Answer: $25$

Using few unknowns

first + (middle 3) = (middle 3) + last $\therefore$ first = last, so the numbers indicated are equal

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The code is:   

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Repeat for the 5s:

 

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The code is:   

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Fours: $3a+x=16$

Fives: $4a+x=19$

$\therefore a=3$

All seven: $6a+x = 2a + (4a+x) = 6+19=25$

Using seven unknowns

Let the $7$-digit code be $abcdefg$.

We know that

$(1)$ $a+b+c+d=16$

$(2)$      $b+c+d+e=16$

$(3)$           $c+d+e+f=16$

$(4)$                $d+e+f+g=16$

$(5)$ $a+b+c+d+e=19$

$(6)$      $b+c+d+e+f=19$

$(7)$           $c+d+e+f+g=19$

If we take equation $(1)$ away from equation $(5)$ we obtain $e=3$.

Similarly:

$(6)-(2)$ gives that $f=3$,

$(7)-(3)$ gives that $g=3$,

$(5)-(2)$ gives that $a=3$,

$(6)-(3)$ gives that $b=3$ and

$(7)-(4)$ gives that $c=3$.

Then using equation $(1)$ we find that $d=7$.

So the code is $3337333$ and the sum of the digits is $25$.