Let a, b, c represent the three numbers. The problem statement gives rise to three equations:
- a +b +c = 26
- 2a -b = c -2
- -3a +b = c
Adding the first two equations gives
... (a +b +c) +(2a -b) = (26) +(c -2)
... 3a +c = 24 +c . . . . . simplify
... 3a = 24 . . . . . . . . . . subtract c
... a = 8 . . . . . . . . . . . . divide by 3
Adding the second and third equations gives ...
... (2a -b) +(-3a +b) = (c -2) +(c)
... -a = 2c -2 . . . . simplify
... -6 = 2c . . . . . . add 2, substitute for a
... -3 = c . . . . . . . . divide by 2
Using the third equation we can find b.
... -3·8 +b = -3 . . . . substitute for a and c
... b = 21 . . . . . . . . .add 24
The numbers are 8, 21, -3.
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The method above is sort of "ad hoc", taking advantage of the numbers in this particular set of equations. You can use more formal methods of Gaussian elimination or Cramer's Rule to solve these by just following a procedure. Or, your graphing calculator can do it for you.