Find the value of $\lambda$ if the plane $2x - 2y + z - 3 + \lambda = 0$ contains the point $(3, 1, 1)$.
Step-by-Step Solution
Key Concept: A point lies on a plane if and only if its coordinates satisfy the plane equation; substitute to solve for unknown parameters.
Since the plane contains the point $(3, 1, 1)$, we substitute these coordinates into the plane equation: $2(3) - 2(1) + 1 - 3 + \lambda = 0$. This gives $6 - 2 + 1 - 3 + \lambda = 0$, which simplifies to $2 + \lambda = 0$, so $\lambda = -2$.
Correct Answer: -2