Easiest way (no algebra).  Each of the 4 will be 6 years younger 7 years
from now than they will be 13 years from now, so that will take four 6's
or 24 years off the sum  of their ages, and 97 - 24 = 73. 

Another way (without algebra):
Let's make up some ages. Let's suppose in 13 years their ages will be 
25,25,25 and 22. (Notice those four ages have sum 97.)  Then today they 
are 13 years younger than they will be then.  So they are 12,12,12,and 9.
In 7 years they will be 19,19,19, and 16, so their sum in 7 years will
be 19+19+19+16 = 73.

Now let's use algebra.

Suppose their ages today are A, B, C, and D.  Then in 13 years their
ages will be A+13, B+13, C+13, and D+13, so the sum of their ages in
13 years will be

A+13 + B+13 + C+13 + D+13

We are told this must equal to 97, so we have the equation

A+13 + B+13 + C+13 + D+13 = 97

       A + B + C + D + 52 = 97

            A + B + C + D = 45

Then in 7 years their ages will be A+7, B+7, C+7, and D+7, 
so the sum of their ages in 7 years will be

A+7 + B+7 + C+7 + D+7

which equals to 

A + B + C + D + 28

Since A + B + C + D = 45, 

A + B + C + D + 28 = 45 + 28 or 73.

Edwin