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Question Number 191595 by MATHEMATICSAM last updated on 26/Apr/23

$$a=\frac{xy}{x+y},b=\frac{xz}{x+z}\mathrm{and}c=\frac{yz}{y+z}.$$$$\mathrm{Represent}x\mathrm{in}a,b,c\mathrm{form}.[x,y,z\ne 0]$$

Answered by mehdee42 last updated on 26/Apr/23

$$\frac{1}{a}=\frac{1}{x}+\frac{1}{y}\left(i\right),\frac{1}{b}=\frac{1}{x}+\frac{1}{z}\left(ii\right),\frac{1}{c}=\frac{1}{y}+\frac{1}{z}\left(iii\right)$$$$\left(i\right),\left(ii\right)\Rightarrow \frac{1}{a}+\frac{1}{b}=\frac{2}{x}+\frac{1}{y}+\frac{1}{z}\stackrel{\left(iii\right)}{\Rightarrow }\frac{1}{a}+\frac{1}{b}=\frac{2}{x}+\frac{1}{c}$$$$\Rightarrow x=\frac{ac+bc-ab}{2abc}✓$$

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