Vì x,y,z tỉ lệ với các số 2,3,4.

⇒x2=y3=z4=k⇒x2=y3=z4=k

⇒⎧⎪⎨⎪⎩x=2ky=3kz=4k⇒{x=2ky=3kz=4k

Thay x = 2k; y = 3k ; z = 4k vào M, ta được:

M=5x+2y+zx+4y−3zM=5x+2y+zx+4y−3z

M=5.(2k)+2.(3k)+4k2k+4.(3k)−3.(4k)M=5.(2k)+2.(3k)+4k2k+4.(3k)−3.(4k)

M=10k+6k+4k2k+12k−12kM=10k+6k+4k2k+12k−12k

M=20k2kM=20k2k

M=202M=202

M=10M=10

Vậy M = 10.

Vì x,y,z tỉ lệ với các số 2,3,4.

⇒\(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)

Đặt \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}=k\)

⇒\(\hept{\begin{cases}x=2k\\y=3k\\z=4k\end{cases}}\)

Thay x = 2k; y = 3k ; z = 4k vào M, ta được:

\(M=\frac{5x+2y+z}{x+4y+3z}\)

\(=\frac{5.2k+2.3k+4k}{2k+4.3k+3.4k}=\frac{10k+6k+4k}{2k+12k+12k}\)\(=\frac{20k}{26k}=\frac{5}{9}\)

Vậy \(M=\frac{5}{9}\)

\(\dfrac{3x-2y}{4}=\dfrac{2z-4x}{3}=\dfrac{4y-3z}{2}\)

\(\Leftrightarrow\dfrac{4\left(3x-2y\right)}{16}=\dfrac{3\left(2z-4x\right)}{9}=\dfrac{2\left(4y-3z\right)}{4}\)

\(\Leftrightarrow\dfrac{12x-8y}{16}=\dfrac{6z-12x}{9}=\dfrac{8y-6z}{4}\)

Áp dụng t,c dãy tỉ số bằng nhau ta có :

\(\dfrac{12x-8y}{16}=\dfrac{6z-12x}{9}=\dfrac{8y-6z}{4}=\dfrac{12x-8y+6z-12x+8y-6z}{16+9+4}=\dfrac{0}{29}=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{12x-8y}{16}=0\\\dfrac{2z-4x}{3}=0\\\dfrac{4y-3z}{2}=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}12x-8y=0\\2x-4z=0\\4y-3z=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x}{2}=\dfrac{y}{3}\\\dfrac{y}{3}=\dfrac{z}{4}\\\dfrac{z}{4}=\dfrac{x}{2}\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}\left(đpcm\right)\)

Từ giả thiết \(\dfrac{3x-2y}{4}=\dfrac{2z-4x}{3}=\dfrac{4y-3z}{4}\)

\(\Rightarrow\dfrac{3xz-2yz}{4z}=\dfrac{2yz-4xy}{3y}=\dfrac{4xy-3xz}{4x}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{3xz-2yz}{4z}=\dfrac{2yz-4xy}{3y}=\dfrac{4xy-3xz}{4x}=\dfrac{3xz-2yz+2yz-4xy+4xy-3xz}{4z+3y+4x}=\dfrac{0}{4z+3y+4x}=0\)

\(\Rightarrow3xz=2yz=4xy\)

\(\Rightarrow\dfrac{3xyz}{y}=\dfrac{2xyz}{x}=\dfrac{4xyz}{z}\)

\(\Rightarrow\dfrac{3}{y}=\dfrac{2}{x}=\dfrac{4}{z}\)

\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}\left(đpcm\right)\)

Chúc bạn học tốt!

Vì x,y,z tỉ lệ với các số 2,3,4.

\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}=k\)

\(\Rightarrow\left\{{}\begin{matrix}x=2k\\y=3k\\z=4k\end{matrix}\right.\)

Thay x = 2k; y = 3k ; z = 4k vào M, ta được:

\(M=\dfrac{5x+2y+z}{x+4y-3z}\)

\(M=\dfrac{5.\left(2k\right)+2.\left(3k\right)+4k}{2k+4.\left(3k\right)-3.\left(4k\right)}\)

\(M=\dfrac{10k+6k+4k}{2k+12k-12k}\)

\(M=\dfrac{20k}{2k}\)

\(M=\dfrac{20}{2}\)

\(M=10\)

Vậy M = 10.

Vì x,y,z tỉ lệ nghịch với các số 2;3;4 nên \(\dfrac{x}{\dfrac{1}{2}}=\dfrac{y}{\dfrac{1}{3}}=\dfrac{z}{\dfrac{1}{4}}\)

Đặt x = \(\dfrac{1}{2}k\), y = \(\dfrac{1}{3}k\), z = \(\dfrac{1}{4}k\)

\(\Rightarrow M=\dfrac{\dfrac{1}{2}k.5+\dfrac{1}{3}k.2+\dfrac{1}{4}k}{\dfrac{1}{2}k+\dfrac{1}{3}k.4-\dfrac{1}{4}k.3}\)

\(M=\dfrac{\dfrac{5}{2}k+\dfrac{2}{3}k+\dfrac{1}{4}k}{\dfrac{1}{2}k+\dfrac{4}{3}k-\dfrac{3}{4}k}\)

\(M=\dfrac{k\left(\dfrac{5}{2}+\dfrac{2}{3}+\dfrac{1}{4}\right)}{k\left(\dfrac{1}{2}+\dfrac{4}{3}-\dfrac{3}{4}\right)}\)

\(M=\dfrac{\dfrac{41}{12}}{\dfrac{13}{12}}\)

\(M=\dfrac{41}{13}\)

Vậy \(M=\dfrac{41}{13}\)

Lời giải:

Vì $x,y,z$ tỉ lệ nghịch với $8,9,10$ nên:

$8x=9y=10z$

$\Rightarrow \frac{8x}{360}=\frac{9y}{360}=\frac{10z}{360}$

$\Rightarrow \frac{x}{45}=\frac{y}{40}=\frac{z}{36}$

Đặt $\frac{x}{45}=\frac{y}{40}=\frac{z}{36}=k$

$\Rightarrow x=45k; y=40k; z=36k$

Khi đó:

\(P=\frac{x+4y-6z}{x+2y-3z}=\frac{45k+4.40k-6.36k}{45k+2.40k-3.36k}=\frac{-11k}{17k}=\frac{-11}{17}\)

Giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{3x-2y}{4}=\dfrac{2z-4x}{3}=\dfrac{4y-3z}{2}=\dfrac{12x-8y}{16}=\dfrac{6z-12x}{9}=\dfrac{8y-6z}{4}\)

\(=\dfrac{12x-8y+6z-12x+8y-6z}{16+9+4}=\dfrac{0}{16+9+4}=0\)

\(\Rightarrow\left\{{}\begin{matrix}\dfrac{12x-8y}{16}=0\\\dfrac{6z-12x}{9}=0\\\dfrac{8y-6z}{4}=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}12x-8y=0\\6z-12x=0\\8y-6z=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}12x=8y\\6z=12x\\8y=6z\end{matrix}\right.\)

\(\Rightarrow12x=8y=6z\)

\(\Rightarrow6x=4y=3z\)

\(\Rightarrow\dfrac{6x}{12}=\dfrac{4y}{12}=\dfrac{3z}{12}\)

\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}\)

Vậy x, y, z tỉ lệ với 2, 3, 4

voi x,y,z ti le voi 8,9,10 =>8x=9y=10z=>y=9/8x,z=5/4x=>(x+9/2x-15/2x)/(x+9/4x-15/4x) hay-2x/-0,5x=>P=4