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Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

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title = "Linear programming"
description = "Graph linear inequalities, identify feasible regions, and optimise objective functions in two variables."
template = "chapter.html"
weight = 19
[extra]
subject = "math"
chapter_id = "linear-programming"
strand = "Number & Algebra"
+++
<div class="book-prose">
<p>
Linear programming finds the best value (maximum or minimum) of a linear objective under linear
constraints. In high school, we solve two-variable problems graphically.
</p>
<h2 id="core-idea" class="book-prose__heading">Core idea</h2>
<p>
Optimize an objective like
\[
P=ax+by
\]
subject to inequalities such as \(x+y\le 12\), \(x\ge0\), \(y\ge0\).
</p>
<h2 id="modeling" class="book-prose__heading">Modeling from context</h2>
<p>
Typical setup:
</p>
<ol>
<li>Define variables (\(x,y\)).</li>
<li>Write constraints from limits/resources.</li>
<li>Write objective function (profit, cost, time, etc.).</li>
<li>Graph constraints and find feasible region.</li>
</ol>
<h2 id="graphing-constraints" class="book-prose__heading">Graphing constraints</h2>
<p>
For each inequality:
</p>
<ul>
<li>Draw boundary line (use solid line for \(\le,\ge\); dashed for \(<,>\)).</li>
<li>Test a point (often origin) to choose correct side.</li>
<li>Combine all valid half-planes.</li>
</ul>
<p>
Include non-negativity constraints \(x\ge0,y\ge0\) unless problem states otherwise.
</p>
<h2 id="feasible-region" class="book-prose__heading">Feasible region</h2>
<p>
The feasible region is where all constraints hold at once. In school-level cases, it is usually a
polygon (possibly unbounded).
</p>
<h2 id="corner-point-principle" class="book-prose__heading">Corner-point principle</h2>
<p>
If an optimum exists for a bounded feasible region, it occurs at a vertex (corner point).
</p>
<p>
So compute all feasible vertices, then evaluate objective at each.
</p>
<h2 id="worked-example" class="book-prose__heading">Worked example</h2>
<p>
Maximize \(P=3x+2y\) subject to:
</p>
<p class="book-formula">\[
x+y\le 8,\quad x+3y\le 12,\quad x\ge0,\ y\ge0.
\]</p>
<p>
Vertices are \((0,0)\), \((8,0)\), \((0,4)\), and intersection of
\(x+y=8\) with \(x+3y=12\), which is \((6,2)\).
</p>
<p>
Evaluate:
</p>
<ul>
<li>\((0,0)\Rightarrow P=0\)</li>
<li>\((8,0)\Rightarrow P=24\)</li>
<li>\((0,4)\Rightarrow P=8\)</li>
<li>\((6,2)\Rightarrow P=22\)</li>
</ul>
<p>
Maximum is \(24\) at \((8,0)\).
</p>
<h2 id="special-cases" class="book-prose__heading">Special cases</h2>
<ul>
<li><strong>No feasible region:</strong> constraints are inconsistent.</li>
<li><strong>Unbounded region:</strong> objective may have no maximum/minimum.</li>
<li><strong>Multiple optima:</strong> objective line parallel to a feasible edge.</li>
</ul>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>Label all boundary lines and intercepts clearly.</li>
<li>Show shading logic for each inequality.</li>
<li>List vertex coordinates before evaluating objective.</li>
<li>Give final answer with units and variable interpretation.</li>
</ul>
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Linear programming developed in the 20th century for resource allocation and planning. George
Dantzig's simplex method made large optimization problems practical in economics and operations
research.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-corners" class="book-prose__heading">Why corner points matter</h3>
<p>
Objective lines \(ax+by=c\) are parallel for different \(c\). Sliding this line across a polygonal
feasible region reaches the last contact at a vertex (or an entire edge in tie cases).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">
Plot \(x+y\le6,\ x\ge0,\ y\ge0\). List feasible vertices.
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Vertices: \((0,0)\), \((6,0)\), \((0,6)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Maximize \(P=2x+y\) on vertices \((0,0)\), \((4,0)\), \((1,3)\), \((0,3)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Values \(0,8,5,3\). Maximum \(8\) at \((4,0)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Minimize \(C=5x+3y\) with feasible vertices \((0,2)\), \((2,1)\), \((3,4)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(C=6,13,27\). Minimum \(6\) at \((0,2)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
In a model, why are \(x\ge0,\ y\ge0\) often included?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Quantities like items, time, or material cannot be negative.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
How can you detect possible multiple optimal solutions graphically?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Objective line is parallel to a boundary edge where both edge endpoints give the same optimum value.</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Production planning:</strong> maximize profit under labor and material limits.</li>
<li><strong>Transport/logistics:</strong> minimize cost with capacity constraints.</li>
<li><strong>Scheduling:</strong> optimize time allocation under availability rules.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school extension units on graphical linear programming.</li>
<li>Introductory operations research texts on feasible regions and corner-point optimization.</li>
</ul>
</div>

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<div class="book-prose">
<p>
Linear programming finds the best value (maximum or minimum) of a linear objective under linear
constraints. In high school, we solve two-variable problems graphically.
</p>
<h2 id="core-idea" class="book-prose__heading">Core idea</h2>
<p>
Optimize an objective like
\[
P=ax+by
\]
subject to inequalities such as \(x+y\le 12\), \(x\ge0\), \(y\ge0\).
</p>
<h2 id="modeling" class="book-prose__heading">Modeling from context</h2>
<p>
Typical setup:
</p>
<ol>
<li>Define variables (\(x,y\)).</li>
<li>Write constraints from limits/resources.</li>
<li>Write objective function (profit, cost, time, etc.).</li>
<li>Graph constraints and find feasible region.</li>
</ol>
<h2 id="graphing-constraints" class="book-prose__heading">Graphing constraints</h2>
<p>
For each inequality:
</p>
<ul>
<li>Draw boundary line (use solid line for \(\le,\ge\); dashed for \(<,>\)).</li>
<li>Test a point (often origin) to choose correct side.</li>
<li>Combine all valid half-planes.</li>
</ul>
<p>
Include non-negativity constraints \(x\ge0,y\ge0\) unless problem states otherwise.
</p>
<h2 id="feasible-region" class="book-prose__heading">Feasible region</h2>
<p>
The feasible region is where all constraints hold at once. In school-level cases, it is usually a
polygon (possibly unbounded).
</p>
<h2 id="corner-point-principle" class="book-prose__heading">Corner-point principle</h2>
<p>
If an optimum exists for a bounded feasible region, it occurs at a vertex (corner point).
</p>
<p>
So compute all feasible vertices, then evaluate objective at each.
</p>
<h2 id="worked-example" class="book-prose__heading">Worked example</h2>
<p>
Maximize \(P=3x+2y\) subject to:
</p>
<p class="book-formula">\[
x+y\le 8,\quad x+3y\le 12,\quad x\ge0,\ y\ge0.
\]</p>
<p>
Vertices are \((0,0)\), \((8,0)\), \((0,4)\), and intersection of
\(x+y=8\) with \(x+3y=12\), which is \((6,2)\).
</p>
<p>
Evaluate:
</p>
<ul>
<li>\((0,0)\Rightarrow P=0\)</li>
<li>\((8,0)\Rightarrow P=24\)</li>
<li>\((0,4)\Rightarrow P=8\)</li>
<li>\((6,2)\Rightarrow P=22\)</li>
</ul>
<p>
Maximum is \(24\) at \((8,0)\).
</p>
<h2 id="special-cases" class="book-prose__heading">Special cases</h2>
<ul>
<li><strong>No feasible region:</strong> constraints are inconsistent.</li>
<li><strong>Unbounded region:</strong> objective may have no maximum/minimum.</li>
<li><strong>Multiple optima:</strong> objective line parallel to a feasible edge.</li>
</ul>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>Label all boundary lines and intercepts clearly.</li>
<li>Show shading logic for each inequality.</li>
<li>List vertex coordinates before evaluating objective.</li>
<li>Give final answer with units and variable interpretation.</li>
</ul>
</div>

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<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Linear programming developed in the 20th century for resource allocation and planning. George
Dantzig's simplex method made large optimization problems practical in economics and operations
research.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-corners" class="book-prose__heading">Why corner points matter</h3>
<p>
Objective lines \(ax+by=c\) are parallel for different \(c\). Sliding this line across a polygonal
feasible region reaches the last contact at a vertex (or an entire edge in tie cases).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">
Plot \(x+y\le6,\ x\ge0,\ y\ge0\). List feasible vertices.
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Vertices: \((0,0)\), \((6,0)\), \((0,6)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Maximize \(P=2x+y\) on vertices \((0,0)\), \((4,0)\), \((1,3)\), \((0,3)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Values \(0,8,5,3\). Maximum \(8\) at \((4,0)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Minimize \(C=5x+3y\) with feasible vertices \((0,2)\), \((2,1)\), \((3,4)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(C=6,13,27\). Minimum \(6\) at \((0,2)\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
In a model, why are \(x\ge0,\ y\ge0\) often included?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Quantities like items, time, or material cannot be negative.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
How can you detect possible multiple optimal solutions graphically?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Objective line is parallel to a boundary edge where both edge endpoints give the same optimum value.</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Production planning:</strong> maximize profit under labor and material limits.</li>
<li><strong>Transport/logistics:</strong> minimize cost with capacity constraints.</li>
<li><strong>Scheduling:</strong> optimize time allocation under availability rules.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school extension units on graphical linear programming.</li>
<li>Introductory operations research texts on feasible regions and corner-point optimization.</li>
</ul>
</div>