1. ## Minimizing

Hard Math Question?

Minimize Z=9x1 + 4x2 + 12x3

The 1,2, and 3 are superscript.

Subject to the constraints:

3x1+x2+4x3 larger or equal to 24
3x1 + x2 + 2x3 larger or equal to 18
x1 >= 0, x2 >=0, x3 larger or equal to 0

What is Z, x1, x2, and x3?

All numbers after letter are superscript. Thanks!

2. Are you sure you don't mean SUBscript?

Did you set up a tableau?

Can you find the intersections of the constraints?

You may wish to hunt around x1 = 0 and x2 = 12, I suppose.

3. $\displaystyle x_{1}, x_{2}, x_{3} \geq 0$

$\displaystyle 3x_{1} +x_{2} +4x_{3} \geq 24$
$\displaystyle 3x_{1} + x_{2} + 2x_{3} \geq 18$

Subtract both inequalities to get: $\displaystyle 2x_{3} \geq 6 \: \Rightarrow \: x_{3} \geq 3$

Since we're looking for the minimum values that'll satisfy both inequalities, we'll set $\displaystyle x_{3} = 3$ and find that: $\displaystyle {\color{red}3x_{1} + x_{2} \geq 12}$

We want to somehow use this inequality in your equation for z to simplify your equation. You can do that by modifying the equation a bit (we also substitute $\displaystyle x_{3} = 3$ as that'll minimize the value of z):

$\displaystyle z = 9x_{1} + {\color{blue}4x_{2}} + 12x_{3}$
$\displaystyle z = 9x_{1} + {\color{blue}3x_{2} + x_{2}} + 12(3)$
$\displaystyle z = 3({\color{red}3x_{1} + x_{2}}) + x_{2} + 36 \: \: \geq \: \: 3({\color{red}12}) + x_{2} + 36$
$\displaystyle z \geq 72 + x_{2}$

And since $\displaystyle x_{2} \geq 0$, you could probably figure out what value of $\displaystyle x_{2}$ to choose and solve for $\displaystyle x_{1}$

4. Hello, Greenbaumenom!

This is a Linear Programming problem.
. . Are you familiar with the techniques?
I'll solve it by "graphing".

Minimize: .$\displaystyle P \:=\:9x + 4y + 12z$
Subject to the constraints:. $\displaystyle \begin{array}{cccc} x &\geq& 0 & {\color{blue}[1]} \\ y &\geq& 0 & {\color{blue}[2]} \\ z & \geq & 0 & {\color{blue}[3]} \\ 3x+y+4z & \geq & 24 & {\color{blue}[4]} \\ 3x + y + 2z & \geq & 18 & {\color{blue}[5]} \end{array}$
Find $\displaystyle x, y, z \text{ and }P.$

[1], [2], [3] places us in the first octant.

[4] is a plane with intercepts: .$\displaystyle (8,0,0),\;(0,24,0),\;(0,0,6)$
Graph the plane and shade the space above the plane.

[5] is a plane with intercepts: .$\displaystyle (6,0,0),\;(0,18,0),\;(0,0,9)$
Graph the plane and shade the space above the plane.

The solution set is a polyhderon with vertices at:
. . $\displaystyle (6,0,0),\;(0,18,0),\;(0,0,9),\;(4,0,3),\;(0,12,3)$

Test these vertices in the $\displaystyle P$-function to find the minimum $\displaystyle P.$

5. Grrr.... Am I maximizing again when I should be minimizing?!