Linear, Quadratic, Cubic Constraints

Algebra Level 5

Three real numbers \(x, y,\) and \(z\) such that \(x < y < z\) satisfy the following three equations:

\[\begin{align} x + y + z &= 4 \\ x^2 + y^2 + z^2 &= 10 \\ x^3 + y^3 + z^3 &= 22. \end{align}\]

Find the value of \(z.\)

This problem is not original.

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