Using matrices and a computer.
Dear i_zz_y_ill,
x+y+z=1------------(1)
----------(2)
----------(3)
----------(4)
First substract equation (4) and equation (2) ; you could find kz
Then substract equation (3) and equation (2); you could find ky
By substitution kz and ky to one of the equaitons (2),(3) or (4) you could find kx.
Now you could find the value of k. Since you know the values of kx,ky and kz now you can find x,y and z
Hope this will help you.
The first thing I would do is multiply all except the first equation by 100!
k(x+ y+ z)= 2
k(x+ 4y+ z)= 3
k(x+ y+ 9z)= 4
Since the first equation was x+ y+ z= 1, k(x+ y+ z)= k= 2.
Now, x+ 4y+ z= 3/2 and subtracting x+ y+ z= 1 from that
3y= -1/2 so y= -1/6.
x+ y+ 9z= 2 and subtracting x+ y+ z= 1 from that, 8z= 1 so z= 1/8.
x- 1/6+ 1/8= x- 4/24+ 3/24= x- 1/24= 1 so x= 25/24.