x +  y +  z = a             (1)
x + 2y + 2z = a + b         (2)
x + 2y + 3z = a + b + c     (3)


The Cramer's rule is very bad choice as the method for solving this system.

The best choice is the Elimination method. Using the Elimination method, first subtract eq(2) from eq(3). You will get

z = c,     and one unknown is just found.


Next, multiply eq(1) by 2 and then write this modified eq(1) with eq(2) together:

2x + 2y + 2z = 2a,          (4)
 x + 2y + 2z =  a + b       (5)


Subtract eq(5) from eq(4). You will get

x = a - b,     and one more unknown is just found.


Now substitute the found values (expressions) for x and z into equation (1). You will get

(a-b) + y + c = a,

which implies

y = a - (a-b) - c = b - c.


Answer.  x = a - b;  y = b - c,  and  z = c.