simple algebraic method to solve magic square for children sorry if the question is too noob for this forum.. my 11 years old son was assigned to the task of solve a (kind of) magic square, that can be represented like the following system of equations:


*

*row $1: 6 + a + b = 11$

*row $2: c + d + 5 = 15$

*row $3: e + 4 + f = 19$

*col $\;\,  1: 6 + c + e = 15$

*col $\;\, 2: a + d + 4 = 16$

*col $\;\, 3: b + 5 + f = 14$
We solved it by writing a small program with a brute-force approach, but I wonder if there is some algebraic method, something more elegant that brute-force but simple enough to be explained to a kid (i. e. simplex matrix manipulations are too complex). Thanks for any suggestion! 
 A: From row 1: $b = 5 - a$.
Substitute this into column 3: $5 - a + 5 + f = 14$, i.e. $f = a + 4$.
Substitute this into row 3: $e + 4 + a + 4 = 19$, i.e. $e = 11 - a$.
Substitute this into column 1: $ 6 + c + 11 - a = 15$, i.e. $c = a - 2$.
Substitute this into row 2: $a - 2 + d + 5 = 15$, i.e. $d = 12 - a$.
But this is just what column $2$ says.
So we can take anything for $a$, and we can get $b, c, d, e, f$ from the equations above.
EDIT: Here's an alternative, maybe better for an $11$-year-old.  To find one solution, guess a value for $a$, say $a=0$.  Then one by one we see what values of $b, f, e, c, d$ would get.  
Now what would happen if we increased $a$ by $1$?  From row $1$, this would make $b$ decrease by $1$.  From column $3$, that would make $f$ increase by $1$.
From row $3$, that would make $e$ decrease by $1$.  From column $1$, that would make $c$ increase by $1$.  From row $2$, that would make $d$ decrease by $1$.
And column $2$ would still be satisfied.  So you'd have another solution, with each variable increased or decreased by $1$.  
Now what if you added an arbitrary amount to $a$?  Again, each of the other variables would increase or decrease by that same arbitrary amount.  So you can make $a$ whatever you want, and find what the other variables would be.
A: So we have this magic square (the left with letters, the right with empty spaces):

Remaining numbers for filling the empty spaces are $1, 2, 3, 7, 8, 9$.
Let's begin with the first row. The sum of empty spaces in it must be $5 \ (=11-6)$. With the remaining numbers there are only two possibilities how to write $5$ as a sum of two numbers:
$$5 = 2 + 3 \\
  5 = 3 + 2 $$
The first one is impossible:

because in this case must be in the place of "?" number $10$.
So there is only the second possibility:

No problem for $11$-year-old child to finish the task now.
