# I have an exam in a few hours. I need to understand the solution to the following question Find

I have an exam in a few hours. I need to understand the solution to the following question

Find the Maximal to the the following $2{x}_{1}+3{x}_{2}$ is the objective function.

The constraints are
$4{x}_{1}+3{x}_{2}\le 600;$
${x}_{1}+{x}_{2}\le 160;$
$3{x}_{1}+7{x}_{2}\le 840;$
${x}_{1},{x}_{2}\ge 0.$
Also the answer should be in the "Dual". If its possible please do it in the Algebraic method. If not I would just like the solution using the tableau method and how do you arrive to the solution.(PS: Any help would be great. )
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Jenna Farmer
First of all, to solve this with the simplex method (tableau method) the inequalities of the contraints should be equalities. So you have to use slack variables.

$4{x}_{1}+3{x}_{2}\le 600$
${x}_{1}+{x}_{2}\le 160$
$3{x}_{1}+7{x}_{2}\le 840$
${x}_{1},{x}_{2}\ge 0$
$⇓$
$4{x}_{1}+3{x}_{2}+{x}_{3}=600$
${x}_{1}+{x}_{2}+{x}_{4}=160$

$3{x}_{1}+7{x}_{2}+{x}_{5}=840$
${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}\ge 0$
Then can you continue by creating the tableau?

As regards the Algebraic method:

Draw the ${x}_{1}$-axis and the ${x}_{2}$-axis. Then from the first contraint you get:

. So you get the line as below, and since it is $\le 600$ the feasible region is below this line

Can you do this for the other contraints and find the solution? You should also keep in mind that ${x}_{1},{x}_{2}\ge 0$