Linear programming is a mathematical technique used to manage and distribute limited resources—such as money, materials, and manpower—in the most efficient way to maximise profits or minimise costs.
Distinguish solid and dotted lines, and determine the correct side to shade based on the sign of coefficient \(b\).
Use a solid line for \(\le\) or \(\ge\).Use a dotted line for \(<\) or \(>\).
For \(ax + by = c\):If \(b > 0\), \(\ge\) is above and \(\le\) is below.If \(b < 0\), the directions reverse.
Model real-world problems, graph constraints, and optimise the objective function.
Variables, usually \(x\) and \(y\), that represent the decisions to be made.
A linear expression written as \(k = ax + by\) that needs to be maximised or minimised.
Restrictions or limitations expressed as linear equations or linear inequalities using symbols such as \(=\), \(\le\), \(<\), \(\ge\), and \(>\).
The common area on a Cartesian graph where all given constraints are simultaneously satisfied.
Assign \(x\) and \(y\) to the two quantities being measured.
Translate written conditions into mathematical symbols.
Write the objective function \(k = ax + by\).
List all constraints as linear inequalities or equations.
Objective function in two variables
Maximise profit or minimise cost within the feasible region.
Answer in your mind, then press "Check Answer".
A tailor sews \(x\) shirts and \(y\) trousers. The total number of garments sewn is at most 60, and the number of trousers must be at least twice the number of shirts. Write two linear inequalities representing these conditions.
For the inequality \(4x - 2y \ge 8\), state whether the boundary line is solid or dotted, and whether the shaded region lies above or below the line \(4x - 2y = 8\).
The shaded region of a linear programming problem has vertices at \((10, 10)\), \((10, 50)\), and \((40, 20)\). Find the maximum value of the objective function \(k = 2x + 3y\).
The optimal point of an objective function can occur in the middle of the shaded region.
In linear programming, the maximum or minimum value always occurs at one of the boundary vertices of the feasible region.
Test points can be chosen directly on the boundary line.
A test point must strictly lie off the boundary line, such as the origin \((0,0)\) when the line does not pass through it.
Objective: Identify constraints, boundary lines, objective functions and key terms in linear programming.
1 Which inequality represents the statement: "The number of units of car N (\(y\)) produced is not more than three times the number of units of car M (\(x\)) produced"?
2 When graphing the inequality \(x - 2y \ge -4\), why does the shaded region lie below the boundary line \(x - 2y = -4\)?
3 When representing a linear inequality graphically, a ________ line is used if the inequality involves the symbols \(\le\) or \(\ge\).
4 In linear programming, an objective function in two variables is generally written in the form \(k =\) ________.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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