I don't know how to get equation 4.41 here in the image

[Shuvro1994]

Homework Statement

I don't know how to get equation 4.41 here in the image.

Relevant Equations

Please see the image I shared and help me to get equation 4.41 from equation 4.39 and 4.40

 

[mfb]

That's a lengthy process. Note that A still appears, so only B,C,D have been eliminated using the four equations. One way to approach this:
Solve the first equation for B, use that in the second equation. Now B is gone. Solve the second equation for C, plug that into the remaining two. Now C is gone as well. Solve one of the last two equations for D, plug it into the other one, solve for E, simplify.
Probably a bit less messy: Solve the first equation for B, use that in the second equation. Write it as $Ce^{ik_2a}=...$ and use that in the other two equations. Solve one of them for $De^{-ik_2a}=...$, plug it into the other one, and then solve for E.

 

[Abhishek11235]

Eq 4.40 is easy to solve for both C and D. Now use Eq 4.39 equation to eliminate B from 4.39(i) equation by putting into 4.39(ii) equation. Now simply put value of C and D in this equation and you will easily get value of E as advertised!

 

[vela]

Cramer’s rule may help lessen the amount of algebra you need to do.