Question 95382
This problem is a long tough grind but we can do it.
Because so there is so much manipulation it is easy to make an error.
The technical term for what we will do is Gaussian Elimination.
Let's line up all the unknowns, one under the other.
X.3a-2b+2c=1
Y.2a+5b-5c=7
Z.4a-3b+c=-3
Multiply -2 times equation Z and add Z to  X.
X.3a-2b+2c=1
Z.-8a+6b-2c=6
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X.-5a+4b=7 this is the sum of X and Z and is the new X. Notice c=0 and the equation is simpler
X.-5a+4b=7
Y.2a+5b-5c=7
Z.4a-3b+c=-3
Multiply 5 times Z and add it to Y
Y. 2a+5b-5c=7
Z. 20a-15b+5c=-15
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=>22a-10b=8 this is the new Y. 
X.-5a+4b=7
Y.22a-10b=8
Z.4a-3b+c=-3
Multiply 2.5 times X and add X and Y
X.-12.5a+10b=17.5
Y.22a-10b=8
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=>9.5a=9.5
a=1 Eureka! the rest is easy.
X. -5+4b=7 4b=12 b=3
Z. 4-9+c=-3 c=-3+5=2
Answer a=1, b=3, c=2
Check: (plug into the original equations)
3a-2b+2c=1 3-6+4=1
2a+5b=7+5c 2+15=7+10
c+3+4a=3b 2+3+4=9
EdJones