Firstly, we need to write down the information given in the form of algebraic equations. We can represent our two unknown numbers, the greater and the lesser, as x and y respectively.
From the given information, we can form two equations. The first equation is from "the difference between the two numbers is 15", which gives us x - y = 15, where x is the larger number and y is the smaller one.
Also, we know that "If 8 is added to twice the greater number, the result is 4 less than 3 times the lesser number", this gives us the second equation: 2*x + 8 = 3*y - 4.
We now have a system of two equations that we can solve simultaneously:
1) x - y = 15
2) 2*x + 8 = 3*y - 4
To make it easier, let's multiply the first equation by 2, which gives us 2*x - 2*y = 30.
Now subtract the second equation (2*x + 8 = 3*y - 4) from the first (2*x - 2*y = 30), it yields: -2*y - 3*y = 30 - (-4 - 8).
This simplifies to -5y = 30 + 4 + 8, or -5*y = 42.
To get the value of y, divide by -5, therefore y = -42 / -5 = 42.
Now to find x, we can substitute y = 42 in the first equation (x - y = 15), so we get x = 42 + 15 = 57.
Hence, the greater number is 57 and the lesser number is 42.