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A student proposes \(X_p = (1, 1, 1)^\top\) as a particular solution of the system \(\{x + y + z = 3 \;;\; 2 x + 3 y + z = 5\}\) in \((x, y, z) \in \mathbb{R}^3\), and claims that the homogeneous system has solution set \(\mathrm{Vect}((-2, 1, 1)^\top)\). Verify first that the proposed \(X_p\) is indeed a particular solution; if not, find a correct one. Then write the full solution set both as a parametric family and in the form \(X_p + \mathrm{Vect}(\ldots)\).
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