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I have a system of equations for my Operations Research class, and the book is solving them by using algebra. However, I think it would be easier to solve them using linear algebra, and will also serve as a powerful tool. The only problem the system has is that it is has constraints. Here is the system:

$Z = 3 x_1 + 5 x_2$, where $Z$ is the total profit. $$ \begin{cases} 1 x_1 + 0 x_2 \leq 4 \\ 0 x_1 + 2 x_2 \leq 12 \\ 3 x_1 + 2 x_2 \leq 18 \end{cases} $$ Also, $x_1$ and $x_2$ are $\geq 0$.

Answer: $x_1 = 2$,
$x_2 = 6$ and
$Z = 36$.

How would I solve it using linear algebra? Thanks

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Instead of an inquality use equality by adding a variable $u,v,w\ge 0 $ s.t : $$x_1 + u = 4$$ $$2x_2+ v = 12$$ $$3x_1+2x_2+w = 18$$

And solve the system.

If you were to minimize the profit function or maximize (as it's more understandable) you would use the simplex method. Read about it in wikipedia.

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The set of constraints generates a compact space, and the function you want to maximize is continuous and linear. By the extreme value theorem, the maximum will be on one of the extreme points. You can check all of them by hand and find the proposed answer.

If you don't want to check all the possibilities by hand, a quick drawing of the level sets will directly point to the right extreme point.

This technique however, is more of analytical nature (rather than algebraic).

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