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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 = "Variations"
description = "Model direct, inverse, joint, and partial variation and solve combined variation problems."
template = "chapter.html"
weight = 18
[extra]
subject = "math"
chapter_id = "variations"
strand = "Number & Algebra"
+++
<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>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Variation models grew from early physics and astronomy, where scientists needed formulas for how
one quantity depends on another. The language of proportionality became a bridge between algebra
and real measurements.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-constant-k" class="book-prose__heading">Why we use a constant \(k\)</h3>
<p>
“Directly proportional” means ratio \(\frac{y}{x}\) stays constant, so \(\frac{y}{x}=k\) and
\(y=kx\). For inverse variation, product \(xy\) stays constant, so \(xy=k\).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">If \(y\propto x\) and \(y=15\) when \(x=3\), find \(y\) when \(x=10\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(k=5\Rightarrow y=5x\Rightarrow y=50\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">If \(y\propto \frac1x\) and \(y=8\) when \(x=5\), find \(y\) when \(x=20\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(k=40\Rightarrow y=\frac{40}{x}\Rightarrow y=2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">\(y\) varies as \(x^2\). If \(y=27\) when \(x=3\), find \(y\) when \(x=5\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(y=kx^2,\ k=3\Rightarrow y=3(25)=75\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">\(y\) varies jointly as \(x\) and \(z\). If \(y=30\) when \(x=3,z=2\), find \(y\) when \(x=5,z=4\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(y=kxz,\ k=5\Rightarrow y=5(5)(4)=100\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">A taxi fare \(F\) varies partly as distance \(d\): \(F=md+c\). If \(F=26\) at \(d=8\), and \(F=38\) at \(d=14\), find \(m,c\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Solve two equations: \(8m+c=26,\ 14m+c=38\).</p>
<p>\(6m=12\Rightarrow m=2,\ c=10\). So \(F=2d+10\).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Physics:</strong> pressure, speed, and force models often use direct/inverse variation.</li>
<li><strong>Business:</strong> cost models combine fixed and variable components.</li>
<li><strong>Engineering:</strong> scaling laws use power variation relations.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school algebra units on direct and inverse variation.</li>
<li>Elementary modeling texts for joint and partial variation.</li>
</ul>
</div>

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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>

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<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Variation models grew from early physics and astronomy, where scientists needed formulas for how
one quantity depends on another. The language of proportionality became a bridge between algebra
and real measurements.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-constant-k" class="book-prose__heading">Why we use a constant \(k\)</h3>
<p>
“Directly proportional” means ratio \(\frac{y}{x}\) stays constant, so \(\frac{y}{x}=k\) and
\(y=kx\). For inverse variation, product \(xy\) stays constant, so \(xy=k\).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">If \(y\propto x\) and \(y=15\) when \(x=3\), find \(y\) when \(x=10\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(k=5\Rightarrow y=5x\Rightarrow y=50\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">If \(y\propto \frac1x\) and \(y=8\) when \(x=5\), find \(y\) when \(x=20\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(k=40\Rightarrow y=\frac{40}{x}\Rightarrow y=2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">\(y\) varies as \(x^2\). If \(y=27\) when \(x=3\), find \(y\) when \(x=5\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(y=kx^2,\ k=3\Rightarrow y=3(25)=75\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">\(y\) varies jointly as \(x\) and \(z\). If \(y=30\) when \(x=3,z=2\), find \(y\) when \(x=5,z=4\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(y=kxz,\ k=5\Rightarrow y=5(5)(4)=100\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">A taxi fare \(F\) varies partly as distance \(d\): \(F=md+c\). If \(F=26\) at \(d=8\), and \(F=38\) at \(d=14\), find \(m,c\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Solve two equations: \(8m+c=26,\ 14m+c=38\).</p>
<p>\(6m=12\Rightarrow m=2,\ c=10\). So \(F=2d+10\).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<ul>
<li><strong>Physics:</strong> pressure, speed, and force models often use direct/inverse variation.</li>
<li><strong>Business:</strong> cost models combine fixed and variable components.</li>
<li><strong>Engineering:</strong> scaling laws use power variation relations.</li>
</ul>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school algebra units on direct and inverse variation.</li>
<li>Elementary modeling texts for joint and partial variation.</li>
</ul>
</div>