Mathematics
Grade 11
15 min
Price lists
Price lists
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1
Introduction & Learning Objectives
Learning Objectives
Translate real-world constraints from a price list into a system of linear inequalities.
Define variables to represent quantities of items from a price list.
Graph the feasible region for a system of inequalities derived from a price list scenario.
Interpret the vertices of a feasible region in the context of a price list problem.
Determine if a given combination of items is a possible purchase based on budget and other constraints.
Formulate an objective function to maximize or minimize a quantity, such as total items or total cost.
Use systems of inequalities to find optimal solutions for purchasing decisions.
Ever tried to buy as much as possible from a menu with only $20 in your pocket? 🍔🥤 How do you figure out the best combination?
This tutorial...
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Key Concepts & Vocabulary
TermDefinitionExample
Price ListA list of items available for purchase and their corresponding costs.A coffee shop menu: Latte - $4.50, Muffin - $3.00, Tea - $2.50.
ConstraintA limitation or restriction in a problem, expressed as a linear inequality.If you have a budget of $20, the total cost of your purchase must be less than or equal to $20. This is a budget constraint.
System of Linear InequalitiesTwo or more linear inequalities involving the same variables. The solution is the set of all ordered pairs that satisfy all inequalities simultaneously.x + y > 5 and 2x - y <= 10, where x and y represent quantities of two different items.
Feasible RegionThe solution set of a system of inequalities, represented by the overlapping shaded area on a graph. It contains all possible combinati...
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Core Formulas
Standard Form of a Linear Inequality
Ax + By \leq C or Ax + By \geq C
Used to represent a single constraint. 'x' and 'y' are the quantities of two items, 'A' and 'B' are their respective prices or values (e.g., calories, weight), and 'C' is the total limit (e.g., budget, total calories).
Non-Negativity Constraints
x \geq 0 and y \geq 0
In most real-world price list problems, the quantities of items purchased cannot be negative. These constraints restrict the solution to the first quadrant of the coordinate plane.
Objective Function Form
Z = ax + by
Represents the quantity to be optimized (maximized or minimized), where 'Z' is the total value. For example, if 'a' and 'b' are the prices...
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Challenging
A farmer is buying feed. Corn feed (x) is $10/bag and soy feed (y) is $15/bag. The farmer needs at least 3000 lbs of protein and 800 lbs of fiber. Corn has 100 lbs of protein and 20 lbs of fiber per bag. Soy has 50 lbs of protein and 80 lbs of fiber per bag. The optimal solution to minimize cost is found at the vertex (200/7, 20/7). At this point, which statement is true?
A.Only the protein constraint is binding.
B.Only the fiber constraint is binding.
C.Both the protein and fiber constraints are binding.
D.Neither constraint is binding; there is a surplus of both.
Challenging
A shop sells shirts (x) and hats (y). The constraints on weekly inventory are: x + y ≤ 100 (total items), x ≥ 20 (min shirts), and y ≥ 10 (min hats). The profit is P = 5x + 8y. If the supplier increases the total item capacity so the first constraint becomes x + y ≤ 120, how does the maximum possible profit change?
A.The maximum profit will decrease.
B.The maximum profit will not change.
C.The maximum profit will increase.
D.It is impossible to determine without re-calculating all vertices.
Challenging
A factory produces widgets (x) and gadgets (y). Constraints are 3x + 2y ≤ 18, x ≥ 0, and y ≥ 0. The goal is to maximize the number of items, N = x + y. The optimal solution occurs at the vertex (2, 6). If the factory can only produce whole numbers of items, what is the best integer solution?
A.(2, 6)
B.The question is flawed, (2, 6) is not a vertex of the feasible region.
C.(1, 7)
D.(3, 4)
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