Question

If 2x = 3y =2z,then 1/z-1/y is equal to ?

Answer 1

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$\tt{\huge{\purple{ ................... }}}$

QUESTION -

If 2x = 3y =2z,then 1/z-1/y is equal to ?

SOLUTION -

$2x = 3y = 2z \\ \\ => Common \: Factor \: i.e, \: LCM \: - \: 2 \times 3 = 6 \\ \\ => Dividing \: By \: Six \\ \\ => \dfrac{ x}{3} = \dfrac{ y}{ 2 } = \dfrac{ z}{ 3 } \\ \\ => Reciprocal \::- \\ \\ => \dfrac{ 3}{x} = \dfrac{ 2}{ y } = \dfrac{3}{ z } \\ \\ => \dfrac{2}{y} = \dfrac{3}{z} \\ \\ =</p><p> y = \dfrac {2}{3} z \\ \\ => Given \::> \dfrac{1}{z} - \dfrac{1}{y} \\ \\ => \dfrac{ z-y}{yz} \\ \\ => Substituting \: The \: Required \: Values \::- \\ \\ => \dfrac{ \dfrac{z}{3} }{ \dfrac{2}{3} {z} ^ 2 } \\ \\ => \dfrac{1}{2z} = \dfrac{ 2}{x } \: ( Substituting \: This \: In \: Original \: Equation ) ........... [ A ]$

Answer 2

Correct question :

$\sf{If\ 2^x=3^y=12^z,\ then\ \frac{1}{z}-\frac{1}{y} =? }$

Given that,

$\sf{2^x=3^y=12^z}$

To find :-

The value of 1/z - 1/y.

Solution :-

Let $\sf{2^x=3^y=12^z=k}$.

$\sf{2^x=k,\ then\ 2=k^\frac{1}{x} }\\\\\sf{3^y=k,\ then\ 3=k^\frac{1}{y} }\\\\\sf{12^z=k,\ then\ 12=k^\frac{1}{z} }$

Now,

$\sf{12=k^{\frac{1}{z}}\\\sf{\implies 2^{2}\times 3=k^{\frac{1}{z}}}$

$\displaystyle{\sf{\implies k^{\frac{2}{x}}\times k^{\frac{1}{y}}=k^{\frac{1}{z}}}}\\\\\displaystyle{\sf{\implies k^{\frac{2}{x}+\frac{1}{y}}=k^{\frac{1}{z}}}}$

We know,

$\sf{a^{mn}=a^m+a^n}$

So,

$\sf{\frac{2}{x}+\frac{1}{y}=\frac{1}{z} }$

$\boxed{\sf{\implies \frac{1}{z}-\frac{1}{y}= \frac{2}{x} }}$