Question

Need help ASAP
Solve:
$\sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x}$
​

Answer 1

Question: Solve for x:-

1/p + 1/q + 1/x = 1/(p + q + x)

Answer:

- q or - p

Step-by-step explanation:

$\sf{ \frac{1}{p} + \frac{1}{q} + \frac{1}{x} = \frac{1}{p + q + x}} \\ \\ \sf{ \frac{1}{p} + \frac{1}{q} = \frac{1}{p + q + x} - \frac{1}{x} \: } \\ \\ \sf{ \frac{p + q}{pq} = \frac{x - (p + q + x)}{x(p + q + x)} } \\ \\ \sf{ \frac{p + q}{pq} = \frac{ - (p + q)}{x(p + q + x)} } \\ \\ \sf{ \frac{1}{pq} = \frac{-1}{xp + xq + {x}^{2} } } \\ \\ \sf{xp + xq + {x}^{2} = - pq} \\ \\\sf{xp +pq + xq + {x}^{2} = 0} \\ \\ \sf{ p(x + q) + x(q + x) = 0 } \\ \\ \sf{(x + q)(x + p) = 0} \\ \\ \sf{ x = - q \: \: \: \: or \: \: \: \: x = - p}$

Answer 2

Given that , $\sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x}$.

Need To Find : Value of x ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀Given that ,

$\dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x} \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Solving \: the \: Given \::}}$

$\dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x} \:\\\\ \dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} - \dfrac{1}{p + q + x} = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{1}{p} + \dfrac{1}{q}\bigg] +\bigg[ \dfrac{1}{x} - \dfrac{1}{p + q + x} \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{(x + p + q )- x }{x ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{x + p + q - x }{x ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{x + p + q - x }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{ p + q }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{1}{pq} + \dfrac{ 1 }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x^2 + xp + xq + pq }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x^2 + xq + xp + pq }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x( x + q ) + p ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = \dfrac{0}{( p + q )} \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( x + p ) ( x + q ) = 0 \times (pq) (x) ( p + q + x ) \:\\\\ \dashrightarrow \sf ( x + p ) ( x + q ) = 0 \:\\\\ \dashrightarrow \sf \:x\:=\:-p \:\:or\:\: x \:=\:-q \:$

$\dashrightarrow \:\underline {\boxed{\purple {\pmb{\frak{\:\:x\:=\:-p \:\:or\:\: \:-q \:}}}}}\:\:\bigstar \:\:$

$\qquad \therefore \underline {\sf Hence, The \:value \:of \: x \:can \: be \:\pmb{\bf{ - p \: or \:-q \:}}.}$