Problem
Minimize the objective function $3 x+3 y$ subject to the constraints. \[ \left\{\begin{array}{l} 2 x+y \geq 12 \\ x+2 y \geq 12 \\ x \geq 0, y \geq 0 \end{array}\right. \] The minimum value of the function is $\square$. (Simplify your answer.) The value of $x$ is $\square$. (Simplify your answer.) The value of $y$ is $\square$. (Simplify your answer.) Don't post this question on Chegg. Doing so violates the UVic Policy on Academic Integrity. 8576
Solution
Step 1 ：Define the objective function as \(3x + 3y\).
Step 2 ：The constraints are given by the inequalities \(2x + y \geq 12\), \(x + 2y \geq 12\), and \(x, y \geq 0\).
Step 3 ：Solve the linear programming problem using these constraints.
Step 4 ：The minimum value of the objective function is \(\boxed{24}\).
Step 5 ：The values of \(x\) and \(y\) at which this minimum occurs are both \(\boxed{4}\).
