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29, 66, 22

Step-by-step explanation:

Let a, b, c represent the three numbers, in order. The problem statement tells us ...

... a + b + c = 117

... c = a - 7

... b = 3c

Solving these equations

There are a variety of ways that three equations in three unknowns can be solved, including elimination, matrix methods, Cramer's rule, and substitution.

We can use substitution to find the values. Using the third equation in the first, we have ...

... a + 3c + c = 117

... a + 4c = 117

Now, using the second equation to substitute for c in this, we have

... a + 4(a -7) = 117

... 5a -28 = 117 . . . . simplify

... 5a = 145 . . . . . . add 28

... a = 29 . . . . . . . . divide by 5

Using the second equation, ...

... c = 29 -7 = 22

Using the third equation, ...

... b = 3·22 = 66

The three numbers are 29, 66, and 22.