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content/math/variations/core.html

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+<div class="book-prose">
+  <p>
+    Variation describes how one quantity changes with another. It turns word statements into useful
+    algebraic models.
+  </p>
+
+  <h2 id="direct-variation" class="book-prose__heading">Direct variation</h2>
+  <p>
+    \(y\) varies directly as \(x\):
+    \[
+    y\propto x \quad\Rightarrow\quad y=kx
+    \]
+    where \(k\) is the constant of variation.
+  </p>
+  <p><strong>Example:</strong> If \(y=18\) when \(x=6\), then \(k=3\), so \(y=3x\).</p>
+
+  <h2 id="inverse-variation" class="book-prose__heading">Inverse variation</h2>
+  <p>
+    \(y\) varies inversely as \(x\):
+    \[
+    y\propto\frac1x \quad\Rightarrow\quad y=\frac{k}{x}.
+    \]
+  </p>
+  <p><strong>Example:</strong> If \(y=5\) when \(x=4\), then \(k=20\), so \(y=\frac{20}{x}\).</p>
+
+  <h2 id="power-variation" class="book-prose__heading">Variation with powers</h2>
+  <p>
+    You may see statements like:
+  </p>
+  <ul>
+    <li>\(y\propto x^2 \Rightarrow y=kx^2\)</li>
+    <li>\(y\propto \sqrt{x} \Rightarrow y=k\sqrt{x}\)</li>
+    <li>\(y\propto \frac{1}{x^2} \Rightarrow y=\frac{k}{x^2}\)</li>
+  </ul>
+
+  <h2 id="joint-variation" class="book-prose__heading">Joint variation</h2>
+  <p>
+    If \(y\) varies jointly as \(x\) and \(z\):
+    \[
+    y\propto xz \Rightarrow y=kxz.
+    \]
+  </p>
+  <p><strong>Example:</strong> \(y=24\) when \(x=2,z=3\Rightarrow k=4\), so \(y=4xz\).</p>
+
+  <h2 id="partial-variation" class="book-prose__heading">Partial variation</h2>
+  <p>
+    If \(y\) varies partly as \(x\), model has constant and variable part:
+    \[
+    y=mx+c,\quad c\ne0.
+    \]
+  </p>
+  <p>
+    This appears in pricing: fixed fee + per-unit charge.
+  </p>
+
+  <h2 id="solving-workflow" class="book-prose__heading">Solving workflow</h2>
+  <ol>
+    <li>Translate words into proportional equation with \(k\).</li>
+    <li>Use given data to find \(k\).</li>
+    <li>Substitute required values to answer question.</li>
+    <li>Check units and reasonableness.</li>
+  </ol>
+
+  <h2 id="word-problems" class="book-prose__heading">Common word-problem forms</h2>
+  <ul>
+    <li>“\(A\) is directly proportional to \(B\)”</li>
+    <li>“\(A\) varies inversely as \(B\)”</li>
+    <li>“\(A\) varies jointly as \(B\) and \(C\)”</li>
+    <li>“\(A\) varies directly as square/cube of \(B\)”</li>
+  </ul>
+
+  <h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
+  <ul>
+    <li>Always introduce \(k\) first before substituting numbers.</li>
+    <li>Keep symbolic form until \(k\) is found.</li>
+    <li>For inverse models, domain often excludes zero.</li>
+    <li>State final answer with units when applicable.</li>
+  </ul>
+</div>