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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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<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>