Will the sum be prime, too?

Algebra Level 3

x 2 + x y + y z = 13 x^2 + xy + yz = 13 y 2 + y z + x z = 17 y^2 + yz + xz = 17 z 2 + x z + x y = 19 z^2 + xz + xy = 19

From the system of linear equations above, compute ∣ x + y + z ∣ \mid x+y+z\mid .


The answer is 7.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

By adding the three equations together, we will get:

x 2 x^2 + y 2 y^2 + z 2 z^2 + 2xy + 2xz + 2yz = 13 + 17 + 19 = 49.

This is actually the expansion of ( x + y + z ) 2 (x+y+z)^2 = x 2 x^2 + y 2 y^2 + z 2 z^2 + 2xy + 2xz + 2yz.

Therefore, ∣ x + y + z ∣ \mid x+y+z \mid = 49 \sqrt{49} = 7.

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