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\(\frac{2}{x+1}=\frac{8}{x-2}\)
\(\Rightarrow2\left(x-2\right)=8\left(x+1\right)\)
\(\Rightarrow2x-4=8x+8\)
\(\Rightarrow2x-8x=8+4\)
\(\Rightarrow-6x=12\)
\(\Rightarrow x=12:\left(-6\right)\)
\(\Rightarrow x=-2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{1}{8}< \frac{x}{12}< \frac{y}{9}< \frac{1}{4}\)
=> x = 2, y = 45
Bài này có thể thử chọn
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<=> \(\frac{1}{x}=8-\frac{1}{y}=\frac{8y-1}{y}\)
=> x=\(\frac{y}{8y-1}\)
<=> 8x=\(\frac{8y}{8y-1}=\frac{8y-1+1}{8y-1}=1+\frac{1}{8y-1}\)
=> để x nguyên thì 1 phải chia hết cho (8y-1) và \(1+\frac{1}{8y-1}\)phải chia hết cho 8.
=> 8y-1={-1; 1}
+/ 8y-1=-1=> y=0 => x=0
+/ 8y-1=1 => y=2/8=1/4 => Loại
ĐS: x=0; y=0
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\(\frac{1}{x}\)+\(\frac{y}{2}\)=\(\frac{5}{8}\)
=> \(\frac{5}{8}\)-\(\frac{y}{2}\)= \(\frac{1}{x}\)
=> \(\frac{5}{8}\)-\(\frac{4y}{8}\)=\(\frac{1}{x}\)
=>\(\frac{5-4y}{8}\)=\(\frac{1}{x}\)
=> (5-4y).x=1.8
=>(5-4y).x=8
=> 8 \(⋮\) (5-4y) , 8 \(⋮\)x
=> 5-4y \(\in\)Ư(8) , x\(\in\)Ư(8)
mà Ư(8)= { 1, -1, 2, -2, 4, -4, 8, -8 }
Ta có bảng
5-4y | 1 | -1 | 2 | -2 | 4 | -4 | 8 | -8 |
x | 8 | -8 | 4 | -4 | 2 | -2 | 1 | -1 |
y | 1 | \(\frac{3}{2}\) | \(\frac{3}{4}\) | \(\frac{7}{2}\) | \(\frac{1}{4}\) | \(\frac{9}{4}\) | - \(\frac{3}{4}\) | \(\frac{13}{4}\) |
mà x,y \(\in\)z nên (x,y)=(8,1)
Vậy x=8, y=1
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Bài 3
\(\frac{x-1}{9}=\frac{8}{3}\)
\(\Rightarrow\left(x-1\right).3=8.9\)
\(\Rightarrow\left(x-1\right).3=72\)
\(\Rightarrow x-1=24\)
\(\Rightarrow x=25\)
\(\frac{-x}{4}=\frac{-9}{x}\)
\(\Rightarrow\left(-x\right).x=\left(-9\right).4\)
\(\Rightarrow-x=-36\)
\(\Rightarrow x=36\)
\(\frac{x}{4}=\frac{18}{x+1}\)
\(\Rightarrow x.\left(x+1\right)=4.18\)
\(\Rightarrow x.\left(x+1\right)=72\)
Vì x và x + 1 là 2 số tự nhiên liên tiếp
\(\Rightarrow x\left(x+1\right)=8.9\)
\(\Rightarrow\orbr{\begin{cases}x=8\\x=8\end{cases}}\)
Bài 4
\(\frac{x-4}{y-3}=\frac{4}{3},x-y=5\)
Ta có :
\(x-y=5\)
\(\Rightarrow x=5+y\)
\(\Rightarrow\frac{y+5-4}{y-3}=\frac{4}{3}\)
\(\Rightarrow\frac{y+1}{y-3}=\frac{4}{3}\)\(\)
\(\Rightarrow\left(y+1\right).3=\left(y-3\right).4\)
\(\Rightarrow y.3+1.3=y.4-3.4\)
\(\Rightarrow y.3+3=y.4-12\)
\(\Rightarrow y.3-y.4=-12-3\)
\(\Rightarrow-1y=-15\)
\(\Rightarrow y=\left(-15\right):\left(-1\right)\)
\(\Rightarrow y=15\)
Vì x = y + 5
\(\Rightarrow x=15+4\)
\(\Rightarrow x=19\)
Vậy x = 19 , y = 15
\(\frac{-x}{4}=\frac{-9}{x}\)
\(\Rightarrow\left(-x\right).x=4.\left(-9\right)\)
\(\Rightarrow-x=-9;x=4\)
\(\Rightarrow x=9;x=4\)
Ta có :\(\frac{x-1}{-2}=\frac{-8}{x-1}\Rightarrow\left(x-1\right)^2=16\Rightarrow\orbr{\begin{cases}x-1=4\\x-1=-4\end{cases}\Rightarrow\orbr{\begin{cases}x=5\\x=-3\end{cases}}}\)
\(\frac{x-1}{-2}=\frac{-8}{x-1}\left(x\ne1\right)\)
\(\Leftrightarrow\left(x-1\right)\left(x-1\right)=-2\cdot\left(-8\right)\)
\(\Leftrightarrow\left(x-1\right)^2=16\)
\(\Leftrightarrow\orbr{\begin{cases}x-1=4\\x-1=-4\end{cases}\Leftrightarrow\orbr{\begin{cases}x=5\\x=-3\end{cases}\left(tm\right)}}\)
Vậy x=5; x=-3