You will notice that I have changed Jean's name to Marie, since Joe and Jean
start with the same letter.  It makes age problems easier to think through
if you can use people's initials to represent their ages.

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In addition to their ages, we must introduce additional
variables for the two 'numbers of years ago' implied in the
problem.  We will let them be X years ago and Y years ago,
respectively.

We begin by interpreting these words:

X YEARS AGO WAS when Joe was half as old as Kate WAS X YEARS AGO:

Joe was J-X then and Kate was K-X then, so

J-X = (K-X)

Next we interpret the words:

We are still talking about X years ago:

X YEARS AGO WAS WHEN his cousin Marie was 38.

Marie was M-X years then, so

M-X = 38  

Now we look at this:

Now we're talking about Y years ago.

Y YEARS AGO WAS when Kate was half as old as Marie. 

Kate was K-Y then and Marie was M-Y 

So

K-Y = (M-Y)

Next we look at this:

We are still talking about Y years ago.

so Y YEARS AGO, Joe himself was 17.  

Joe was J-Y then so,

J-Y = 17

Now we look at this:

That's easy!

Their ages NOW total 113   

So

J+K+M = 113

So we have this system of 5 equations in
5 unknowns:

J-X = (K-X)
M-X = 38
K-Y = (M-Y)
J-Y = 17
J+K+M = 113

Simplify the first equation:

J-X = (K-X)

Multiply both sides by 2

    2J - 2X = K - X
2J - K - X = 0

Simplify the third equation

K-Y = (M-Y}}}

Multiply both sides by 2

2K - 2Y = M - Y
2K - M - Y = 0

So here are the 5 equations:

2J - K - X = 0
M - X = 38
2K - M - Y = 0
J - Y = 17
J + K + M = 113

Now rearrange the terms so that like letters 
are in the same columns:

(1)  2J -  K     - X     =   0
(2)            M - X     =  38
(3)       2K - M     - Y =   0
(4)   J              - Y =  17
(5)   J +  K + M         = 113

Solve (4) for Y 

J - 17 = Y

and substitute in (3)

  2K - M - (J - 17) = 0
    2K - M - J + 17 = 0

Rearrange as

(6)  -J + 2K - M         = -17

So take out (3) and (4) and put in (6)

(1)  2J -  K     - X  =   0
(2)            M - X  =  38
(6)  -J + 2K - M      = -17
(5)   J +  K + M      = 113

Solve (2) for X

M - 38 = X

and substitute in (1)

2J - K - (M - 38) = 0
2J - K - M + 38 = 0

Rearrange as 

(7) 2J - K - M = -38

Take out (1) and (2) and put in (7)

(7)  2J -  K - M = -38
(6)  -J + 2K - M = -17
(5)   J +  K + M = 113

Add equations (6) and (5) and get

          3K     =  96
               K = 32, so Kate is 32

Add equations (7) and (5) and get

     3J          =  75
               J = 25, so Joe is 25

Substitute J = 25 and K = 32 in (5)

     25 + 32 + M = 113
          57 + M = 113
               M =  56 so Marie is 56.

That's the answer.  Now to check it, we will 
need X and Y, which we can get by going
back where we solved for them.  

 M - 38 = X and  J - 17 = Y
56 - 38 = X     25 - 17 = Y 
     18 = X          8 = Y

So

X = 18 YEARS AGO and  Y = 8 YEARS AGO.

Let's check the words, not the equations:

18 years ago Joe was 25-18=7 years old, Kate was 32-18=14, 
so it checks that that was when Joe was half Kate's age.
It also checks that Marie was 38 because 56-18=38.

8 years ago Kate was 32-8=24, and Marie was 56-8=48,
so it checks that that was when Kate was half of Marie's age.
It also checks that Joe was 17 because 25-8=17.

That's easy to check: 25+32+56=113.

Edwin