2x - y + 2z = 15   (1)
-x + y +  z =  3   (2)
3x - y + 2z = 18   (3)


Distract equation (1) from equation (2) (both sides). You will get

x = 18 - 15  --->  x = 3. So, you just solved for x.

Next substitute x=3 into (1) and (2). You will get

2*3 - y + 2z = 15    (4)   ( instead of (1) )
-3  + y +  z =  3    (5)   ( instead of (2) )

Simplify (4) and (5). You will get

-y + 2z = 15 - 6,
 y +  z =  3 + 3.

Simplify 

-y + 2z = 9,        (6)
 y +  z = 6.        (7)

Add equations (6) and (7) (both sides). You will get

3z = 9 + 6  --->  3z = 15  --->  z =  = 5.  So, you just solved for z.

Now find y from (7). Please do it on your own.