Use the simplex method to solve the following linear programming problem. Find $y_{1} \geq 0$ and $y_{2} \geq 0$ such that $4 y_{1}+y_{2} \geq 14, y_{1}+4 y_{2} \geq 14$, and $w=3 y_{1}+15 y_{2}$ is minimized.
The minimum is $w=\square$ at $\mathrm{y}_{1}=\square$ and $\mathrm{y}_{2}=\square$. (Simplify your answers.)
Answer
Final Answer: The minimum is \(w=\boxed{42}\) at \(y_{1}=\boxed{14}\) and \(y_{2}=\boxed{0}\).
Step 1 ：The problem is a linear programming problem and can be solved using the simplex method. The simplex method is an algorithm for solving linear programming problems. In this case, we are trying to minimize the function \(w=3y_1+15y_2\) subject to the constraints \(4y_1+y_2 \geq 14\) and \(y_1+4y_2 \geq 14\) with \(y_1 \geq 0\) and \(y_2 \geq 0\).
Step 2 ：By applying the simplex method, we find that the minimum value of \(w\) is 42.0 when \(y_1\) is 14.0 and \(y_2\) is 0.0.
Step 3 ：Final Answer: The minimum is \(w=\boxed{42}\) at \(y_{1}=\boxed{14}\) and \(y_{2}=\boxed{0}\).
