Your starting equations are

    2x +  y = 8    (1)

    3x - 2y = 5.   (2)


From equation (1), express y = 8 - 2x  and substitute it into equation (2)

    3x - 2(8-2x) = 5.


You have now one single equation for one unknown x.
Simplify and find x

    3x - 16 + 4x = 5,

    3x + 4x = 5 + 16

       7x   =   21

        x   =   21/7 = 3.


So, x is just found, x= 3.


Now substitute x = 3 in the first equation to find y

    2*3 + y = 8,

    y = 8 - 2*3 = 8 - 6 = 2.


ANSWER.  The solution is this pair  (x,y) = (3,2).


CHECK.  Substitute these values x and y into equations to make sure
        that left sides are equal to their right sides.