For an elementary chemical reaction, A _ 2 ⇄ _ k _ - 1 ^ k _ 1 2A ,...

For the reaction, ${A}_{2}\overset{{K}_{1}}{\to}2A$

${r}_{f} = - \dfrac{d\left[{A}_{2}\right]}{dt} = \dfrac{1}{2}{\left(\dfrac{d\left[A\right]}{dt}\right)}_{1} = {K}_{1}\left[{A}_{2}\right]$

${\left(\dfrac{d\left[A\right]}{dt}\right)}_{1} = 2{K}_{1}\left[{A}_{2}\right]$

For the reaction, $2A\overset{{k}_{- 1}}{\to}{A}_{2}$

${r}_{b} = - \dfrac{1}{2}{\left(\dfrac{d\left[A\right]}{dt}\right)}_{2} = {K}_{- 1}{\left[A\right]}^{2}$

${\left(\dfrac{d\left[A\right]}{dt}\right)}_{2} = - 2{K}_{- 1}{\left[A\right]}^{2}$

Hence, the overall rate is given by:

$\dfrac{d\left[A\right]}{dt} = 2{k}_{1}\left[{A}_{2}\right] - 2{k}_{- 1}{\left[A\right]}^{2}$