Solve-for-a-b-c-R-1-a-1-b-c-1-2-1-b-1-c-a-1-3-1-c-1-a-b-1-4-

Question Number 215048 by Hanuda354 last updated on 27/Dec/24

Solve for a, b, c \in R

\frac{1}{a} + \frac{1}{b + c} = \frac{1}{2}

\frac{1}{b} + \frac{1}{c + a} = \frac{1}{3}

\frac{1}{c} + \frac{1}{a + b} = \frac{1}{4}

Commented by Abdullahrussell last updated on 27/Dec/24

l e t, a + b + c = x

w e g e t f r o m e q n . (1), (2), (3) r e s p e c t i v l y

a b + a c = 2 x, b c + a b = 3 x, a c + b c = 4 x

a d d i n g a l l w e g e t : a b + b c + c a = \frac{9 x}{2}

\Rightarrow a b = \frac{x}{2}, b c = \frac{5 x}{2}, c a = \frac{3 x}{2}

\Rightarrow \frac{a c}{b c} = \frac{a}{b} = \frac{3 x}{2} \times \frac{2}{5 x} = \frac{3}{5} \Rightarrow a : b = 3 : 5

\Rightarrow \frac{a b}{a c} = \frac{b}{c} = \frac{x}{2} \times \frac{2}{3 x} = \frac{1}{3} \Rightarrow b : c = 1 : 3

\Rightarrow a : b : c = 3 : 5 : 15 [3 k : 5 k : 15 k]

\Rightarrow a + b + c = 23 k = x

a b = \frac{x}{2} \Rightarrow 3 k \times 5 k = \frac{23 k}{2} \Rightarrow k = \frac{23}{30}

\Rightarrow a = \frac{23}{10}, b = \frac{23}{6}, c = \frac{23}{2} (A n s) .

Commented by Hanuda354 last updated on 27/Dec/24

T h a n k s