Constrained Optimization: Substitution and Lagrange

The problem economics actually asks

Method one: substitution

Method two: Lagrange

The multiplier has a meaning

In the substitution method for maximizing $U = x_1 x_2$ subject to $x_1 + x_2 = 20$, what exactly does the first move accomplish?

For $\mathcal{L} = U(x_1, x_2) + \lambda(m - p_1 x_1 - p_2 x_2)$, what do the conditions $\frac{\partial U}{\partial x_1} = \lambda p_1$ and $\frac{\partial U}{\partial x_2} = \lambda p_2$ state economically?

At the optimum $(10, 10)$ of $U = x_1 x_2$ with $p_1 = p_2 = 1$, what is the marginal rate of substitution $MU_1 / MU_2$?

The Lagrange multiplier at an optimum equals 10. Which statement interprets it correctly?

Maximizing $x_1 (20 - x_1)$ by the first-order condition, compute the optimal $x_1^*$, and write the number.

For the problem maximize $x_1 x_2$ subject to $x_1 + x_2 = 20$, compute the Lagrange multiplier $\lambda$ at the optimum, and write the number.

If income rises from 20 to 21 with both prices at one, where does the optimum of x1 x2 move?

The condition MU1 over p1 equals MU2 over p2 says which practical sentence?

Why divide one first-order condition by the other in the Lagrange setup?

When is the Lagrange method the more convenient route over substitution?

At the budget of 21 the optimum is (10.5, 10.5). Compute the new maximum utility and write the number with two decimals.

Lambda equals 10 at the optimum. Predict the utility gain from one more dollar and write that prediction.