Minimize $Q=2 x^{2}+3 y^{2}$, where $x+y=5$
\[
\begin{array}{l}
x= \\
y=
\end{array}
\]
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Step 1 ：We are given the problem to minimize the function \(Q=2 x^{2}+3 y^{2}\) under the constraint \(x+y=5\).

Step 2 ：We can solve this problem using the method of Lagrange multipliers. The Lagrange function is \(L(x, y, \lambda) = 2x^2 + 3y^2 - \lambda(x + y - 5)\).

Step 3 ：We need to find the partial derivatives of L with respect to x, y, and \(\lambda\), set them equal to zero, and solve the resulting system of equations.

Step 4 ：The partial derivative of L with respect to x is \(Lx = 4x - \lambda\).

Step 5 ：The partial derivative of L with respect to y is \(Ly = 6y - \lambda\).

Step 6 ：The partial derivative of L with respect to \(\lambda\) is \(L\lambda = -x - y + 5\).

Step 7 ：Solving these equations, we find that the solution to the system of equations is \(x = 3\), \(y = 2\), and \(\lambda = 12\).

Step 8 ：We can substitute these values back into the original equation to verify that they satisfy the constraint \(x + y = 5\).

Step 9 ：Substituting \(x = 3\) and \(y = 2\) into the constraint equation, we find that \(3 + 2 = 5\), which satisfies the constraint.

Step 10 ：\(\boxed{x=3, y=2}\) are the values that minimize the function \(Q=2 x^{2}+3 y^{2}\) under the constraint \(x+y=5\).