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\(\frac{x}{5}-\frac{1}{y+2}=\frac{1}{10}\)
\(\frac{1}{y+2}=\frac{x}{5}-\frac{1}{10}=\frac{2x}{10}-\frac{1}{10}=\frac{2x-1}{10}\)
\(\Rightarrow\left(y+2\right).\left(2x-1\right)=1.10=10\)
\(\Rightarrow2x-1\inƯ\left(10\right)\)
Mà 2x - 1 là lẻ
\(\Rightarrow2x-1\in\left[1;5;-1;-5\right]\)
Xét \(2x-1=1\Rightarrow x=1\)
\(\Rightarrow y+2=10\Rightarrow y=8\)
Xét \(2x-1=5\Rightarrow x=3\)
\(\Rightarrow y+2=2\Rightarrow y=0\)
Xét \(2x-1=-1\Rightarrow x=0\)
\(\Rightarrow y+2=-10\Rightarrow y=-12\)
Xét \(2x-1=-5\Rightarrow x=-2\)
\(\Rightarrow y+2=-2\Rightarrow y=-4\)
tính: \(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{1982}+\frac{1}{1984}+\frac{1}{1986}\)
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Giải
Ta có:\(\frac{x}{5}+1=\frac{1}{y-1}\)
\(\Rightarrow\frac{x}{5}+\frac{5}{5}=\frac{1}{y-1}\)
\(\Rightarrow\frac{x+5}{5}=\frac{1}{y-1}\)
\(\Leftrightarrow\left(x+5\right).\left(y-1\right)=5\)
Vì \(x;y\in Z\)
\(\Rightarrow x+5;y-1\in Z\)
\(\Rightarrow x+5;y-1\inƯ\left(5\right)=\left\{-5;-1;1;5\right\}\)
Ta lập bảng:
x + 5 | -5 | -1 | 1 | 5 |
y - 1 | -1 | -5 | 5 | 1 |
x | -10 | -6 | -4 | 0 |
y | 0 | -4 | 6 | 2 |
Vậy có 4 cặp ( x ; y ) cần tìm.
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![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{x}{5}+1=\frac{x+5}{5}=\frac{1}{y-1}\)
=>\(\left(x+5\right)\left(y-1\right)=5.1=\left(-5\right)\left(-1\right)\)
TH1: \(x+5=5\)=> \(x=0\)
\(y-1=1\)=> \(y=2\)
TH2: \(x+5=1\)=> \(x=-4\)
\(y-1=5\)=> \(y=6\)
TH3: \(x+5=-5\)=> \(x=-10\)
\(y-1=-1\)=> \(y=0\)
TH4: \(x+5=-1\)=> \(x=-6\)
\(y-1=-5\)=> \(y=-4\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Sau đó ta có: x(y-1)+(y-1)=5
(x+1)(y-1)=5=1.5=5.1=-1.(-5)=-5.(-1)
Tự xét lập bảng...
Vậy x=0 thì y=6
x=4 thì y=2
x=-2 thì y=-4
x=-6 thì y=0
![](https://rs.olm.vn/images/avt/0.png?1311)
a) Ta có : \(\frac{x}{3}-\frac{4}{y}=\frac{1}{5}\)
\(\Rightarrow\frac{x}{3}-\frac{1}{5}=\frac{4}{y}\)
\(\Rightarrow\frac{x.5}{15}-\frac{3}{15}=\frac{4}{y}\)
\(\Rightarrow\frac{x.5-3}{15}=\frac{4}{y}\)
\(\Rightarrow\left(x.5-3\right).y=15.4\)
\(\Rightarrow x.5.y-3.5=60\)
\(\Rightarrow xy5-15=60\)
\(\Rightarrow xy5=60+15\)
\(\Rightarrow xy5=75\)
\(\Rightarrow xy=75\div5\)
\(\Rightarrow xy=15\)
\(\Rightarrow xy=1.15=3.5=\left(-15\right)\left(-1\right)=\left(-3\right)\left(-5\right)=\left(-5\right)\left(-3\right)=\left(-1\right)\left(-15\right)=5.3=15.1\)
Do đó x = 1 thì y = 15
x = 3 thì y =5
x = -15 thì y = -1
x = -3 thì y = -5
x = -5 thì y = -3
x = -1 thì y = -15
x = 5 thì y = 3
x = 15 thì y = 1
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
\(\frac{x}{5}\)+\(\frac{5}{5}\)= \(\frac{1}{y-1}\)
\(\frac{x+5}{5}\)= \(\frac{1}{y-1}\)
=>\(\frac{x+5}{y-1}\)= \(\frac{1}{5}\)
=> x+5=1
x=1-5
x=(-4)
=>y-1=5
y= 5+1
y=6
Vậy x=(-4)
y=(6)