This is the matrix equation

Ax = b

It can be shown that the proportionality constant is 44.

This is the matrix A

-44 0 0 1 0 0
0 -44 0 0 1 0
0 0 -44 0 0 1
0 0 0 7 0 -6
0 0 0 -3 4 0
1 1 1 0 0 0

This is the RHS of the equation (ie the "vector" b)

0
0
0
0
0
17.5

The unknown "vector" x is

ea      Joe age
ka      Jack age
ma      Jim age
en      Joe number of spitachios
kn      Jack number of spitachios
mn      Jim number of spitachios

For example, row 1 of matrix A captures the fact that

-44ea + 0ka + 0ma + 1en + 0kn + 0mn = 0

This is the solution (ie the "vector" x)

   6.000
   4.500
   7.000
264.000
198.000
308.000

This is the inverse of A

-0.014935064935 0.007792207792 0.007792207792 0.001298701299 -0.001948051948 0.342857142857

0.005844155844 -0.016883116883 0.005844155844 0.000974025974 0.004220779221 0.257142857143

0.009090909091 0.009090909091 -0.013636363636 -0.002272727273 -0.002272727273 0.4

0.342857142857 0.342857142857 0.342857142857 0.057142857143  -0.085714285714 15.085714285714

0.257142857143 0.257142857143 0.257142857143 0.042857142857  0.185714285714 11.314285714286

0.4 0.4 0.4 -0.1 -0.1 17.6

which is what we use on the RHS to get the solution. Since all but the
last element of the RHS is 0, the solution boils down to multiplying the
right-most elements of the above matrix by 17.5.

e.g.

0.342857142857 * 17.5 = 6

...

17.6 * 17.5 = 308