Substitution in Systems of Nonlinear Equations

Suppose that $x$, $y$, and $z$ are real numbers such that $$x + y + z = 3$$ and $$x^2 + y^2 + z^2 = 6.$$ What is the largest possible value of $z$?

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(Hint: For which values of $z$ can you solve the system of equations in $x$ and $y$?)

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Source: Math Prize for Girls