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Worksheet: Linear Programming

In this worksheet, we will practice finding the optimal solution of a linear system that has an objective function and multiple constraints.

Q1:

Find the maximum value of the objective function 𝑝 = 2 π‘₯ + 6 𝑦 given the constraints π‘₯ β‰₯ 0 , 𝑦 β‰₯ 0 , π‘₯ + 𝑦 ≀ 6 , 3 π‘₯ + 𝑦 ≀ 9 , and π‘₯ + 2 𝑦 ≀ 8 .

Q2:

Find the maximum value of the objective function 𝑝 = 2 π‘₯ + 5 𝑦 given the constraints π‘₯ β‰₯ 0 , 𝑦 β‰₯ 0 , π‘₯ + 𝑦 ≀ 6 , π‘₯ + 2 𝑦 ≀ 8 , and 3 π‘₯ + 𝑦 ≀ 9 .

Q3:

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

  • A ( 2 , βˆ’ 2 )
  • B ( 2 , 2 )
  • C ( 0 , 0 )
  • D ( 1 , 6 )
  • E ( 2 , βˆ’ 6 )

Q4:

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

  • Aminimum: 0, maximum: 7
  • Bminimum: 1, maximum: 7
  • Cminimum: 1, maximum: 14
  • Dminimum: 0, maximum: 14

Q5:

Consider the following inequalities in the nonnegative variables π‘₯ 1 , π‘₯ 2 , and π‘₯ 3 : Find the maximum and minimum possible values of 𝑧 = π‘₯ βˆ’ 2 π‘₯ 1 2 subject to these constraints.

  • Aminimum: βˆ’ 7 2 , maximum: 7
  • Bminimum: βˆ’ 1 3 2 , maximum: 6
  • Cminimum: βˆ’ 7 , maximum: 6
  • Dminimum: βˆ’ 7 , maximum: 7

Q6:

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

  • Aminimum: 0, maximum: 2 7 2
  • Bminimum: 1, maximum: 2 7 2
  • Cminimum: 1, maximum: 7
  • Dminimum: 0, maximum: 7

Q7:

Consider the following inequalities in the nonnegative variables π‘₯ 1 , π‘₯ 2 , and π‘₯ 3 : Find the maximum and minimum possible values of 𝑧 = π‘₯ βˆ’ 2 π‘₯ βˆ’ 3 π‘₯ 1 2 3 subject to these constraints.

  • Aminimum: βˆ’ 2 0 , maximum: 7
  • Bminimum: βˆ’ 2 0 , maximum: 6
  • Cminimum: βˆ’ 2 1 , maximum: 6
  • Dminimum: βˆ’ 2 1 , maximum: 7

Q8:

Which of the following points is a solution to the system of inequalities π‘₯ β‰₯ 0 , 𝑦 β‰₯ 0 , 3 π‘₯ βˆ’ 4 𝑦 β‰₯ βˆ’ 8 , and π‘₯ + 𝑦 β‰₯ βˆ’ 9 ?

  • A ( βˆ’ 4 , 5 )
  • B ( βˆ’ 9 , 8 )
  • C ( βˆ’ 5 , βˆ’ 2 )
  • D ( 8 , 1 )
  • E ( βˆ’ 1 , 0 )

Q9:

Given the graph below and that π‘₯ β‰₯ 0 , 𝑦 β‰₯ 0 , π‘₯ + 𝑦 ≀ 7 , and 𝑦 β‰₯ 5 , determine at which point the function 𝑝 = 3 π‘₯ βˆ’ 𝑦 has its maximum using linear programming.

  • A 𝐢
  • B 𝐴
  • C 𝐷
  • D 𝐡

Q10:

Minimise 𝑧 = π‘₯ + π‘₯ 1 2 subject to the constraints π‘₯ + π‘₯ β‰₯ 2 1 2 , π‘₯ + 3 π‘₯ ≀ 2 0 1 2 , and π‘₯ + π‘₯ ≀ 1 8 1 2 .

  • A9
  • B7
  • C1
  • D2
  • E16

Q11:

Which of the following points belongs to the solution set of the two inequalities π‘₯ > βˆ’ 1 and 𝑦 > βˆ’ 4 ?

  • A ( βˆ’ 1 , 7 )
  • B ( βˆ’ 4 , βˆ’ 4 )
  • C ( βˆ’ 2 , 4 )
  • D ( 8 , 3 )
  • E ( 1 , βˆ’ 7 )

Q12:

If the point ( βˆ’ 1 3 βˆ’ π‘₯ , βˆ’ 1 1 βˆ’ π‘₯ ) , where π‘₯ ∈ 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 𝑧 = π‘₯ βˆ’ 2 π‘₯ + π‘₯ 1 2 3 subject to these constraints.

  • Aminimum: βˆ’ 1 1 2 , maximum: 5
  • Bminimum: βˆ’ 7 , maximum: 5
  • Cminimum: βˆ’ 7 2 , maximum: 7
  • Dminimum: βˆ’ 7 , maximum: 7
  • Eminimum: βˆ’ 1 3 2 , maximum: 5

Q14:

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

  • Athird
  • Bsecond
  • Cforth
  • Dfirst

Q15:

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

  • Option A: a base salary of $ 1 4 0 0 0 a year with a commission of 1 0 % of your sales
  • Option B: a base salary of $ 1 9 0 0 0 a year with a commission of 4 % of your sales

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

  • Amore than $ 5 5 0 0 0 0
  • Bmore than $ 3 5 7 1 4 . 2 9
  • Cmore than $ 2 3 5 7 1 4 . 2 9
  • Dmore than $ 8 3 3 3 3 . 3 3

Q16:

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

  • A 3 r d
  • B 1 s t
  • C 4 t h
  • D 2 n d

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