Solve the linear programming problem by the simplex method. Minimize $4 x+y$ subject to the constraints shown on the right. \[ \left\{\begin{array}{l} x+y \geq 2 \\ 3 x \quad \geq 5 \\ x \geq 0, y \geq 0 \end{array}\right. \] The minimum value of $\mathrm{M}$ is which is attained for $\mathrm{x}=$ and $y=$ (Type integers or fractions.)

Step 1 ：Given a linear programming problem, the objective is to minimize the function \(4x + y\) subject to the constraints \(x + y \geq 2\), \(3x \geq 5\), and \(x, y \geq 0\).

Step 2 ：To solve this problem, we can use the simplex method. The simplex method is an algorithm for solving linear programming problems. It starts with a feasible solution and iteratively improves it until it finds the optimal solution.

Step 3 ：First, we need to convert the inequalities into equalities by introducing slack variables. Then, we can set up the initial simplex tableau and perform the simplex algorithm to find the optimal solution.

Step 4 ：The optimal solution is \(x = 1.667\) and \(y = 0.333\), and the minimum value of \(M\) is \(7.0\).

Step 5 ：Final Answer: The minimum value of \(M\) is \(\boxed{7.0}\), which is attained for \(x = \boxed{1.667}\) and \(y = \boxed{0.333}\).