Ta có: \(\dfrac{y-z}{\left(x-y\right)\left(x-z\right)}=\dfrac{y-x+x-z}{\left(x-y\right)\left(x-z\right)}\)\(=\dfrac{y-x}{\left(x-y\right)\left(x-z\right)}+\dfrac{x-z}{\left(x-y\right)\left(x-z\right)}\) \(=\dfrac{1}{z-x}+\dfrac{1}{x-y}\)

Tương tự:

\(\dfrac{z-x}{\left(y-z\right)\left(y-x\right)}=\dfrac{1}{x-y}+\dfrac{1}{y-z}\)

\(\dfrac{x-y}{\left(z-x\right)\left(z-y\right)}=\dfrac{1}{y-z}+\dfrac{1}{z-x}\)

\(\Rightarrow\dfrac{y-z}{\left(x-y\right)\left(x-z\right)}+\dfrac{z-x}{\left(y-z\right)\left(y-x\right)}+\dfrac{x-y}{\left(z-x\right)\left(z-y\right)}\) \(=\dfrac{2}{x-y}+\dfrac{2}{y-z}+\dfrac{2}{z-x}\) \(\left(đpcm\right)\)

Câu 1: 

c: 2x=3y

nên x/3=y/2

=>x/9=y/6

5y=3z

nên y/3=z/5

=>y/6=z/10

=>x/9=y/6=z/10

Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{x}{9}=\dfrac{y}{6}=\dfrac{z}{10}=\dfrac{3x+3y-7z}{3\cdot9+3\cdot6-7\cdot10}=\dfrac{35}{-25}=-\dfrac{7}{5}\)

Do đó: x=-63/5; y=-42/5; z=-14

Bài 2:

Gọi ba số lần lượt là a,b,c

Theo đề, ta có: 4/3a=b=3/4c

\(\Leftrightarrow\dfrac{a}{\dfrac{3}{4}}=\dfrac{b}{1}=\dfrac{c}{\dfrac{4}{3}}\)

\(\Leftrightarrow\dfrac{a}{9}=\dfrac{b}{12}=\dfrac{c}{16}\)

Đặt \(\dfrac{a}{9}=\dfrac{b}{12}=\dfrac{c}{16}=k\)

=>a=9k; b=12k; c=16k

Theo đề, ta có: \(a^2+b^2+c^2=481\)

\(\Leftrightarrow81k^2+144k^2+256k^2=481\)

=>k²=1

Trường hợp 1: k=1

=>a=9; b=12; c=16

Trường hợp 2: k=-1

=>a=-9; b=-12; c=-16

Sửa đề: Cho \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\) . CMR: \(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)

Giải:

\(\dfrac{b.z-x.y}{a}=\dfrac{c.x-a.z}{b}=\dfrac{a.y-b.x}{c}\)

\(\Rightarrow\dfrac{a\left(bz-cy\right)}{a^2}=\dfrac{b\left(cx-az\right)}{b^2}=\dfrac{c\left(ay-bz\right)}{c^2}\)

\(\Rightarrow\dfrac{abz-acy}{a^2}=\dfrac{bcx-abz}{b^2}=\dfrac{acy-bcx}{c^2}\)

\(\Rightarrow\dfrac{abz-acy+bcx-abz+acy-bcx}{a^2+b^2+c^2}\)

\(\Rightarrow\dfrac{0}{a^2+b^2+c^2}\)

\(=0\)

\(\dfrac{bz-cy}{a}=0\)

\(\Rightarrow bz-cy=0\)

\(\Rightarrow\dfrac{z}{c}=\dfrac{y}{b}\left(1\right)\)

\(\dfrac{cx-az}{b}=0\)

\(\Rightarrow cx-az=0\)

\(\Rightarrow cx=az\)

\(\Rightarrow\dfrac{x}{a}=\dfrac{z}{c}\left(2\right)\)

Từ (1) và (2) suy ra:

\(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)

Áp dụng tc dtsbn:

\(\dfrac{x}{2013}=\dfrac{y}{2014}=\dfrac{z}{2015}=\dfrac{x-z}{-2}=\dfrac{y-z}{-1}=\dfrac{x-y}{-1}\\ \Leftrightarrow\dfrac{x-z}{2}=\dfrac{y-z}{1}=\dfrac{x-y}{1}\\ \Leftrightarrow x-z=2\left(y-z\right)=2\left(x-y\right)\\ \Leftrightarrow\left(x-z\right)^3=8\left(x-y\right)^3=8\left(x-y\right)^2\left(x-y\right)=8\left(x-y\right)^2\left(y-z\right)\)

Sửa lại đề nha: x+y+z=0

a)

Xét x+y+z=0

(x+y+z)²=0²

x²+y²+z²+2xy+2yz+2zx=0

=> x²+y²+z²=-2xy-2yz-2zx

Xét \(\dfrac{x^2+y^2+z^2}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)

= \(\dfrac{x^2+y^2+z^2}{\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(z^2-2zx+x^2\right)}\)

=\(\dfrac{x^2+y^2+z^2}{x^2-2xy+y^2+y^2-2yz+z^2+z^2-2zx+x^2}\)

=\(\dfrac{x^2+y^2+z^2}{2x^2+2y^2+2z^2-2xy-2yz-2zx}\)(1)

Thay x²+y²+z²=-2xy-2yz-2zx vào (1)

=>\(\dfrac{x^2+y^2+z^2}{2x^2+2y^2+2z^2+x^2+y^2+z^2}\\=\dfrac{x^2+y^2+z^2}{3x^2+3y^2+3z^2}\\ =\dfrac{x^2+y^2+z^2}{3\left(x^2+y^2+z^2\right)}\\ =\dfrac{1}{3}\)

b)

Xét x+y+z=0 ba lần:

- Lần 1:x+y+z=0

<=> x+y=0-z

<=>(x+y)²=(0-z)²

<=>x²+2xy+y²=z²

<=>x²+y²-z²=-2xy(1)

-Lần 2: x+y+z=0

<=> y+z=0-x

<=>(y+z)²=(0-x)²

<=>y²+2yz+z²=x²

<=>y²+z²-x²=-2yz(2)

-Lần 3: x+y+z=0

<=>z+x=0-y

<=>(z+x)²=(0-y)²

<=>z²+2zx+x²=y²

<=> z²+x²-y²=-2zx(3)

Thay (1),(2),(3) vào Q, ta có:

=>\(\dfrac{\left(x^2+y^2-z^2\right)\left(y^2+z^2-x^2\right)\left(z^2+x^2-y^2\right)}{16xyz}=\dfrac{\left(-2xy\right)\left(-2yz\right)\left(-2zx\right)}{16xyz}\\=\dfrac{\left(-2yz\right)\left(-2zx\right)}{-8z}\\ =\dfrac{y\left(-2zx\right)}{4}\\ =\dfrac{-2xyz}{4}\\ =-\dfrac{xyz}{2}\)

\(A=\left(\dfrac{-3}{7}.x^3.y^2\right).\left(\dfrac{-7}{9}.y.z^2\right).\left(6.x.y\right)\)

\(A=\left(\dfrac{-3}{7}x^3y^2\right).\left(\dfrac{-7}{9}yz^2\right).6xy\)

\(A=\left(\dfrac{-3}{7}.\dfrac{-7}{9}.6\right).\left(x^3.x\right)\left(y^2.y.y\right).z^2\)

\(A=2x^4y^4z^2\)

\(B=-4.x.y^3\left(-x^2.y\right)^3.\left(-2.x.y.z^3\right)^2\)

\(B=\left[\left(-4\right).\left(-2\right)\right].\left(x.x^6.x^2\right)\left(y^3.y^3.y^2\right)\left(z^6\right)\)

\(B=8x^7y^{y^8}z^6\)