{y}=\frac12-\frac{1}{x}\end{cases}\Rightarrow\begin{cases}x=7\\ \frac{1}{y}=\frac12-\frac17=\frac{5}{14}\end{cases}\)

=>x=7(nhận ) và y=14/5(nhận)

b: ĐKXĐ: 2x-y<>0 và x+y<>0

\(\begin{cases}\frac{3}{2x-y}-\frac{6}{x+y}=-1\\ \frac{1}{2x-y}-\frac{1}{x+y}=0\end{cases}\Rightarrow\begin{cases}\frac{1}{2x-y}=\frac{1}{x+y}\\ \frac{3}{x+y}-\frac{6}{x+y}=1\end{cases}\)

=>\(\begin{cases}2x-y=x+y\\ \frac{-3}{x+y}=1\end{cases}\Rightarrow\begin{cases}2x-y=x+y\\ x+y=-3\end{cases}\)

=>\(\begin{cases}2x-y=-3\\ x+y=-3\end{cases}\Rightarrow\begin{cases}2x-y+x+y=-3-3\\ x+y=-3\end{cases}\Rightarrow\begin{cases}3x=-6\\ y=-3-x\end{cases}\)

=>x=-2(nhận) và y=-3-(-2)=-3+2=-1(nhận)

c: ĐKXĐ: x<>-1 và y<>3

\(\begin{cases}\frac{5x}{x+1}+\frac{y}{y-3}=27\\ \frac{2x}{x+1}-\frac{3y}{y-3}=4\end{cases}\Rightarrow\begin{cases}\frac{y}{y-3}=27-\frac{5x}{x+1}\\ \frac{2x}{x+1}-3\left(27-\frac{5x}{x+1}\right)=4\end{cases}\)

=>\(\begin{cases}\frac{2x}{x+1}-81+\frac{15x}{x+1}=4\\ \frac{y}{y-3}=27-\frac{5x}{x+1}\end{cases}\Rightarrow\begin{cases}\frac{17x}{x+1}=85\\ \frac{y}{y-3}=27-\frac{5x}{x+1}\end{cases}\)

=>\(\begin{cases}\frac{x}{x+1}=5\\ \frac{y}{y-3}=27-5\cdot5=2\end{cases}\Rightarrow\begin{cases}5x+5=x\\ 2y-6=y\end{cases}\)

=>4x=-5 và y=6

=>x=-5/4(nhận) và y=6(nhận)

d: ĐKXĐ: x<>-2; y<>0

\(\begin{cases}\frac{7}{x+2}+\frac{3}{y}=2\\ \frac{4}{x+2}-\frac{1}{y}=\frac52\end{cases}\Rightarrow\begin{cases}\frac{1}{y}=\frac{4}{x+2}-\frac52\\ \frac{7}{x+2}+3\left(\frac{4}{x+2}-\frac52\right)=2\end{cases}\)

=>\(\begin{cases}\frac{1}{y}=\frac{4}{x+2}-\frac52\\ \frac{19}{x+2}-\frac{15}{2}=2\end{cases}\Rightarrow\begin{cases}\frac{19}{x+2}=\frac{19}{2}\\ \frac{1}{y}=\frac{4}{x+2}-\frac52\end{cases}\)

=>\(\begin{cases}x+2=2\\ \frac{1}{y}=\frac42-\frac52=\frac{-1}{2}\end{cases}\Rightarrow\begin{cases}x=0\\ y=-2\end{cases}\) (nhận)

\(\left[\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{2}{x+y}.\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\right]:\dfrac{x^3+y^3}{x^2y^2}-\dfrac{x+y}{x^2-xy+y^2}\)

\(=\left[\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{2}{x+y}.\dfrac{x+y}{xy}\right].\dfrac{x^2y^2}{x^3+y^3}-\dfrac{x+y}{x^2-xy+y^2}\)

\(=\left[\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{2}{xy}\right].\dfrac{x^2y^2}{\left(x+y\right)\left(x^2-xy+y^2\right)}-\dfrac{x+y}{x^2-xy+y^2}\)

\(=\dfrac{y^2+x^2+2xy}{x^2y^2}.\dfrac{x^2y^2}{\left(x+y\right)\left(x^2-xy+y^2\right)}-\dfrac{x+y}{x^2-xy+y^2}\)

\(=\dfrac{\left(x+y\right)^2}{\left(x+y\right)\left(x^2-xy+y^2\right)}-\dfrac{x+y}{x^2-xy+y^2}\)

=\(=\dfrac{x+y}{x^2-xy+y^2}-\dfrac{x+y}{x^2-xy+y^2}=0\)

a:

ĐKXĐ: x<>0; y<>0

\(\frac{2}{x}+\frac{1}{y}=3\)

=>\(\frac{2y+x}{xy}=3\)

=>3xy=x+2y

=>3xy-x-2y=0

=>x(3y-1)-\(2y+\frac23=\frac23\)

=>\(3x\left(y-\frac13\right)-2\left(y-\frac13\right)=\frac23\)

=>\(\left(3x-2\right)\left(y-\frac13\right)=\frac23\)

=>(3x-2)(3y-1)=2

=>(3x-2;3y-1)∈{(1;2);(2;1);(-1;-2);(-2;-1)}

=>(3x;3y)∈{(3;3);(4;2);(1;-1);(0;0)}

=>(x;y)∈{(1;1);(4/3;2/3);(1/3;-1/3);(0;0)}

mà x,y nguyên

nên x=1; y=1

b: ĐKXĐ: x<>0; y<>0

\(\frac{2}{y}-\frac{1}{x}=\frac{8}{xy}+1\)

=>\(\frac{2x-y}{xy}=\frac{8+xy}{xy}\)

=>xy+8=2x-y

=>xy-2x+y+8=0

=>x(y-2)+y-2+10=0

=>(x+1)(y-2)=-10

=>(x+1;y-2)∈{(1;-10);(-10;1);(-1;10);(10;-1);(2;-5);(-5;2);(-2;5);(5;-2)}

=>(x;y)∈{(0;-8);(-11;3);(-2;12);(9;1);(1;-3);(-6;4);(-3;7);(4;0)}

mà x<>0; y<>0

nên (x;y)∈{(-11;3);(-2;12);(9;1);(1;-3);(-6;4);(-3;7)}

d: ĐKXĐ: x<>0; y<>0

\(-\frac{3}{y}-\frac{12}{xy}=1\)

=>\(\frac{-3x-12}{xy}=1\)

=>xy=-3x-12

=>xy+3x=-12

=>x(y+3)=-12

=>(x;y+3)∈{(1;-12);(-12;1);(-1;12);(12;-1);(2;-6);(-6;2);(-2;6);(6;-2);(3;-4);(-4;3);(-3;4);(4;-3)}

=>(x;y)∈{(1;-15);(-12;-2);(-1;9);(12;-4);(2;-9);(-6;-1);(-2;3);(6;-5);(3;-7);(-4;0);(-3;1);(4;-6)}

mà y<>0

nên (x;y)∈{(1;-15);(-12;-2);(-1;9);(12;-4);(2;-9);(-6;-1);(-2;3);(6;-5);(3;-7);(-3;1);(4;-6)}

e: ĐKXĐ: y<>0

\(\frac{x}{8}-\frac{1}{y}=\frac14\)

=>\(\frac{x}{8}-\frac14=\frac{1}{y}\)

=>\(\frac{x-2}{8}=\frac{1}{y}\)

=>(x-2)y=8

=>(x-2;y)∈{(1;8);(8;1);(-1;-8);(-8;-1);(2;4);(4;2);(-2;-4);(-4;-2)}

=>(x;y)∈{(3;8);(10;1);(1;-8);(-6;-1);(4;4);(6;2);(0;-4);(-2;-2)}

mà y<>0

nên (x;y)∈{(3;8);(10;1);(1;-8);(-6;-1);(4;4);(6;2);(0;-4);(-2;-2)}