MHB System of three equations x^2 + (xz) - y - z = 0

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The discussion revolves around solving a system of three nonlinear equations involving variables x, y, and z. The equations presented are y^2 + (yx) - z - x = 0, z^2 + (zy) - x - y = 0, and x^2 + (xz) - y - z = 0. A proposed solution is x = y = z = 1, which satisfies all three equations. Participants are encouraged to verify this solution or explore other potential solutions. The focus remains on finding valid values for the variables that meet the criteria set by the equations.
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Find x, y, z solving all the three equations:

y^2 + (y*x) - z - x = 0z^2 + (z*y) - x - y = 0x^2 + (x*z) - y - z = 0I would be very grateful!
 
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x = y = z = 1
 

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