Menu Close

If-a-b-c-b-c-a-c-a-b-1-and-a-b-c-0-then-prove-that-1-a-1-b-1-c-




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

Leave a Reply

Your email address will not be published. Required fields are marked *