Projectile motion, simultaneous equations, arbitrary values, physics homework help

From

Physics for Scientists and Engineers, 6th Edition 6th Edition by Paul A. Tipler (Author), Gene Mosca (Author)
https://gyazo.com/5c59508ac1591d40ece62ab248ae37b2

Example 3-10: The Ranger and the Monkey, part A
https://gyazo.com/959989443146fba7739df09c06a18bd0

I do not find the book’s answer satisfactory as they do not strictly show mathematically how the equations to produce the end co-ordinates are equivalent with arbitrary values.

i.e. I feel like they are making assumptions that are not shown mathematically and I would like to know how to express the entire answer mathematically in the form “equation 1 = equation 2”

Any advice specifically or in terms of an approach would be welcomed