In order to solve for a set of three equations with three variables, you start by picking any two of the three sets of equations. That means you can pick equations 1 and 2, then 1 and 3, or 2 and 3. You now eliminate one of the variables from each pair using the elimination method.
We shall begin with equations 1 and 2.
2a + b - c = -17 -----------(1)
a + 5b - 3c = -13 --------(2)
Multiply equaton (1) by 1 and then equation (2) by 2, to eliminate the a variable. You now have;
2a + b - c = -17 --------(3)
2a + 10b - 6c = -26 ---(4)
Next you subtract equation (3) from equation (4)
9b - 5c = -9
We shall now move onto equations 2 and 3.
a + 5b - 3c = -13 ----------------(1)
3a - 2b + 6c = 28 --------------(2)
Multiply equation (1) by 3 and then equation (2) by 1 to eliminate the a variable. You now have;
3a + 15b - 9c = -39 ----------------(3)
3a - 2b + 6c = 28 ------------------(4)
Next you subtract equation (3) from equation (4)
-17b +15c = 67
We now have two new equations which are as follows;
9b - 5c = -9 -----------------(a)
-17b + 15c = 67 -------------(b)
We shall now solve for variables b and c by using elimination method. To begin, multiply equation (a) by 15 and equation (b) by -5 to eliminate the c variable.
135b - 75c = -135 ----------(1)
85b -75c = -335 -----------(2)
Subtract equation (2) from equation (1)
50b = 200
Divide both sides of the equation by 50 and you'll have,
b = 4
Next you substitute for the value of b into equation (a) which is one of the two new equations we just derived and you now have;
9b - 5c = -9
9(4) -5c = -9
36 - 5c = -9
Subtract 36 from both sids of the equation and you now have
- 5c = - 45
Divide both sides of the equation by - 5 and you now arrive at,
c = 9
Having calculated b = 4 and c = 9, we shall now go back to the "ORIGINAL" equations which are,
2a + b - c = -17 ------------------(1)
a + 5b - 3c = -13 ----------------(2)
3a - 2b + 6c = 28 --------------(3)
Substitute for the values of b and c into ANY of the equations, and so we shall take equation (1),
2a + b - c = -17
2a + 4 - 9 = -17
2a - 5 = -17
Add 5 to both sides of the equation,
2a = -12
Next you divide both sides of the equation by 2 and you arrive at,
a = - 6
Therefore your answer is,
a = - 6
b = 4
c = 9