a:
ĐKXĐ: x<>0; y<>0
\(\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{5}\)
=>\(\dfrac{x+y}{xy}=\dfrac{1}{5}\)
=>5x+5y=xy
=>5x-xy+5y=0
=>x(5-y)+5y-25=-25
=>-x(y-5)+5(y-5)=-25
=>(-x+5)(y-5)=-25
=>(x-5)(y-5)=25
=>\(\left(x-5;y-5\right)\in\left\{\left(1;25\right);\left(25;1\right);\left(-1;-25\right);\left(-25;-1\right);\left(5;5\right);\left(-5;-5\right)\right\}\)
=>(x;y)\(\in\left\{\left(6;30\right);\left(30;6\right);\left(4;-20\right);\left(-20;4\right);\left(10;10\right);\left(0;0\right)\right\}\)
mà x<>0;y<>0
nên \(\left(x;y\right)\in\left\{\left(6;30\right);\left(30;6\right);\left(4;-20\right);\left(-20;4\right);\left(10;10\right)\right\}\)
b:
ĐKXĐ: x<>0; y<>0
\(\dfrac{2}{x}+\dfrac{1}{y}=3\)
=>\(\dfrac{2y+x}{xy}=3\)
=>3xy=x+2y
=>3xy-x-2y=0
=>\(x\left(3y-1\right)-2y+\dfrac{2}{3}=\dfrac{2}{3}\)
=>\(3x\left(y-\dfrac{1}{3}\right)-2\left(y-\dfrac{1}{3}\right)=\dfrac{2}{3}\)
=>\(\left(3x-2\right)\left(y-\dfrac{1}{3}\right)=\dfrac{2}{3}\)
=>(3x-2)(3y-1)=2
=>\(\left(3x-2;3y-1\right)\in\left\{\left(1;2\right);\left(2;1\right);\left(-1;-2\right);\left(-2;-1\right)\right\}\)
=>\(\left(3x;3y\right)\in\left\{\left(3;3\right);\left(4;2\right);\left(1;-1\right);\left(0;0\right)\right\}\)
=>\(\left(x;y\right)\in\left\{\left(1;1\right);\left(\dfrac{4}{3};\dfrac{2}{3}\right);\left(\dfrac{1}{3};-\dfrac{1}{3}\right);\left(0;0\right)\right\}\)
mà x,y nguyên và x<>0; y<>0
nên \(\left(x;y\right)\in\left\{\left(1;1\right)\right\}\)
c:
ĐKXĐ: x<>0; y<>0
\(\dfrac{2}{y}-\dfrac{1}{x}=\dfrac{8}{xy}+1\)
=>\(\dfrac{2x-y}{xy}=\dfrac{8}{xy}+\dfrac{xy}{xy}\)
=>2x-y=8+xy
=>2x-y-xy=8
=>2x-xy-y=8
=>x(2-y)-y+2=10
=>(2-y)(x+1)=10
=>(x+1)(y-2)=-10
=>\(\left(x+1;y-2\right)\in\){(1;-10);(-10;1);(-1;10);(10;-1);(2;-5);(-5;2);(-2;5);(5;-2)}
=>(x;y)\(\in\){(0;-8);(-11;3);(-2;12);(9;1);(1;-3);(-6;4);(-3;7);(4;0)}
mà x<>0; y<>0
nên (x;y)\(\in\){(-11;3);(-2;12);(9;1);(1;-3);(-6;4);(-3;7)}
d: \(\dfrac{2x}{3}-\dfrac{2}{y}=\dfrac{1}{3}\)
=>\(\dfrac{2xy-6}{3y}=\dfrac{y}{3y}\)
=>2xy-6=y
=>2xy-y=6
=>y(2x-1)=6
mà 2x-1 lẻ(do x là số nguyên)
nên \(\left(2x-1;y\right)\in\left\{\left(1;6\right);\left(-1;-6\right);\left(3;2\right);\left(-3;-2\right)\right\}\)
=>\(\left(x;y\right)\in\left\{\left(1;6\right);\left(0;-6\right);\left(2;2\right);\left(-1;-2\right)\right\}\)