Problem Solving

If x^2-yz=10, y^2+xz=10, z^2 +xy=10, and z not=y, what is the value of x^2+y^2+z^2


10
15
20
25
30

C

is there an easy way to do this problem? shortcut?

x^2 - yz = 10 (eqn a)
y^2 + xz = 10 (eqn b)
z^2 + xy = 10 (eqn c)

x^2 + y^2 + z^2 = ?

(eqn a) - (eqn b):

x^2 - yz - y^2 - xz = 0
x^2 - y^2 = xz + yz
(x + y)(x - y) = z(x + y)
x - y = z (eqn 1)

Add equations a, b and c:

x^2 + y^2 + z^2 - yz + xz + xy = 30
x^2 + y^2 + z^2 - yz + xz + (10 - z^2) = 30 (xy substituted with equation c)
x^2 + y^2 + z^2 - yz + xz - z^2 = 20
x^2 + y^2 - yz + xz = 20
x^2 + y^2 + z(x - y) = 20
x^2 + y^2 + z(z) = 20 (insert equation 1)
x^2 + y^2 + z^2 = 20

Choose C

-BM-

How do we know that (x + y) ≠ 0, BM?

sanju09 wrote:How do we know that (x + y) ≠ 0, BM?

You are right, i didnt consider (x+y) = 0 in my solution.

However, with (x+y) = 0, can equations a, b and c all be satisfied? I cannot find a solution for x, y and z under this condition.

How would you solve this problem, sanju?

-BM-