Solving Linear Inequalities for Optimization

I want the max of:

$100-(2x_1+3x_2+4x_3+5x_4+6x_5+7x_6)$

I am given 5 inequalities:

$x_1+x_4\le6$

$x_2+x_5\le8$

$x_3+x_6\le7$

$x_1+x_2+x_3\le9$

$x_4+x_5+x_6\le11$

and

$x_1+x_2+x_3+x_4+x_5+x_6\ge0$

It seems obvious that i want the most of $x_1$, and the least of $x_6$ within my constraints, but I am having trouble setting up the set of equations properly. I read about using an extra variable for each inequality to turn it into an equality, but then I think I'll have 12 or 13 equations to solve.

Is there some next step that I should take to solve this?

• Have a look at the Simplex Method. – Daryl Sep 26 '12 at 3:58
• What do you know at this point in your course? Daryl is right that the simplex method will solve this problem for you. Are you allowed to use that? – Mike Spivey Sep 26 '12 at 4:26
• All your variables are positive? – cactus314 Sep 26 '12 at 13:28
• The simplex method is what I needed – Jimmy Pitts Sep 29 '12 at 16:43