Your starting system of equations is

    4x^2 - y^2 = 15       (1)

     2x  - y   = 5        (2)


Decompose first equation as the difference of squares

    (2x-y)*(2x+y) = 15    (3)


In this equation, replace (2x-y) by 5, based on equation (2)

    5*(2x+y) = 15

which gives

    2x + y = 15/5 = 3.


So, now you have two linear equations instead of (1) and (2)

    2x + y = 3    (4)

    2x - y = 5    (5)


To solve, add the equations.  You will get

    4x = 3 + 5 = 8,  hence  x = 8/4 = 2.


Then from equation (3)

    y = 3 - 2x = 3 - 2*2 = -1.


ANSWER.  x = 2;  y = -1.