Tìm x, y, z, t biết \(\dfrac{1}{4x}=\dfrac{1}{2y}=\dfrac{3}{4z}=\dfrac{1}{t}\) và \(x+y+z+t=10\) \(\left(x,y,z,t\ne0\right)\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
1) Phân số đầu nhân 2.
_ Phân số thứ 2 nhân 3, p/s thứ 3 giữ nguyên.
_ Lấy phân số đầu + p/s thứ 2 - p/s thứ 3.
_ Dựa vào dãy tỉ số bằng nhau tìm x, y, z.
2) \(x-y-z=0\Rightarrow x=y+z\)
Khi đó thay vào B được:
\(B=\left(1-\dfrac{z}{y+z}\right)\left(1-\dfrac{y+z}{y}\right)\left(1+\dfrac{y}{z}\right)\)
\(=\dfrac{y}{y+z}.\dfrac{z}{y}.\dfrac{y+z}{z}\)
\(=1\)
Vậy B = 1.
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có :
\(\dfrac{x+y-z}{z}=\dfrac{y+z-x}{x}=\dfrac{z+x-y}{y}\\ \Leftrightarrow\dfrac{x+y+z}{z}=\dfrac{x+y+z}{x}=\dfrac{x+y+z}{y}\left(cùngcộngthêm2\right)\)
TH1: \(x+y+z\ne0\)
\(\Rightarrow x=y=z\)
\(\Rightarrow P=\left(1+1\right)\left(1+1\right)\left(1+1\right)\\ =2\cdot2\cdot2=8\)
TH2: \(x+y+z=0\Rightarrow\left\{{}\begin{matrix}x=-\left(y+z\right)\\y=-\left(x+z\right)\\z=-\left(y+x\right)\end{matrix}\right.\)(*)
\(\Rightarrow P=\left(1+\dfrac{-\left(y+z\right)}{y}\right)\left(1+\dfrac{-\left(z+x\right)}{z}\right)\left(1+\dfrac{-\left(x+y\right)}{z}\right)\\ =\left(1-1-\dfrac{z}{y}\right)\left(1-1-\dfrac{x}{z}\right)\left(1-1-\dfrac{y}{z}\right)\\ =\left(-\dfrac{z}{y}\right)\left(-\dfrac{x}{z}\right)\left(-\dfrac{y}{z}\right)\\ =-1\)
Vậy P=8 hoặc P=-1
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\dfrac{y+z+t-nx}{x}=\dfrac{z+t+x-ny}{y}=\dfrac{t+x+y-nz}{z}=\dfrac{x+y+z-nt}{t}\)
\(=\dfrac{y+z+t-nx+z+t+x-ny+t+x+y-nz+x+y+z-nt}{x+y+z+t}\)
\(=\dfrac{3x+3y+3z+3t-n\left(x+y+z+t\right)}{x+y+z+t}\)
\(=\dfrac{3\left(x+y+z+t\right)-n\left(x+y+z+t\right)}{x+y+z+t}=\dfrac{\left(3-n\right)\left(x+y+z+t\right)}{x+y+z+t}=3-n\)
Nên \(\left\{{}\begin{matrix}y+z+t-nx=3x-nx\\z+t+x-ny=3y-ny\\t+x+y-nz=3z-nz\\x+y+z-nt=3t-nt\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}y+z+t=3x\\z+t+x=3y\\t+x+y=3z\\x+y+z=3t\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{y+z+t}{3}\\y=\dfrac{z+t+x}{3}\\z=\dfrac{t+x+y}{3}\\t=\dfrac{x+y+z}{3}\end{matrix}\right.\)
Thay vào \(P\) ta có:
\(P=x+2y-3z+t\)
\(P=\dfrac{y+z+t}{3}+\dfrac{2\left(z+t+x\right)}{3}-\dfrac{3\left(t+x+y\right)}{3}+\dfrac{x+y+z}{3}\)
\(P=\dfrac{y+z+t+2z+t+x-3t-3x-3y+x+y+z}{3}\)
\(P=\dfrac{\left(x+x-3x\right)+\left(y+y-3y\right)+\left(z+z+2z\right)+\left(t+t-3t\right)}{3}\)
\(P=\dfrac{-x-y-z+4t}{3}\)
\(P=\dfrac{-\left(x+y+z+t\right)+5t}{3}\)
\(P=\dfrac{-2012+5t}{3}\)
Tốn sức quá T^T
![](https://rs.olm.vn/images/avt/0.png?1311)
a)
\(\dfrac{x^2+x-6}{x^3-4x^2-18x+9}=\dfrac{x^2+3x-2x-6}{x^3+3x^2-7x^2-21x+3x+9}\)
\(=\dfrac{x\left(x+3\right)-2\left(x+3\right)}{x^2\left(x+3\right)-7x\left(x+3\right)+3\left(x+3\right)}\)
\(=\dfrac{\left(x-2\right)\left(x+3\right)}{\left(x^2-7x+3\right)\left(x+3\right)}=\dfrac{x-2}{x^2-7x+3}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
Ta có :
\(B=\left(1-\frac{z}{x}\right)\left(1-\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\)
\(B=\frac{(x-z)(y-x)(z+y)}{xyz}\)
Vì \(x-y-z=0\Rightarrow x=y+z\). Do đó:
\(B=\frac{(y+z-z)[y-(y+z)](z+y)}{yz(y+z)}\)
\(B=\frac{y(-z)(z+y)}{yz(y+z)}=\frac{-yz(y+z)}{yz(y+z)}=-1\)