Suppose a^x = (x + y + z)^y, a^y = (x + y + z)^z and a^z = (x + y + z)^x, where a 0 and a is not equal to 1. What is the relationship between x, y and z?

Question

Suppose a^x = (x + y + z)^y, a^y = (x + y + z)^z and a^z = (x + y + z)^x, where a > 0 and a is not equal to 1. What is the relationship between x, y and z?

Accepted Answer

x = y = z. Let s = x + y + z and t = logₐ(s). Taking log base a of each equation gives x = y*t, y = z*t, z = x*t. Multiplying the three: x*y*z = x*y*z*t³, so t³ = 1, giving t = 1 (for real values), i.e. logₐ(s) = 1. Then x = y, y = z, z = x, so x = y = z.

Suppose a^x = (x + y + z)^y, a^y = (x + y + z)^z and a^z = (x + y + z)^x, where a > 0 and a is not equal to 1. What is the relationship between x, y and z?

Solution

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