x(y+z)
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78 + yz = yz + 78
(z + 1)(y + x)
The idea here is to transfer to the other side everything you don't need. x = yz yz = x (switching both sides) y = x/z (you divide both sides by z, since you want to get rid of the z on the left side).
y(z+x) + 4(x+z)
There appear to be 10 terms in the determinant. A determinant can only have a perfect number of terms. So something has gone wrong with the question. 1: x2 plus 1 2: xy 3: xz 4: xy 5: y2 plus 1 6: yz 7: 1 plus x2 plus y2 plus z2 8: xz 9: yz 10: z2 plus 1