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\(3x-y=3z\Rightarrow-y=3z-3x\Rightarrow y=3x-3z\)
\(2x+y=7z\Rightarrow y=7z-2x\)\(\Rightarrow3x-3z=7z-2x=y\Rightarrow3x-3z-7z+2x=5x-10z=0\Rightarrow x-2z=0\Rightarrow x=2z\)
\(2x+y=7z\Rightarrow2\cdot2z+y=7z\Rightarrow4z+y=7z\Rightarrow y=3z\)
\(M=\frac{x^2-2xy}{x^2+y^2}=\frac{\left(2z\right)^2-2\cdot2z\cdot3z}{\left(2z\right)^2+\left(3z\right)^2}=\frac{4z^2-12z^2}{4z^2+9z^2}=-\frac{8z^2}{13z^2}=-\frac{8}{13}\)
ta có 5x=10z=> x=2z=> y=3z
Tháy vào, ta có \(M=\frac{4z^2-12z^2}{4z^2+9z^2}=\frac{-8z^2}{13z^2}=-\frac{8}{13}\)
Ta có:
\(3x-y+2x+y=3z+7z\)
\(5x=10z\)
\(x=2z\)
thay:\(4z+y=7z\) \(\Rightarrow y=3z\)
Thay vào M ta đc:M=\(\frac{4z^2-12z^2}{4z^2+9z^2}\) =\(\frac{-8z^2}{13z^2}=\frac{-8}{13}\)
vậy\(M=\frac{-8}{13}\) nếu\(3x-y=3z;2x+y=7z\)
Ta có:
\(\left\{{}\begin{matrix}3x-y=3z\\2x+y=7z\end{matrix}\right.\) ⇔ \(\left\{{}\begin{matrix}5x=10z\\2x+y=7z\end{matrix}\right.\) ⇔\(\left\{{}\begin{matrix}x=2z\\y=3z\end{matrix}\right.\)
Thay x = 2z và y = 3z vào biểu thức M ta được:
M = \(\dfrac{\left(2z\right)^2-2.2z.3z}{\left(2z\right)^2+\left(3z\right)^2}\)
= \(\dfrac{4z^2-12z^2}{4z^2+9z^2}\)
= \(\dfrac{-8z^2}{13z^2}\)
= \(\dfrac{-8}{13}\)
Vậy...
Bạn tham khảo tại đây:
Câu hỏi của trieu dang - Toán lớp 8 - Học toán với OnlineMath
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Rightarrow\frac{\left(yz+xz+xy\right)}{xyz}=0\)
\(\Rightarrow yz+zx+xy=0\)
Ta có : \(x^2+2yz=x^2+yz+yz\)
\(=x^2+yz-zx-xy\)
\(=x\left(x-z\right)-y\left(x-z\right)\)
\(=\left(x-y\right)\left(x-z\right)\)
Tương tự : \(y^2+2xz=y^2+xz+xz\)
\(=y^2+xz-xy-yz\)
\(=y\left(y-x\right)+z\left(x-y\right)\)
\(=\left(x-y\right)\left(z-y\right)\)
\(z^2+2xy=\left(x-z\right)\left(y-z\right)\)
\(\Rightarrow M=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(x-y\right)\left(z-y\right)}+\frac{xy}{\left(x-z\right)\left(y-z\right)}\) \(M=\frac{yz\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}-\frac{xz\left(x-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\frac{xy\left(x-y\right)}{\left(x-z\right)\left(y-z\right)\left(x-y\right)}\)
\(M=\frac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{yz\left(y-z\right)-xz\left(x-y+y-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(A=\frac{\left(yz-xz\right)\left(y-z\right)+\left(xy-xz\right)\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=\frac{\left(x-y\right)\left(x-z\right)\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=1\)
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)nhân lần lượt với x; y; z, ta có:
\(1+\frac{x}{y}+\frac{x}{z}=0\)(1)
\(1+\frac{y}{z}+\frac{y}{x}=0\)(2)
\(1+\frac{z}{x}+\frac{z}{y}=0\)(3)
Từ: (1); (2) và (3) => \(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+\frac{x}{z}+\frac{y}{x}+\frac{z}{y}=-3\)(*)
Mặt khác: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)quy đồng ta có:
\(\frac{\left(xy+yz+zx\right)}{xyz}=0\)hay xy + yz + zx = 0
Hay: \(\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right).\left(xy+yz+zx\right)=0\)
Khai triển, ta có:
\(\frac{yz}{x^2}+\frac{zx}{y^2}+\frac{xy}{z^2}+\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+\frac{z}{x}+\frac{y}{x}+\frac{z}{y}=0\)
Vậy: \(\frac{yz}{x^2}+\frac{zx}{y^2}+\frac{xy}{z^2}=-\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+\frac{x}{z}+\frac{y}{x}+\frac{z}{y}\right)=3\)
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\Rightarrow xy+yz+xz=0\) (nhân 2 vế với\(xyz\ne0\))
=> x2 + 2yz = x2 + 2yz - xy - yz - xz = x2 - xz - xy + yz = x(x - z) - y(x - z) = (x - y)(x - z).
Tương tự,y2 + 2xz = (y - x)(y - z) ; z2 + 2xy = (z - x)(z - y)
\(\Rightarrow\frac{yz}{x^2+2yz}+\frac{xz}{y^2+2xz}+\frac{xy}{z^2+2xy}=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(y-x\right)\left(y-z\right)}+\frac{xy}{\left(z-x\right)\left(z-y\right)}=1\)
Mình sửa lại đề cho đúng nhé
\(\hept{\begin{cases}3x-y=3z\\2x+y=7z\end{cases}}\Leftrightarrow\hept{\begin{cases}x=2z\\y=3z\end{cases}}\)
Thế vô M ta được
\(M=\frac{x^2-2xy}{x^2+y^2}=\frac{4z^2-2.2z.3z}{4z^2+9z^2}=-\frac{8}{13}\)