Question

what expression must the center cell of the table below contain so that the sums of each row, each column, and each diagonal are equivalent?

  x       8x      -3x
-2x      ?        6x
7x      -4x       3x

Answer 1

Oh, I adore magic squares!

We need all rows to be equivalent, so try adding up the bottom (complete) row to get 7x-4x+3x=10x-4x=6x.

To solve the middle number, we have -2x+X+6x=6x, so X=2x.
To solve the top number, we have X+8x-3x=6x, so X=6x-8x+3x=x

Now check first column, 
x-2x+7x=6x   ok
Second column
8x+2x-4x=6x  ok
Top-right to bottom-right diagonal: x+2x+3x=6x, ok   
Bottom-left to top right diagonal: 7x+2x-3x=6x

So everything is ok!