Question Number 211485 by MATHEMATICSAM last updated on 10/Sep/24
$$\mathrm{If}\:\frac{{a}\:−\:{b}}{{c}}\:+\:\frac{{b}\:−\:{c}}{{a}}\:+\:\frac{{c}\:+\:{a}}{{b}}\:=\:\mathrm{1}\:\mathrm{and}\: \\ $$$${a}\:−\:{b}\:+\:{c}\:\neq\:\mathrm{0}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}}\:=\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:. \\ $$
Answered by Frix last updated on 10/Sep/24
$$\frac{{a}−{b}}{{c}}+\frac{{b}−{c}}{{a}}+\frac{{c}+{a}}{{b}}=\mathrm{1} \\ $$$$\left({a}−{b}+{c}\right)\left({ab}+{ac}−{bc}\right)=\mathrm{0} \\ $$$${a}−{b}+{c}\neq\mathrm{0}\:\Rightarrow\:{ab}+{ac}−{bc}=\mathrm{0} \\ $$$$\frac{{ab}}{{abc}}+\frac{{ac}}{{abc}}−\frac{{bc}}{{abc}}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{c}}+\frac{\mathrm{1}}{{b}}−\frac{\mathrm{1}}{{a}}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{a}}=\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}} \\ $$