Question

1, 1/x+1/y=1/4 2,1/x+1/y=1/5 Tìm x,y thuộc Z

Answer 1

1:

Điều kiện: x<>0; y<>0

\(\frac{1}{x}+\frac{1}{y}=\frac14\)

=>\(\frac{x+y}{xy}=\frac14\)

=>xy=4(x+y)

=>xy-4x-4y=0

=>x(y-4)-4y+16=16

=>(x-4)(y-4)=16

mà x-4<>-4 và y-4<>-4

nên (x-4;y-4)∈{(1;16);(16;1);(-1;-16);(-16;-1);(2;8);(8;2);(-2;-8);(-8;-2);(4;4)}

=>(x;y)∈{(5;20);(20;5);(3;-12);(-12;3);(6;12);(12;6);(2;-4);(-4;2);(8;8)}

2: ĐKXĐ: y<>0; x<>0

\(\frac{1}{x}+\frac{1}{y}=\frac15\)

=>\(\frac{x+y}{xy}=\frac15\)

=>xy=5(x+y)

=>xy-5x-5y=0

=>x(y-5)-5y+25=25

=>(x-5)(y-5)=25

mà x-5<>-5; y-5<>-5

nên (x-5;y-5)∈{(1;25);(25;1);(-1;-25);(-25;-1);(5;5)}

=>(x;y)∈{(6;30);(30;6);(4;-20);(-20;4);(10;10)}