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Suppose (x, y, z) is a point in R³ satisfying both x + 2y - 3z = 6 and x + y - z = 4. Among all such points, find the minimum value of the expression x² + y² + z² - 2x + 2y - 2z + 3, and then compute 15(x + y + z).
- 30
- 45
- 60
- 75
Correct answer: 45
Solution
The expression equals the squared distance from (x,y,z) to the point (1,-1,1). The two plane equations define a line, and minimizing distance from (1,-1,1) to that line gives the optimal (x,y,z).
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