Question

[LÊ DƯƠNG QUỲNH TRÂM]

tìm x,y thuộc Z biết

1/x+y/3=5/6

Answer 1

[Nguyễn Lê Phước Thịnh]

\(\dfrac{1}{x}+\dfrac{y}{3}=\dfrac{5}{6}\)

=>\(\dfrac{3+xy}{3x}=\dfrac{5}{6}\)

=>\(6\left(xy+3\right)=5\cdot3x\)

=>\(2\left(xy+3\right)=5x\)

=>2xy-5x=-6

=>x(2y-5)=-6

mà 2y-5 lẻ

nên \(\left(x;2y-5\right)\in\left\{\left(2;-3\right);\left(-2;3\right);\left(-6;1\right);\left(6;-1\right)\right\}\)

=>\(\left(x;y\right)\in\left\{\left(2;1\right);\left(-2;4\right);\left(-6;3\right);\left(6;2\right)\right\}\)

Answer 2

[Tran Thu]

 $\frac{1}{x}+\frac{y}{3}=\frac{5}{6}$

$\Rightarrow \frac{1}{x}=\frac{5}{6}-\frac{y}{3}$

$\Rightarrow \frac{1}{x}=\frac{5}{6}-2\frac{y}{6}$

$\Rightarrow \frac{1}{x}=\frac{5-2y}{6}$

$\Rightarrow x(5-2y)=6$

$Vì$ $x,y\in Z\Rightarrow 5-2y\in Ư\left(6\right)=\{\pm 1;\pm 2;\pm 3;\pm 6\}$

mà $5-2y$ luôn lẻ 

$\Rightarrow 5-2y\in \{\pm 1;\pm 3\}$

Ta có bảng sau : 

$$\begin{array}{ccccc}\text{5 - 2y}& \text{-1}& \text{1}& \text{-3}& \text{3}\\ \text{x}& \text{-6}& \text{6}& \text{-2}& \text{2}\\ \text{2y}& \text{6}& \text{4}& \text{8}& \text{2}\\ \text{y}& \text{3}& \text{2}& \text{4}& \text{1}\end{array}$$

$Vậy$ $(x;y)\in \{(-6;3);(6;2);(-2;4);(2;1)\}$