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In this lesson, we will learn how to find the optimal solution of a linear system that has an objective function and multiple constraints.

Q1:

Find the maximum value of the objective function π = 2 π₯ + 6 π¦ given the constraints π₯ β₯ 0 , π¦ β₯ 0 , π₯ + π¦ β€ 6 , 3 π₯ + π¦ β€ 9 , and π₯ + 2 π¦ β€ 8 .

Q2:

Find the maximum value of the objective function π = 2 π₯ + 5 π¦ given the constraints π₯ β₯ 0 , π¦ β₯ 0 , π₯ + π¦ β€ 6 , π₯ + 2 π¦ β€ 8 , and 3 π₯ + π¦ β€ 9 .

Q3:

Use the graph to identify which of the following points is NOT a solution to the given set of inequalities.

Q4:

Consider the following inequalities in the nonnegative variables π₯ 1 , π₯ 2 , and π₯ 3 : Find the maximum and minimum possible values of π§ = 2 π₯ + π₯ 1 2 subject to these constraints.

Q5:

Consider the following inequalities in the nonnegative variables π₯ 1 , π₯ 2 , and π₯ 3 : Find the maximum and minimum possible values of π§ = π₯ β 2 π₯ 1 2 subject to these constraints.

Q6:

Consider the following inequalities in the nonnegative variables π₯ 1 , π₯ 2 , and π₯ 3 : Find the maximum and minimum possible values of π§ = π₯ + 2 π₯ 1 2 subject to these constraints.

Q7:

Consider the following inequalities in the nonnegative variables π₯ 1 , π₯ 2 , and π₯ 3 : Find the maximum and minimum possible values of π§ = π₯ β 2 π₯ β 3 π₯ 1 2 3 subject to these constraints.

Q8:

Which of the following points is a solution to the system of inequalities π₯ β₯ 0 , π¦ β₯ 0 , 3 π₯ β 4 π¦ β₯ β 8 , and π₯ + π¦ β₯ β 9 ?

Q9:

Given the graph below and that π₯ β₯ 0 , π¦ β₯ 0 , π₯ + π¦ β€ 7 , and π¦ β₯ 5 , determine at which point the function π = 3 π₯ β π¦ has its maximum using linear programming.

Q10:

Minimize π§ = π₯ + π₯ ο§ ο¨ subject to the constraints π₯ + π₯ β₯ 2 ο§ ο¨ , π₯ + 3 π₯ β€ 2 0 ο§ ο¨ , and π₯ + π₯ β€ 1 8 ο§ ο¨ .

Q11:

Which of the following points belongs to the solution set of the two inequalities π₯ > β 1 and π¦ > β 4 ?

Q12:

If the point ( β 1 3 β π₯ , β 1 1 β π₯ ) , where π₯ β β€ is located in the second quadrant, find π₯ .

Q13:

Consider the following inequalities in the nonnegative variables π₯ 1 , π₯ 2 , and π₯ 3 : Find the maximum and minimum possible values of π§ = π₯ β 2 π₯ + π₯ 1 2 3 subject to these constraints.

Q14:

The solution set of the inequalities π₯ > 0 and π¦ > 0 lies in which quadrant?

Q15:

When hired at a new job selling electronics, you are given two pay options:

How many dollarsβ worth of electronics would you need to sell for option A to produce a larger income?

Q16:

Fill in the blank: The quadrant representing the solution set of the inequalities π¦ > 0 and π₯ < 0 is the quadrant.

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