12f-6e=-2 ---> 1.
7e-3f=6 ---> 2.

manipulate 1. 

12f-6e=-2
12f=-2+6e
f= (-1+3e)/6 --> 3.


manipulate 2.

7e-3f=6
-3f=6-7e
3f=7e-6
f=(7e-6)/3 --> 4.

equate 3. and 4.

3. = 4.
(-1+3e)/6 = (7e-6)/3

3(-1+3e)/6 = 7e-6

(-1+3e)/2 = 7e-6

-1+3e = 14e-12
 -11e = -11
    e= 1
e= 1 ---> 5.

substitute 5. into 1.

12f-6e=-2

12f-[6* (1)] = -2

12f- (6) = -2

12f = -2 + (6)

12f = 4

f = 4/12 = 1/3

therefore, e=1 f=1/3

lets check the values by substiting them into 2.

2. : 

7e-3f=6
7(1) - [3(1/3)] = 6
7-1 = 6
6=6 --> correct!

We now have proven that e= 1, f=1/3!