------ eq (i)
 ------ eq (ii)	
 ------ Multiplying eq (i) by - 2 ------ eq (iii)	
 ---------- Adding eqs (iii) & (ii)	
- 9y = - 14 -------- Cross-multiplying


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Instead of substituting the value of y into one of the equations, it's best to repeat the process, as follows:
 ------ eq (i)
 ------ eq (ii)	
 ------ Multiplying eq (ii) by - 4 ------ eq (iii)	
 ---------- Adding eqs (iii) & (i)	
x = - 21(2) ------- Cross-multiplying

Introduce new variables  u = 1/x  and  v = 1/y.

Then the original system of non-linear equations becomes (by a magical way) a linear system relative new variables "u" and "v".

3u + 4v = 5/2,   (1)
6u +  v = 1/2.   (2)


Now you can apply any method you want / you know to solve (1), (2).

I will use Elimination method. 

Multiply equation (1) by 4 (both sides), and equation (2) by 2 (both sides).


You will get

12u + 16v = 10,   (3)
12u +  2v = 1.    (4)


Subtract eq(4) from eq(3) to get

14v = 9.

Then  v = 9/14.

Having this, you easily get (from (4))  12u = 1 - 2*(9/14) = 1 - 18/14 = -4/14 = -2/7.

Hence,  u = -2/(7*12) = -1/42.

Thus you get  u = -1/42,  v = 9/14.


Now recall how you introduced u and v:  u = 1/x = -1/42;  hence,  x = -42.

                                   and   v = 1/y = 9/14;  hence,  y = 14/9.

The original system solved.  The solutions are  x = -42  and  y = 14/9.