Question

Solve for x: 1/a+b+x = 1/a+ 1/b+ 1/x

Answer 1

Answer:

x = -a, -b

Step-by-step explanation:

$\dfrac{1}{a+b+x} = \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{x}$

Transposing x to the LHS we get,

$\dfrac{1}{a+b+x} - \dfrac{1}{x} = \dfrac{1}{a} + \dfrac{1}{b}$

Taking LCM we get,

$\dfrac{ x - ( a + b + x ) }{(a+b+x)(x)} = \dfrac{b + a }{ab}\\\\\\\implies \dfrac{ x - a - b - x}{ax + bx + x^2} = \dfrac{a+b}{ab}\\\\\\\implies \dfrac{ -1 ( a + b )}{ ax + bx + x^2} = \dfrac{a+b}{ab}$

( a + b ) gets cancelled and we get,

$\implies \dfrac{ -1}{ ax + bx + x^2} = \dfrac{1}{ab}\\\\\text{Cross multiplying we get,}\\\\\implies -ab = ax + bx + x^2\\\\\implies x^2 + ax + bx + ab = 0$

Solving this we get,

⇒ x ( x + a ) + b ( x + a ) = 0

⇒ ( x + a ) ( x + b ) = 0

⇒ x = -a, -b

These are the required answers !!

Hope it helped !!