Quantum Mechanic/wave function/junction condition

[GreenTea09]

May i know how do i eliminate C and D and how do i obtain the last two equations? Are there skipping of steps in between 4th to 5th equation? What are the intermediate steps that i should take to transit from 4th equation to the 5th equation?

 

[PeroK]

It looks like quite a few steps have been skipped. You either trust the author or check the algebra yourself.

 

[Haborix]

Or the best of both worlds: trust, but verify. Doing a messy calculation is like eating oatmeal, not pleasant in the moment but makes things easier later.

 

[BvU]

And for the sake of PF readers: provide essential context so we can see where these boundary condition equations are sitting ! And where the wave number is $k_i, k_i', k_i''$ (hey, can we skip the subscript 1 ?