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Algebra Difficulty 4.4 AIME Find the answer

Suppose that x,y,zx, y, z are real numbers such that x=y+z+2x=y+z+2, y=z+x+1y=z+x+1, and z=x+y+4z=x+y+4. Compute x+y+zx+y+z.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Adding all three equations gives x+y+z=2(x+y+z)+7x+y+z=2(x+y+z)+7 from which we find that x+y+z=7x+y+z=-7.

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