Find solutions to magic puzzle with sums

Question

I need help to solve the folowing puzzle using linear algebra (matrix and Gauss-Jordan Method):

(for example the second horinzontal line: w + w + w + z = 45 or the second vertical line y + w + x + w = 46)

x   y   z   w   ?
w   w   w   z   45
y   x   y   x   48
z   w   y   z   53
?   46  59  41

The problem is to find the value of ? (and also the value of x,y,z and w), I tried to solve this, creating the matrix:

1   1   1   1   a
0   0   1   3   45
2   2   0   0   48
0   1   2   1   53

this matrix corresponds to the following systems of linear equations:

x + y + z + w = a
0x + 0y + z + 3w = 45
2x + 2y + 0z + 0w = 48
0x + y + 2z + w = 53

I could not find the values x,y,z and w and the value of a, for solve the vertical and horizontal sums . Appreciate any help, thanks.

Answer 1

Let's look at 3rd row first, y+x+y+x=48, we get y+x=24. Then we look at 2nd column, y+w+x+w=46, so 2w = 46-24 = 22, w = 11. Because w+w+w+z = 45, z = 45-33 = 12. Therefore, x+y+z+w = 24+11+12 = 47.