V= k₃*[ES]

Rate of formation of ES = k₁ * [E]*[S]

Rate of breakdown of ES = (k₂ + k₃) * [ES]

At steady state, the formation and the breakdown are equal. This steady state would only be temporary.

k₁ * [E]*[S] = (k₂ + k₃) * [ES]

rearranging:

[ES] = [E]*[S] / ( (k₂ + k₃)/(k₁))

We can lump these constants to make a new constant called K_M = (k2+k3)/k1

[ES] = [E][S]/ K_M

[E_T] = [E] + [ES] (The total amount of enzyme equals the free and that bound to substrate)

Substituting in [E_T] - [ES] for [E]

[ES] = ([E_T] - [ES]) [S]/ K_M

Solving for [ES] leads to [ES] = ([E_T] (([S]/ K_M)/(1 + [S]/ K_M ))

Which simplifies to

[ES] = ([E_T] *([S]/([S] + K_M )

Multiplying both sides by the kinetic constant k₃ gives the velocity of the reaction

v = k₃ * [ES] = k₃*[E_T] *(([S]/([S] + K_M )

and substituting V_max for k₃*[E_T] leads to the familiar form of the Michaelis Menten Equation

v = V_max *[S]/([S] + K_M )