Solve-the-system-a-b-1-c-1-2-1-c-b-1-a-1-3-1-a-c-1-b-1-4-1-

Question Number 206294 by sniper237 last updated on 11/Apr/24

S o l v e t h e s y s t e m

{(a + b)}^{- 1} + c^{- 1} = 2^{- 1}

{(c + b)}^{- 1} + a^{- 1} = 3^{- 1}

{(a + c)}^{- 1} + b^{- 1} = 4^{- 1}

Commented by MATHEMATICSAM last updated on 11/Apr/24

Q 202400 This quetion is like this one .

Commented by sniper237 last updated on 11/Apr/24

T h a n k s

Answered by mr W last updated on 11/Apr/24

\frac{1}{a + b} + \frac{1}{c} = \frac{1}{2} \Rightarrow 2 (a + b + c) = c a + b c \dots (i)

\frac{1}{b + c} + \frac{1}{a} = \frac{1}{3} \Rightarrow 3 (a + b + c) = a b + c a \dots (i i)

\frac{1}{c + a} + \frac{1}{b} = \frac{1}{4} \Rightarrow 4 (a + b + c) = b c + a b \dots (i i i)

(i) + (i i) + (i i i) :

4.5 (a + b + c) = a b + b c + c a

s a y k = a + b + c

\Rightarrow a b = 2.5 (a + b + c) = 2.5 k

\Rightarrow b c = 1.5 (a + b + c) = 1.5 k

\Rightarrow c a = 0.5 (a + b + c) = 0.5 k

{(a b c)}^{2} = 2.5 \times 1.5 \times 0.5 k^{3} = \frac{15 k^{3}}{8}

\Rightarrow a b c = \sqrt{\frac{15 k^{3}}{8}} = \frac{k \sqrt{30 k}}{4} (“ -^{″} r e j e c t e d)

\Rightarrow c = \frac{k \sqrt{30 k}}{4 \times 2.5 k} = \frac{\sqrt{30 k}}{10}

\Rightarrow a = \frac{k \sqrt{30 k}}{4 \times 1.5 k} = \frac{\sqrt{30 k}}{6}

\Rightarrow b = \frac{k \sqrt{30 k}}{4 \times 0.5 k} = \frac{\sqrt{30 k}}{2}

a + b + c = (\frac{1}{10} + \frac{1}{6} + \frac{1}{2}) \sqrt{30 k} = k

\Rightarrow k = \frac{23^{2}}{30} \Rightarrow \sqrt{30 k} = 23

\Rightarrow a = \frac{23}{6}, b = \frac{23}{2}, c = \frac{23}{10} ✓

Answered by MATHEMATICSAM last updated on 12/Apr/24

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

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

\Rightarrow \frac{a c + b c}{a + b + c} = 2 \dots . (1)

From the other two equations same

way we can get

\frac{a c + a b}{a + b + c} = 3 \dots . (2)

\frac{a b + b c}{a + b + c} = 4 \dots . (3)

(1) + (2) + (3)

\frac{2 (a b + b c + c a)}{a + b + c} = 9

\Rightarrow \frac{a b + b c + c a}{a + b + c} = 4.5 \dots . (4)

(4) - (1)

\frac{a b}{a + b + c} = 2.5 \dots . (5)

(4) - (2)

\frac{b c}{a + b + c} = 1.5 \dots . (6)

(4) - (3)

\frac{c a}{a + b + c} = 0.5 \dots . (7)

(5) \div (6)

\frac{a b}{b c} = \frac{2.5}{1.5} \Rightarrow \frac{a}{c} = \frac{5}{3} \Rightarrow c = \frac{3 a}{5}

(6) \div (7)

\frac{b c}{c a} = \frac{1.5}{0.5} \Rightarrow \frac{b}{a} = 3 \Rightarrow b = 3 a

\frac{a b}{a + b + c} = 2.5 = \frac{5}{2}

\Rightarrow \frac{a \times 3 a}{a + 3 a + \frac{3 a}{5}} = \frac{5}{2}

\Rightarrow \frac{3 a^{2}}{\frac{23 a}{5}} = \frac{5}{2}

\Rightarrow \frac{15 a}{23} = \frac{5}{2}

\Rightarrow a = \frac{5}{2} \times \frac{23}{15} = \frac{23}{6}

b = 3 a = 3 \times \frac{23}{6} = \frac{23}{2}

c = \frac{3 a}{5} = \frac{3}{5} \times \frac{23}{6} = \frac{23}{10}