Your starting equations are

    25x^2 - y^2 = 36      (1)

    5x    + y   =  2      (2)


This system of two equations (one equation of the degree 2 and the other of the degree 1) is very special.


Although one of the starting equations is of the degree 2, the system can be reduced to the system 
of two LINEAR equations, which can be EASILY solved.


The first equation admits factoring

    (5x - y)*(5x + y) = 36    (3)


In this factored equation, we can replace (5x + y) by the value of 2, based on equation (2).
Doing it, we transform equation (3) to the form

    2*(5x - y) = 36,  

or

    5x - y = 18             (4)

(where 18 = 36/2).


Thus from the original non-linear system of equation, we get the linear system

    5x - y = 18             (4)

    5x + y =  2             (5)


The system of linear equations (4), (5) is EQUIVALENT to the nonlinear system of equations (1), (2).


To solve equations (4), (5), use the ELIMINATION method and add equations (4) and (5).  You will get

    10x = 18 + 2 = 20

      x          = 20/10 = 2.


To find y, substitute the found value x= 2 into equation (5).  You will get

    5*2 + y = 2,  y = 2 - 10 = -8.


Thus the original system of equations (1), (2) has a UNIQUE solution (x,y) = (2,-8).