Starting from the partial differential equation #1, differentiate once more with respect to z in Step a.
Interchange the order of differentiation in Step b; we are only interested in solutions which permit this!
Substitute partial differential equation #2 in Step c.
The wave equation results in Step d! Similar equations will result for the other components of E and also for B. The general form of the wave equation in terms of vector quantitites is:
Note that we started with two coupled first-order partial differential equations and ended with an uncoupled second-order differential equation.