algebra 2 worksheet linear programming word problems answers

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Linear Programming Word Problems

Intro How-To Word Problems More Examples Four Variables

What are the constraints they "forget" to mention?

When you are doing a linear programming word problem, you are dealing with a (pretend) real-world situation. In this context, you will need to remember the understood (and thus usually omitted) constraints; namely, that you can't (generally) have negative amounts of inputs.

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This omitted fact requires that you remember to add non-negativity constraints for each of your graphing variables; namely, x  ≥ 0 and y  ≥ 0 .

  • A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day.   If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?

The question asks for the optimal number of calculators, so my variables will stand for numbers of calculators:

x : number of scientific calculators produced

y : number of graphing calculators produced

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Since they can't produce negative numbers of calculators, I have my first two constraints; namely, x  ≥ 0 and y  ≥ 0 . However, as it turns out, I can ignore these constraints in this particular exercise, because they've already given me minimums:

The exercise also gives maximums:

The minimum shipping requirement gives me another constraint:

x + y ≥ 200

For graphing purposes, I'll restate this constraint as:

y ≥ − x + 200

The profit relation will be my optimization equation:

P = −2 x + 5 y

So my entire system is:

P = 2 x + 5 y , subject to:

100 ≤ x ≤ 200

80 ≤ y ≤ 170

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The feasibility region graphs as:

The corner points are at (100, 170) , (200, 170) , (200, 80) , (120, 80) , and (100, 100) . When you test these points in the optimization equation, you should obtain the maximum value of profit P  = 650 at ( x ,  y ) = (100, 170) . Interpreting this within the context of the original word problem, the solution will be " 100 scientific calculators and 170 graphing calculators".

(Yes, you can also find the answer just using logic: If the one kind loses you money and the other makes you money, obviously you'll want to maximize production of what makes you money.)

  • You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?

The question ask for the number of cabinets I need to buy, so my variables will stand for that:

x : number of model X cabinets purchased y : number of model Y cabinets purchased

Naturally, x  ≥ 0 and y  ≥ 0 . I have to consider the costs and the floor space (that is, the footprint of each unit), while maximizing the storage volume, so costs and floor space will be my constraints, while volume will be my optimization equation.

cost: 10 x + 20 y ≤ 140 , or y ≤ −(½) x + 7

space: 6 x + 8 y ≤ 72 , or y ≤ −(¾) x + 9

volume: V = 8 x + 12 y

This system (along with the first two constraints) graphs as:

The corner points are at (0, 0) , (8, 3) , (0, 7) , and (12, 0) . When you plug them into the optimization equation, you should obtain a maximal volume of 100 cubic feet by buying eight of model X and three of model Y.

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algebra 2 worksheet linear programming word problems answers

Lesson LINEAR PROGRAMMING PROBLEMS AND SOLUTIONS 2

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Linear Programming

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Linear Programming: Word Problems and Applications

Several word problems and applications related to linear programming are presented along with their solutions and detailed explanations. Methods of solving inequalities with two variables , system of linear inequalities with two variables along with linear programming and optimization are used to solve word and application problems where functions such as return, profit, costs, etc., are to be optimized.

A store sells two types of toys, A and B. The store owner pays $8 and $14 for each one unit of toy A and B respectively. One unit of toys A yields a profit of $2 while a unit of toys B yields a profit of $3. The store owner estimates that no more than 2000 toys will be sold every month and he does not plan to invest more than $20,000 in inventory of these toys. How many units of each type of toys should be stocked in order to maximize his monthly total profit profit?

Solution to Example 1 Let x be the total number of toys A and y the number of toys B; x and y cannot be negative, hence x ? 0 and y ? 0 The store owner estimates that no more than 2000 toys will be sold every month x + y ? 2000 One unit of toys A yields a profit of $2 while a unit of toys B yields a profit of $3, hence the total profit P is given by P = 2 x + 3 y The store owner pays $8 and $14 for each one unit of toy A and B respectively and he does not plan to invest more than $20,000 in inventory of these toys 8 x + 14 y ? 20,000 What do we have to solve? Find x and y so that P = 2 x + 3 y is maximum under the conditions \[ \begin{cases} \ x \ge 0 \\ \ x \ge 0 \\ \ x + y \le 2000 \\ \ 8 x + 14 y \le 20,000 \\ \end{cases} \]

Vertices of the solution set: A at (0 , 0) B at (0 , 1429) C at (1333 , 667) D at (2000 , 0) Calculate the total profit P at each vertex P(A) = 2 (0) + 3 ()) = 0 P(B) = 2 (0) + 3 (1429) = 4287 P(C) = 2 (1333) + 3 (667) = 4667 P(D) = 2(2000) + 3(0) = 4000 The maximum profit is at vertex C with x = 1333 and y = 667. Hence the store owner has to have 1333 toys of type A and 667 toys of type B in order to maximize his profit.

A company produces two types of tables, T1 and T2. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish.It takes 4 hours to produce the parts of one unit of T2, 2.5 hour to assemble and 1.5 hours to polish. Per month, 7000 hours are available for producing the parts, 4000 hours for assembling the parts and 5500 hours for polishing the tables. The profit per unit of T1 is $90 and per unit of T2 is $110. How many of each type of tables should be produced in order to maximize the total monthly profit?

Solution to Example 2 Let x be the number of tables of type T1 and y the number of tables of type T2. Profit P(x , y) = 90 x + 110 y \[ \begin{cases} \ x \ge 0 \\ \ x \ge 0 \\ \ 2x + 4y \le 7000 \\ \ x + 2.5y \le 4000 \\ \ 2x + 1.5y \le 5500 \\ \end{cases} \]

A at (0,0) B at (0,1600) C at (1500,1000) D at (2300,600) E at (2750,0)

Evaluate profit P(x,y) at each vertex A at (0,0) : P(0 , 0) = 0 B at (0,1600) : P(0 , 1600) = 90 (0) + 110 (1600) = 176000 C at (1500,1000) : P(1500,1000) = 90 (1500) + 110 (1000) = 245000 D at (2300,600): P(2300,600) = 90 (2300) + 110 (600) = 273000 E at (2750,0) : P(2750,0) = 90 (2750) + 110 (0) = 247500

The maximum profit of $273000 is at vertex D. Hence the company needs to produce 2300 tables of type T1 and 600 tables of type T2 in order to maximize its profit.

A farmer plans to mix two types of food to make a mix of low cost feed for the animals in his farm. A bag of food A costs $10 and contains 40 units of proteins, 20 units of minerals and 10 units of vitamins. A bag of food B costs $12 and contains 30 units of proteins, 20 units of minerals and 30 units of vitamins. How many bags of food A and B should the consumed by the animals each day in order to meet the minimum daily requirements of 150 units of proteins, 90 units of minerals and 60 units of vitamins at a minimum cost?

Solution to Example 3 Let x be the number of bags of food A and y the number of bags of food B. Cost C(x,y) = 10 x + 12 y \[ \begin{cases} \ x \ge 0 \\ \ y \ge 0 \\ \ 40x + 30y \ge 150 \\ \ 20x + 20y \ge 90 \\ \ 10x + 30y \ge 60 \\ \end{cases} \]

Vertices: A at intersection of \( 10x + 30y = 60 \) and \( y = 0 \) (x-axis) coordinates of A: (6 , 0) B at intersection of \( 20x + 20y = 90 \) and \( 10x + 30y = 60 \) coordinates of B: (15/4 , 3/4) C at intersection of \( 40x + 30y = 150 \) and \( 20x + 20y = 90 \) coordinates of C : (3/2 , 3) D at at intersection of \( 40x + 30y = 150 \) and \( x = 0 \) (y-axis) coordinates of D: (0 , 5)

Evaluate the cost c(x,y) = 10 x + 12 y at each one of the vertices A(x,y), B(x,y), C(x,y) and D(x,y). At A(6 , 0) : c(6 , 0) = 10 (6) + 12 (0) = 60 At B(15/4 , 3/4) : c(15/4 , 3/4) = 10 (15/4) + 12 (3/4) = 46.5 At C(3/2 , 3) : c(3/2 , 3) = 10 (3/2) + 12 (3) = 51 At D(0 , 5) : c(0 , 5) = 10 (0) + 12 (5) = 60 The cost c(x , y) is minimum at the vertex B(15/4 , 3/4) where x = 15/4 = 3.75 and y = 3/4 = 0.75. Hence 3.75 bags of food A and 0.75 bags of food B are needed to satisfy the minimum daily requirements in terms of proteins, minerals and vitamins at the lowest possible cost.

John has $20,000 to invest in three funds F1, F2 and F3. Fund F1 is offers a return of 2% and has a low risk. Fund F2 offers a return of 4% and has a medium risk. Fund F3 offers a return of 5% but has a high risk. To be on the safe side, John invests no more than $3000 in F3 and at least twice as much as in F1 than in F2. Assuming that the rates hold till the end of the year, what amounts should he invest in each fund in order to maximize the year end return?

Solution to Example 4 Let x be the amount invested in F1, y the amount invested in F2 and z the amount invested in F1. x + y + z = 20,000 z = 20,000 - (x + y) Total return R of all three funds is given by R = 2% x + 4% y + 5% z = 0.02 x + 0.04 y + 0.05 (20,000 - (x + y)) Simplifies to R(x ,y) = 1000 - 0.03 x - 0.01 y : This is the return to maximize Constraints: x, y and z are amounts of money and they must satisfy x ? 0 y ? 0 z ? 0 Substitute z by 20,000 - (x + y) in the above inequality to obtain 20,000 - (x + y) ? 0 which may be written as x + y ? 20,000 John invests no more than $3000 in F3, hence z ? 3000 Substitute z by 20,000 - (x + y) in the above inequality to obtain 20,000 - (x + y) ? 3000 which may be written as x + y ? 17,000 Let us put all the inequalities together to obtain the following system \[ \begin{cases} \ x \ge 0 \\ \ y \ge 0 \\ \ (x + y) \ge 17,000 \\ \ (x + y) \le 20,000 \\ \ x \ge 2 y \\ \end{cases} \]

Vertices: A at intersection of \( x + y = 20000 \) and \( y = 0 \) , coordinates of A: (20000 , 0) B at intersection of \( x+y = 17000 \) and \( y=0 \) , coordinates of B: (17000 , 0) C at intersection of \( x+y = 17000 \) and \( x = 2y \) , coordinates of C : (11333 , 5667) D at at intersection of \( x = 2y \) and \( x + y = 20000 \) , coordinates of D: (13333 , 6667)

Evaluate the return R(x,y) = 1000 - 0.03 x - 0.01 y at each one of the vertices A(x,y), B(x,y), C(x,y) and D(x,y). At A(20000 , 0) : R(20000 , 0) = 1000 - 0.03 (20000) - 0.01 (0) = 400 At B(17000 , 0) : R(17000 , 0) = 1000 - 0.03 (17000) - 0.01 (0) = 490 At C(11333 , 5667) : R(11333 , 5667) = 1000 - 0.03 (11333) - 0.01 (5667) = 603 At D(13333 , 6667) : R(13333 , 6667) = 1000 - 0.03 (13333) - 0.01 (6667) = 533 The return R is maximum at the vertex At C(11333 , 5667) where x = 11333 and y = 5667 and z = 20,000 - (x+y) = 3000 For maximum return, John has to invest $11333 in fund F1, $5667 in fund F2 and $3000 in fund F3.

Each month a store owner can spend at most $100,000 on PC's and laptops. A PC costs the store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while laptop is sold for a profit of $700. The sore owner estimates that at least 15 PC's but no more than 80 are sold each month. He also estimates that the number of laptops sold is at most half the PC's. How many PC's and how many laptops should be sold in order to maximize the profit?

Solution to Example 5 Let x and y be the numbers of PC's and laptops respectively that should be sold. Profit = 400 x + 700 y to maximize Constraints 15 ? x ? 80 "least 15 PC's but no more than 80 are sold each month" y ? (1/2) x 1000 x + 1500 y ? 100,000 "store owner can spend at most $100,000 on PC's and laptops" \[ \begin{cases} \ 15 \le x \le 80 \\ \ y \ge 0 \\ \ y \le (1/2) x \\ \ 1000 x + 1500 y \le 100,000 \\ \end{cases} \]

Vertices: A at intersection of \( x = 15 \) and \( y = 0 \) (x-axis) coordinates of A: (15 , 0) B at intersection of \( x = 15 \) and \( y = (1/2) x \) coordinates of B: (15 , 7.5) C at intersection of \( y = (1/2) x \) and \( 1000 x + 1500 y = 100000 \) coordinates of C : (57.14 , 28.57) D at at intersection of \( 1000 x + 1500 y = 100000 \) and \( x = 80 \) (y-axis) coordinates of D: (80 , 13.3)

Evaluate the profit at each vertex A(15 , 0), P = 400 × 15 + 700 × 0 = 6000 B(15 , 7.5) , P = 400 × 15 + 700 × 7.5 = 11250 C(57.14 , 28.57) , P = 400 × 57.14 + 700 × 28.57 = 42855 D (80 , 13.3) , P = = 400 × 80 + 700 × 13.3 = 41310

The profit is maximum for x = 57.14 and y = 28.57 but these cannot be accepted as solutions because x and y are numbers of PC's and laptops and must be integers. We need to select the nearest integers to x = 57.14 and y = 28.57 that are satisfy all constraints and give a maximum profit x = 57 and y = 29 do not satisfy all constraints x = 57 and y = 28 satisfy all constraints Profit = 400 × 57 + 700 × 28 = 42400 , which is maximum.

Infinite Algebra 2

Test and worksheet generator for algebra 2.

Infinite Algebra 2 covers all typical Algebra 2 material, beginning with a few major Algebra 1 concepts and going through trigonometry. There are over 125 topics in all, from multi-step equations to trigonometric identities. Suitable for any class with advanced algebra content. Designed for all levels of learners, from remedial to advanced.

Included in version 2.70.06 released 1/5/2023:

  • Fixed: Choices could appear incorrectly

Included in version 2.70.05 released 12/21/2022:

  • Fixed: Continuous Exponential Growth and Decay Word Problems - Could generate an incorrect answer

Included in version 2.70 released 7/8/2022:

  • New: Add preferences for Metric/Imperial units
  • New: Add preference to “Prefer x“ as the variable letter
  • New topic: Discrete Relations
  • New topic: Continuous Relations
  • New topic: Evaluating and Graphing Functions
  • New topic: Direct and Inverse Variation
  • New topic: Continuous Exponential Growth and Decay
  • Improved: UI, security, and stability with updated libraries
  • Fixed: Properties of Exponents – Choices could be equivalent
  • Fixed: Properties of Logarithms – Choices could be equivalent
  • Fixed: Solving Exponential Equations With Logarithms – Choices could be equivalent
  • Fixed: Geometric Series – Choices could be equivalent

Included in version 2.62 released 2/8/2022:

  • New: Print questions from Presentation View
  • New Topic: Discrete Exponential Growth and Decay Word Problems
  • Improved: UI, security, and stability with updated framework and libraries
  • Improved: [Mac] Dark mode
  • Fixed: [Windows] - Detached topic list does not close when application is closed
  • Fixed: Probability Mutually Exclusive Word Problems - mutually exclusive case given when inappropriate

Included in version 2.61.03 released 9/10/2021:

  • Improved: Minor UI improvements
  • Fixed: Law of Cosines - Crash when opening topic
  • Fixed: [Windows] Unable to change page layout to landscape
  • Fixed: [Windows] Error 'Unable to load platform plugin'

Included in version 2.61 released 8/27/2021:

  • Fixed: Preferences page for Kuta Works
  • Improved: Security and stability with updated networking libraries

Included in version 2.60 released 8/19/2021:

  • Improved: Site licenses check expiration date more frequently
  • Fixed: Combinatoric Word Problems - Values could overflow
  • Fixed: Properties of Exponents - Answer might not be fully simplified
  • Fixed: Properties of Logs - Possible to have duplicate choices
  • Fixed: Radical Equations - Option to mix radicals and rational exponents had no effect

Included in version 2.52 released 6/14/2019:

  • New: Scramble questions by directions
  • New: Scramble all questions in assignment
  • New: Consolidate question sets with identical options
  • New: Regroup question sets by directions
  • New: Kuta Works - Create a two semester course
  • New: Kuta Works - Clone assignments from a previous course into a new one
  • Fixed: Presentation View - Arrow keys could change zoom at start/end of assignment
  • Fixed: Tight Layout - Directions could be cramped for no reason
  • Fixed: Inverse Trig - Angles limited for tan( x )
  • Fixed: Inverse Trig - Errors in UI logic

Included in version 2.50 released 4/12/2019:

  • New: Kuta Works - Option to hide answers and results from students until after due date
  • New: Kuta Works - Option to control how long choices are hidden
  • Improved: Options windows appear at a better initial size
  • Improved: Window controls shown at their native size and spacing
  • Improved: Network proxy configuration window
  • Improved: License activation process
  • Fixed: Presentation View - Answers could be cut off
  • Fixed: Some window controls cut off on Windows 10 with display scaling
  • Fixed: Spelling errors
  • Fixed: Kuta Works - Course list could display more than just your active courses
  • Fixed: Graphing Rational Functions - Open holes in graph could appear as filled holes

Included in version 2.42 released 12/11/2018:

  • Fixed: Certain symbols incorrect if saved on Mac and loaded on Windows or vice versa
  • Fixed: General Sequences - Recursive formula sometimes wrong
  • Fixed: Properties of Logarithms - Avoid questions with reducible roots

Included in version 2.41 released 9/26/2018:

  • Fixed: Archived courses in Kuta Works are now hidden
  • Fixed: Generating or regenerating questions could cause a crash

Included in version 2.40 released 8/8/2018:

  • New: Integrated with Kuta Works . Now post assignments online.

Included in version 2.25 released 4/27/2018:

  • Improved: Generate questions more quickly
  • Improved: Measurement arrows reach correct endpoints
  • Improved: Absolute Value Equations - Better decimal numbers
  • Improved: Absolute Value Inequalities - Better decimal numbers
  • Improved: Compound Inequalities - Better decimal numbers
  • Improved: Multi-Step Equations - Better decimal numbers and more predictable special cases
  • Improved: Multi-Step Inequalities - Better decimal numbers
  • Improved: Order of Operations - Better decimal numbers
  • Fixed: Trig units for piecewise functions and ordered pairs
  • Fixed: Graphing Rational Functions - Prevent equivalent choices
  • Fixed: Law of Sines - Program could freeze
  • Fixed: Transformations - Prevent equivalent choices

Included in version 2.18 released 7/31/2017:

  • Fixed: Open circles on graphs were filled in when displayed as answer in red
  • Fixed: Typo in sample custom question
  • Fixed: Transformations - Program could freeze
  • Fixed: Systems of Quadratic Equations - Program could freeze

Included in version 2.17 released 4/27/2017:

  • Improved: [Windows] "Export to clipboard as bitmap" renders image with better quality
  • Fixed: Probability with Combinations and Permutations - Could ask about impossible lottery ball

Included in version 2.16.20 released 11/16/2016:

  • Fixed: Solving Quadratic Equations by Taking Square Roots - Option to "Allow fractions" not working as expected

Included in version 2.16 released 9/16/2016:

  • Fixed: Numbers with many significant digits could round incorrectly
  • Fixed: Customized question containing an error could not be modified
  • Fixed: Customized question containing an error could cause a crash
  • Fixed: Feedback tool detects Windows 10 properly
  • Fixed: Mouse cursor when drag-dropping a question
  • Fixed: Keyboard shortcut for changing order of question sets in assignment

Included in version 2.15 released 7/14/2016:

  • Improved: Reorder question sets by dragging and dropping
  • Improved: Move individual questions within a set by dragging and dropping
  • Improved: Presentation View - Commands now on a movable, dockable toolbar
  • Improved: Added option for large toolbar icons
  • Improved: New icons
  • Fixed: Certain options could result in double parenthesis or parenthesis around a negative sign
  • Fixed: Arrow over vector variable name not red when appropriate
  • Fixed: Solving Systems of Equations by Graphing - Answer could be displayed twice
  • Fixed: Equations of Parabolas - Certain givens indicate wrong sign

Included in version 2.11 released 5/13/2016:

  • Improved: [Windows] "Export to clipboard as bitmap" pastes into Microsoft Word at appropriate DPI
  • Fixed: Keyboard shortcut for adding space to question set

Included in version 2.10 released 5/11/2016:

  • New: Preference for including "None of the above" as a choice
  • New: One column layout
  • New: Added support for Infinite Precalculus
  • New: Easily add piecewise functions of graphs in custom questions: Example: piecewise([2x-3] if [x<5], [x-1] if [x >= 5])
  • New: Easily add functions with restricted domains to graphs in custom questions Example: function(x/2, x<0)
  • Improved: More efficient layout of choices
  • Improved: [Mac] Added option for legal paper in page setup
  • Fixed: Spelling and grammatical errors
  • Fixed: Graphing Quadratic Functions - Prevent equivalent choices
  • Fixed: Binomial Theorem - Certain options didn't allow for many questions
  • Fixed: Right Triangle Trig - Couldn't select just one ratio except sine
  • Fixed: Right Triangle Trig - Answer sometimes given in problem
  • Fixed: Law of Cosines - Certain SAS cases could freeze program
  • Note: Older versions will not be able to open assignments saved with this version

Included in version 2.06 released 8/3/2015:

  • Fixed: Crash when saving if previously used folder no longer exists

Included in version 2.05 released 7/20/2015:

  • New Topic: Sample Spaces and The Fundamental Counting Principle
  • New Topic: Independent and Dependent Events
  • New Topic: Independent and Dependent Events Word Problems
  • New Topic: Mutually Exclusive Events
  • New Topic: Mutually Exclusive Events Word Problems
  • Fixed: Classifying Conics - Checkboxes for conic types had no effect
  • Fixed: Presentation View - Displayed wrong directions in 3-up mode
  • Fixed: Presentation View - Lines too long after creating new problems
  • Fixed: [Windows] Uncommon error when using the network

Included in version 2.04.40 released 3/31/2015:

  • Fixed: Activation data unrecognized under special circumstances
  • Fixed: [Windows] Help files broken by previous release
  • Fixed: Order of Operations - Radio buttons reversed

Included in version 2.04.20 released 3/23/2015:

  • Improved: [Windows] "Export to clipboard as bitmap" uses a transparent background

Included in version 2.04 released 3/17/2015:

  • New Topic: Permutations
  • New Topic: Combinations
  • New Topic: Permutations vs Combinations
  • New Topic: Probability with Permutations and Combinations
  • New Topic: Converting Degrees and Degrees-Minutes-Seconds
  • [factorial(n)]
  • [perm(n, r)]
  • [comb(n, r)]
  • Improved: Better support for proxies
  • Improved: Support for HiDPI Mac Retina displays
  • Improved: Print options window - Some settings are now persistent
  • Improved: Presentation View - Use arrow keys to navigate between questions
  • Improved: Question set updates free response/multiple-choice when all questions have changed
  • Improved: [Mac] Activation also checks /Library/Preferences/ks-config.txt for serial numbers
  • Fixed: [Windows] Crash when printing with certain printers
  • Fixed: Crash when loading an assignment with over 80 custom questions
  • Fixed: Custom question answer was never choice A
  • Fixed: Paper size set before printing
  • Fixed: Cramer's Rule - Fixed directions
  • Fixed: Systems of Three Equations - Restored the option to "only solve for x"
  • Fixed: Center of circle wasn't appearing

Included in version 2.03 released 9/19/2014:

  • Fixed: Preference for question layout mode not correctly loaded
  • Fixed: Issues with automatic spacing and highlighting questions when using tight layout
  • Fixed: In Presentation View, graphing problems display choices more compactly
  • Fixed: Printing blank pages on HP LaserJet 1018, 1020, and 1022 among others
  • Fixed: Logs and Exponents as Inverses - problem with slider

Included in version 2.02 released 8/5/2014:

  • New: Software uses system proxy when necessary
  • Improved: Activation window has taskbar entry
  • Improved: Remembers the last directory used when saving and opening files
  • Fixed: [Windows only] Possible crash when loading an assignment with a lot of custom questions
  • Fixed: [Mac only] Application stops responding to hover events
  • Fixed: Activation from before v2.0 sometimes not recognized
  • Fixed: Export to clipboard sometimes cuts off choices B and D
  • Fixed: Help contents could become unusable
  • Fixed: Word wrap when modifying a question
  • Fixed: Memory leak

Included in version 2.01 released 7/28/2014:

  • Fixed: Could crash when opening an assignment from a different program

Included in version 2.00 released 7/23/2014:

  • New: Print to PDF
  • New: Filter for topic list
  • New: Zoom while viewing on screen
  • New: "More like these" button in presentation mode
  • New: Show where punch holes will go on the page
  • New: Print only an answer sheet
  • New: When printing, page range is enabled
  • New: Page elements are outlined when hovered
  • New: Double-click on directions to change them
  • New: When merging assignments, options to put similar topics together or put them end-to-end
  • New: Highlight and go to the questions in a question set (menu command or shift-click)
  • New: Site licenses can be activated per-machine if run by the administrator
  • Ctrl-Click on a question
  • Shift-Click on a question
  • Ctrl-Shift-Click on a question
  • Shift-Click in a empty area of a page
  • Ctrl-Shift-Click in a empty area of a page
  • Shift-Click on a question set
  • Ctrl-Shift-Click on a question set
  • Improved: Assignment files are drastically smaller
  • Improved: Sidebar is dockable on either side, can be floated, and position reset
  • Improved: Auto-spacing doesn't leave dead space at bottom of each page
  • Improved: Preview when editing directions
  • Improved: Keyboard shortcuts work no matter where the focus is
  • Improved: Scale number of questions window more user-friendly
  • Fixed: Minor bugs in seven topics

Included in version 1.56 released 8/14/2013:

  • Fixed: Graphing Exponential Functions - Correct choice used lighter shade of blue
  • Fixed: Radical Equations could be missing a solution
  • Fixed: Graphs of Polynomial Functions - Inflection points sometimes listed as local extrema
  • Fixed: Properties of Exponents - Occasionally answer wasn't fully simplified
  • Fixed: Dividing Polynomials - Occasionally polynomial was evenly divisible when it shouldn't have been
  • Fixed: Graphing Rational Functions - Minor UI issue
  • Fixed: Geometric Sequences - Given information sometimes allowed for two possible common ratios
  • Fixed: Dividing Polynomials could freeze
  • Fixed: Systems of Equations by Elimination - "No solution" occasionally should have been "Infinite solutions"

Included in version 1.55 released 12/11/2012:

  • New: Graphs can be added to custom questions
  • Improved: Graphing and Graph Paper utility more powerful and easier to use
  • Improved: Support for loading files from Infinite Calculus
  • Improved: Faster save/load
  • Fixed: Answer for Factoring Quadratic Expressions sometimes incorrect
  • Fixed: Choices for Evaluating Functions with a variable operand could contain wrong variable
  • Fixed: Choices for Function Operations with a variable operand could contain wrong variable
  • Fixed: Function Operations, functions when composed with self could provide the wrong answer
  • Fixed: Custom questions with an illegal expression could freeze the program
  • Fixed: Certain families of functions graphed incorrectly

Included in version 1.53 released 9/11/2012:

  • New: Preference for notation for greatest integer function
  • New: Added maximize/restore button to Presentation View
  • New: Graphing Absolute Value Equations has options for including coefficients
  • Improved: Help files
  • Improved: Scroll bars
  • Improved: User interface
  • Improved: When choices make a question too tall for a page, some choices are removed
  • Improved: Algebraic simplification routines are now more efficient
  • Improved: Better graphs for: Graphing Absolute Value Equations, Graphing Linear Equations, Graphing Exponential Functions, Graphing Quadratic Functions, Graphing Linear Inequalities, Graphing Systems of Linear Inequalities
  • Improved: Better number lines for: Multi-Step Inequalities, Compound Inequalities, Absolute Value Inequalities
  • Fixed: Wrapping to full-page and half-page could be too wide
  • Fixed: Able to graph certain families of functions
  • Fixed: Issue involving loading & regenerating an assignment and answers being hidden
  • Fixed: Program could ask about changing directions too much
  • Fixed: Graphs for Compound Inequalities could omit interval
  • Fixed: Double & Half Angle Identities: Button to show more examples did not work
  • Fixed: Writing Linear Equations: Answer can't be line given in question

Included in version 1.52 released 5/29/2012:

  • Improved: More professional radical signs
  • Improved: Graphs of discontinuous functions have breaks and open / closed holes
  • Improved: Even less likely to crash when generating questions
  • Fixed: Word wrapping could skip blank lines, not wrap where appropriate
  • Fixed: Wrapping to full page in a custom question was too wide
  • Fixed: Products of powers of e (like 2 e &sup3;) no longer display with a multiplication dot
  • Fixed: Most diagrams appeared incorrectly in print preview
  • Fixed: Random error message on save or load
  • Fixed: Graphing Rational Functions: Open holes in graph could appear as filled holes

Included in version 1.51.02 released on 4/9/2012:

  • Fixed: Minimum Windows version was set incorrectly

Included in version 1.51 released 4/5/2012:

  • New: Added greatest integer function / floor function to custom questions    Example: Find [int(2x)] when [x=3/5]
  • New: Added piecewise functions to custom questions    Example: [eval(f,x)] = [piecewise [x] if [x < 0], [xx] if [x >= 0]]
  • Improved: Faster! Optimized rendering of questions to screen
  • Improved: Faster! Optimized graphing routines
  • Improved: Improved graphing capabilities
  • Improved: Diagrams drawn more smoothly on screen
  • Fixed: Rewriting Logarithms - Numeric problems sometimes had bad values
  • Fixed: Evaluating Logarithms - Problems sometimes had bad values
  • Fixed: Graphs could omit holes
  • Fixed: Graphs of constant functions or those involving e and π could be incorrect
  • Fixed: Minor indentation issue in custom questions
  • Fixed: Horizontal asymptotes could be drawn beyond a graph's area

Included in version 1.50 released 3/15/2012:

  • New: Presentation View window is resizeable
  • New: Presentation View has option to automatically hide the answers when a new question is displayed
  • New: High-level filter to prevent questions from containing an illegal expression
  • Improved: Function notation can now be edited
  • Improved: Faster! Optimized the simplification of mathematical expressions
  • Improved: Faster! Improved undo/redo algorithm
  • Improved: Smaller executable size
  • Improved: "Current Question Sets" list easier to use
  • Improved: Options windows resize more smoothly
  • Fixed: Equation of Parabolas could freeze
  • Fixed: Geometric Series questions could list "Illegal Expression" as a choice
  • Fixed: Graphing Exponential Functions could freeze
  • Fixed: Systems of Equations Word Problems could freeze
  • Fixed: Punctuation in some word problems
  • Fixed: Certain expressions in a custom question would cause the software to crash
  • Fixed: Expressions like root × term would not print a multiplication dot
  • Note: Beginning with this version, Windows XP SP3 is the minimum required version of Windows

Included in version 1.45 released 4/12/2011:

  • New: Assignments from this program can be opened and modified by our other programs          (Assignments saved with v1.45 or greater can be opened by other programs v1.45 or greater)
  • New: Student data fields (name, date, period) can be renamed
  • New: Additional student data field available
  • New: Add & Continue is available when modifying an existing question set
  • New: Link to this details page when a software update is available
  • New: Product serial numbers can be placed in config.txt to facilitate enterprise installations
  • Improved: Better backwards- and future-compatibility
  • Fixed: Assignments with quotes in the title don't prevent Save As
  • Fixed: Quotients could be sometimes be simplified incorrectly
  • Fixed: Choices for dilations sometimes inappropriate
  • Fixed: Dilations rounding x- and y-coordinates
  • Fixed: Function Operations could have "Illegal Expression" as an answer or as a choice
  • Fixed: Multi-Step Equations: Program could freeze
  • Fixed: Dividing Radical Expressions: Program could freeze
  • Fixed: Add & Continue respects change in problem type
  • Fixed: Powers of i are correctly displayed

Included in version 1.42 released on 6/28/2010:

  • New: Automatically checks for updates
  • New: List topics by index order or suggested order

Included in version 1.40 released 2/8/2010:

  • New Preference: 'Spacious' or 'tight' layout of questions on page
  • Improved: Question sets with one question are more intelligently spaced
  • Improved: Graphs with small ranges now look better
  • Improved: Options screen now resizeable
  • Fixed: Directions occasionally orphaned at bottom of page
  • Fixed: Correct numbers changed because zeros at the end of a number were deleted (50 --> 5)

Included in version 1.08.20 released 5/22/2009:

  • Changed: Minor change to license agreement so that renewals extend the termination date instead of beginning a new term.

Included in version 1.08 released 4/10/2009:

  • New: Possible to change directions for a set without regenerating questions
  • Improved: Changing directions cascades changes to appropriate neighbors
  • Correction: Answers to Properties of Exponents could have wrong sign
  • Correction: Inequalities were sometimes solved incorrectly
  • Fixed: Software sometimes failed to renumber questions in assignment

Included in version 1.07 released 1/6/2009:

  • New: Trig. equations that don't require factoring
  • Correction: Function operations / linear combinations answers sometimes wrong
  • Fixed: Software crashes when starting if closed when side panel is minimized

Included in version 1.06 released 12/4/2008:

  • Improved: Isolate up to four questions on screen at once
  • Improved: Support for computers with multiple processors
  • Improved: User interface: Toolbar + Preferences
  • Fixed: Obscure error involving selecting a question and scaling the assignment

Included in version 1.03 released 10/1/2008:

  • New: Export individual questions as bitmaps
  • Improved: Custom questions wrap to whole/half page
  • Improved: Insert special symbols and math in custom questions
  • Improved: Insert 'blanks' into math text
  • Improved: Line thickness preference also now controls darkness of grid lines in graphs
  • Correction: Writing logs in terms of others sometimes created duplicate questions
  • Fixed: Internet calls would sometimes report wrong error

Included in version 1.02 released 9/18/2008:

  • Correction: Question type was not saving all of its options
  • Correction: Equations of Hyperbolas description was sometimes wrong
  • Correction: Geometric Series directions was sometimes wrong
  • Correction: Binomial Theorem type could cause the program to freeze
  • Correction: Quadratic/Linear systems sometimes had the wrong answer
  • Correction: Rational Exponents answers now correct
  • Fixed: Internet calls could freeze program

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Linear Programming Word Problems 2

Linear Programming Word Problems 2 - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Section solving linear programming problems, Pre ap algebra 2 lesson 2 6 linear programming problems, Linear programming some worked examples and exercises, Answer key linear programming practice problems work, Chapter 4 linear programming the simplex method, Linear programming, Linear programming word problems, Work linear programming.

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1. Section 2.1 Solving Linear Programming Problems

2. pre-ap algebra 2 lesson 2-6 linear programming problems, 3. linear programming : some worked examples and exercises ..., 4. answer key linear programming practice problems worksheet, 5. chapter 4: linear programming the simplex method, 6. 0.1 linear programming, 7. linear programming word problems, 8. worksheet 3.2 linear programming.

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Linear Equation Word Problems Worksheet

Students will practice solving linear equation word problems.

Download this web page as a pdf with answer key

Example Questions

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This is a 4 part worksheet:

  • Part I Model Problems
  • Part II Practice
  • Part III Challenge Problems
  • Part IV Answer Key
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Worksheet - Algebra - Linear Functions - Standard Form Word Problems (Day 2)

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Worksheet for Algebra 1 – Linear Functions - Standard Form Word Problems (Day 2)

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Linear Programming Word Problems 2

Displaying all worksheets related to - Linear Programming Word Problems 2 .

Worksheets are Section solving linear programming problems, Pre ap algebra 2 lesson 2 6 linear programming problems, Linear programming some worked examples and exercises, Answer key linear programming practice problems work, Chapter 4 linear programming the simplex method, Linear programming, Linear programming word problems, Work linear programming.

Click on pop-out icon or print icon to worksheet to print or download.

1. Section 2.1 Solving Linear Programming Problems

2. pre-ap algebra 2 lesson 2-6 linear programming problems, 3. linear programming : some worked examples and exercises ..., 4. answer key linear programming practice problems worksheet, 5. chapter 4: linear programming the simplex method, 6. 0.1 linear programming, 7. linear programming word problems, 8. worksheet 3.2 linear programming.

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  3. Linear Programming Worksheet Honors Algebra 2 Answers

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  1. ALGEBRA

  2. College Algebra Lesson #15 _ Linear Equations _ Graphical Solution @ProfDavidJDelosReyes

  3. Linear Equations in Two Variables

  4. Linear Programming Optimization (2 Word Problems)

  5. Linear Equations Worksheet

  6. Linear Algebra 101: Basic Example Solutions

COMMENTS

  1. PDF Linear Programming WS II

    Honors Algebra II Linear Programming Word Problems Worksheet II. You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files.

  2. PDF Infinite Algebra 2

    Worksheet by Kuta Software LLC-6-Answers to Target 1.3 - Section 2 Skill WS - Linear Programming 1) (8, 3) is the solution, which means 8 Cabinet X's and 3 Cabinet Y's for a maximum volume of 100ft3 x y 12345678910111213141516 2 4 6 8 10 12 14 16 2) (40, 50) is the solution, which means that 40 hamburgers and 50 hot dogs should be sold

  3. PDF Section 2.1

    To solve a linear programming problem, we first need to know the Fundamental Theorem of Linear Programming: Given that an optimal solution to a linear programming problem exists, it must occur at a vertex of the feasible set.

  4. PDF LINEAR PROGRAMMING -- ALGEBRA 2

    y = Constraints: Objective Function: Vertices: of Feasible Region: Ordered Pair of Optimal Solution: Maximum Profit: 3: Get the CARrect Answer The Gala Events Center has a rectangular parking lot measuring 20 m by 50 m. Only 60% of the lot is usable space. A car requires 6 square meters of space and a bus requires 30 square meters of space.

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    Recommendations. Skill plans. IXL plans. Virginia state standards. Textbooks. Test prep. Awards. Improve your math knowledge with free questions in "Linear programming" and thousands of other math skills.

  6. Linear-programming word problems

    When you are doing a linear programming word problem, you are dealing with a (pretend) real-world situation. In this context, you will need to remember the understood (and thus usually omitted) constraints; namely, that you can't (generally) have negative amounts of inputs. MathHelp.com

  7. Free Printable Math Worksheets for Algebra 2

    Vertex form. Graphing quadratic inequalities. Factoring quadratic expressions. Solving quadratic equations w/ square roots. Solving quadratic equations by factoring. Completing the square. Solving equations by completing the square. Solving equations with the quadratic formula. The discriminant.

  8. Lesson LINEAR PROGRAMMING PROBLEMS AND SOLUTIONS 2

    Lessons In depth This Lesson (LINEAR PROGRAMMING PROBLEMS AND SOLUTIONS 2) was created by by Theo (13134) : View Source, Show About Theo: this lesson includes problems pertaining to linear programming and their solutions. PROBLEM NUMBER 5 An elementary school wants to send children on a field trip to a museum.

  9. Linear Programming (examples, solutions, videos, worksheets, activities)

    Examples, videos, worksheets, solutions, and activities to help Algebra 2 students learn about linear programming. What is Linear Programming? The purpose of linear programming is to optimize some objective function given a set of constraints on the values of x and y. These constraints are usually provided as a system of inequalities.

  10. Linear Programming Practice Problems and Examples

    This worksheet accompanies an instructional YouTube video to practice setting up and solving problems through linear programming. It comes towards the end of the chapter on Systems of Equations in our flipped class. ... Algebra, Word Problems, Algebra 2. Resource Type. Worksheets, Homework. Standards. ... The worksheet has two problems which ...

  11. Advanced Algebra 2

    linear programming word problems . Linear programming practice test (unit 3b) Matrices (unit 3c test) using determinants to solve for variables. adding and subtracting matrices . using Cramer's rule to use matrices to solve systems. finding inverses of matrices with and without calculators. finding slope of lines and area of triangles using ...

  12. PDF Linear Programming Word Problems

    a. Let x = y = Since you cannot have negative cars, write two inequalities to represent the situation. Since the plant has a capacity of 1200 cars, write an inequality to represent the situation. Since the dealer orders up to 600 Topaz and 800 Mustangs, write two inequalities to represent the situation. Write an equation for the profit.

  13. PDF Infinite Algebra 2

    Worksheet by Kuta Software LLC-3-Section 2: Complete each linear programming problem. 9) A farmer has 7 acres to plant in wheat and rye. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the

  14. Algebra 2 Worksheets (pdf) with answer keys

    Complex Numbers Simplify Imaginary Numbers Adding and Subtracting Complex Numbers Multiplying Complex Numbers Dividing Complex Numbers Dividing Complex Number (advanced) End of Unit, Review Sheet Exponential Growth (no answer key on this one, sorry) Compound Interest Worksheet #1 (no logs) Compound Interest Worksheet (logarithms required)

  15. Algebra 2

    1) Lecture Notes w/Examples 2) Worksheet to practice (can be used as Homework or Quiz) 3) KEY for Worksheet with full solutions, step-by-step What does Lesson 167 cover? In this lesson, we will learn How to Solve Real-life Word Problems that Involve Linear Programming. We will either maximize or minimize a quantity using the given information.

  16. Linear Programming: Word Problems and Applications

    Methods of solving inequalities with two variables, system of linear inequalities with two variables along with linear programming and optimization are used to solve word and application problems where functions such as return, profit, costs, etc., are to be optimized. Example 1 A store sells two types of toys, A and B.

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    Test and worksheet generator for Algebra 2. Create customized worksheets in a matter of minutes. Try for free. ... Continuous Exponential Growth and Decay Word Problems - Could generate an incorrect answer ... Writing Linear Equations: Answer can't be line given in question ; Included in version 1.52 released 5/29/2012:

  18. Linear Programming Worksheets

    Linear Programming Word Problems Worksheet II 1) You need to buy some filing cabinets. ... Section 2 Skill WS - Linear Programming. Answers to Target 1.3 - Section 2 Skill WS - Linear Programming 1) (8, 3) is the solution, which means 8 Cabinet X's and 3 Cabinet Y's for a maximum volume of 100ft3 x y 12345678910111213141516 2 4 6 8 10 12 14 16 ...

  19. Linear Programming Word Problems 2 Worksheets

    Linear Programming Word Problems 2 - Displaying top 8 worksheets found for this concept.. Some of the worksheets for this concept are Section solving linear programming problems, Pre ap algebra 2 lesson 2 6 linear programming problems, Linear programming some worked examples and exercises, Answer key linear programming practice problems work, Chapter 4 linear programming the simplex method ...

  20. Linear Equation Word Problems Worksheet (pdf) and Answer Key. 31

    Free worksheet (pdf) and answer key on the solving word problems based on linear equations and real world linear models. Scaffolded questions that start relatively easy and end with some real challenges. Plus model problems explained step by step

  21. Linear Programming Word Problems 2 Worksheets

    Showing 8 worksheets for Linear Programming Word Problems 2. Worksheets are Section solving linear programming problems, Pre ap algebra 2 lesson 2 6 l...

  22. Worksheet

    Notes, worksheets, and answer keys for Algebra 1 - Linear Functions - Standard Form & Word Problems. For use in an Interactive Math Notebook so all students can have the same Graphic Organizer with the same notes to create equity. The notes, worksheets, and answer keys can be posted online. Stud

  23. Linear Programming Word Problems 2

    Displaying all worksheets related to - Linear Programming Word Problems 2. Worksheets are Section solving linear programming problems, Pre ap algebra 2 lesson 2 6 linear programming problems, Linear programming some worked examples and exercises, Answer key linear programming practice problems work, Chapter 4 linear programming the simplex method, Linear programming, Linear programming word ...