Đáp án: $P=\dfrac43$

 

Giải thích các bước giải:

Ta có:

$\dfrac{xy}{2x+3y}=\dfrac23\to \dfrac{2x+3y}{xy}=\dfrac23\to \dfrac2y+\dfrac3x=\dfrac23$

$\dfrac{yz}{4y+3z}=3\to \dfrac{4y+3z}{yz}=\dfrac13\to \dfrac4z+\dfrac3y=\dfrac13$

$\dfrac{xz}{x+2z}=\dfrac32\to \dfrac{x+2z}{xz}=\dfrac23\to \dfrac1z+\dfrac2x=\dfrac23$

$\to \dfrac2y+\dfrac3x=\dfrac1z+\dfrac2x\to \dfrac1z=\dfrac2y+\dfrac1x$

      $\dfrac2y+\dfrac3x=2( \dfrac4z+\dfrac3y)\to \dfrac3x-\dfrac4y-\dfrac8z=0$

$\to \dfrac3x-\dfrac4y-8(\dfrac2y+\dfrac1x)=0$

$\to -\dfrac{20}y-\dfrac5x=0$

$\to \dfrac{20}y=-\dfrac5x$

$\to \dfrac1x=-\dfrac4y$

$\to y=-4x$

$\to \dfrac1z=\dfrac2{-4x}+\dfrac1x=\dfrac1{2x}$

$\to z=2x$

Lại có:

$\dfrac{xy}{2x+3y}=\dfrac23$

$\to \dfrac{x\cdot (-4x)}{2x+3\cdot (-4x)}=\dfrac23$

$\to x=\dfrac53$

Ta có:

$P=\dfrac{xyz}{xy+yz+zx}=\dfrac{x\cdot (-4x)\cdot 2x}{x\cdot (-4x)+(-4x)\cdot 2x+2x\cdot x}=\dfrac45x=\dfrac45\cdot \dfrac53=\dfrac43$