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<title>Class 9 Maths · Ganita Manjari · Exercise Set 2.6</title>
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<body>
<div class="container">
<!-- ===== HEADER ===== -->
<div class="header">
<h1>
Exercise Set 2.6
<span class="chapter-tag">Class 9 · Ganita Manjari</span>
</h1>
<div class="subhead">
<span>📘 Chapter 2 — Polynomials</span>
<span class="badge">📌 Graphs of Linear Equations</span>
</div>
<div class="info-note">
<span class="icon">📐</span>
<span>
A linear equation <span class="math">y = ax + b</span> represents a straight line.
The value of <strong>a</strong> (coefficient of x) controls the <strong>slope</strong> — its steepness and direction.
The value of <strong>b</strong> (constant term) controls the <strong>y-intercept</strong> — where the line crosses the y-axis.
</span>
</div>
</div>
<!-- ============================================================ -->
<!-- QUESTION 1 -->
<!-- ============================================================ -->
<div class="question-block">
<div class="q-header">
<span class="q-label">1</span>
</div>
<div class="question-text">
Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’.
</div>
<!-- (i) -->
<div class="sub-question">
<div class="q-header">
<span class="q-label sub">(i)</span>
</div>
<div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
y = 4x, y = 2x, y = x
</div>
<div class="solution" style="border-left-color:#6b9fd4;">
<p>All lines are of the form <span class="math">y = ax</span> (b = 0).</p>
<div class="line-cards">
<div class="line-card up">y = 4x</div>
<div class="line-card up">y = 2x</div>
<div class="line-card up">y = x</div>
</div>
<p><strong>Observation:</strong></p>
<ul>
<li>All lines pass through the origin <span class="math">(0, 0)</span>.</li>
<li>The value of ‘a’ (slope) determines steepness.</li>
<li>Larger ‘a’ → steeper line.</li>
</ul>
<div class="result-box info">
<span class="icon">📈</span>
<span>As <strong>a</strong> increases (1 → 2 → 4), the line becomes <strong>steeper</strong>.</span>
</div>
</div>
</div>
<!-- (ii) -->
<div class="sub-question">
<div class="q-header">
<span class="q-label sub">(ii)</span>
</div>
<div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
y = −6x, y = −3x, y = −x
</div>
<div class="solution" style="border-left-color:#6b9fd4;">
<p>All lines are of the form <span class="math">y = ax</span> (b = 0).</p>
<div class="line-cards">
<div class="line-card down">y = −6x</div>
<div class="line-card down">y = −3x</div>
<div class="line-card down">y = −x</div>
</div>
<p><strong>Observation:</strong></p>
<ul>
<li>All lines pass through the origin.</li>
<li>Negative ‘a’ means lines slope downward.</li>
<li>Larger magnitude of ‘a’ → steeper downward slope.</li>
</ul>
<div class="result-box info">
<span class="icon">📉</span>
<span>As <strong>|a|</strong> increases (1 → 3 → 6), the downward slope becomes <strong>steeper</strong>.</span>
</div>
</div>
</div>
<!-- (iii) -->
<div class="sub-question">
<div class="q-header">
<span class="q-label sub">(iii)</span>
</div>
<div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
y = 5x, y = −5x
</div>
<div class="solution" style="border-left-color:#6b9fd4;">
<div class="line-cards">
<div class="line-card up">y = 5x</div>
<div class="line-card down">y = −5x</div>
</div>
<p><strong>Observation:</strong></p>
<ul>
<li>Both lines pass through the origin.</li>
<li><span class="math">y = 5x</span> slopes upward, <span class="math">y = −5x</span> slopes downward.</li>
<li>Same magnitude of ‘a’ → same steepness but opposite direction.</li>
</ul>
<div class="result-box info">
<span class="icon">🔄</span>
<span>The <strong>sign</strong> of <strong>a</strong> determines the direction of the slope, while the <strong>magnitude</strong> determines steepness.</span>
</div>
</div>
</div>
<!-- (iv) -->
<div class="sub-question">
<div class="q-header">
<span class="q-label sub">(iv)</span>
</div>
<div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
y = 3x − 1, y = 3x, y = 3x + 1
</div>
<div class="solution" style="border-left-color:#6b9fd4;">
<p>All lines have the same slope <span class="math">(a = 3)</span>.</p>
<div class="line-cards">
<div class="line-card up">y = 3x − 1</div>
<div class="line-card up">y = 3x</div>
<div class="line-card up">y = 3x + 1</div>
</div>
<p><strong>Observation:</strong></p>
<ul>
<li>Lines are parallel (same slope).</li>
<li>Different values of ‘b’ shift the line up or down.</li>
<li><span class="math">b = −1</span> → line below origin</li>
<li><span class="math">b = 0</span> → passes through origin</li>
<li><span class="math">b = +1</span> → line above origin</li>
</ul>
<div class="result-box info">
<span class="icon">⬆️⬇️</span>
<span><strong>b</strong> shifts the line <strong>vertically</strong> without changing its slope.</span>
</div>
</div>
</div>
<!-- (v) -->
<div class="sub-question">
<div class="q-header">
<span class="q-label sub">(v)</span>
</div>
<div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
y = −2x − 3, y = −2x, y = 2x + 3
</div>
<div class="solution" style="border-left-color:#6b9fd4;">
<div class="line-cards">
<div class="line-card down">y = −2x − 3</div>
<div class="line-card down">y = −2x</div>
<div class="line-card up">y = 2x + 3</div>
</div>
<p><strong>Observation:</strong></p>
<ul>
<li><span class="math">y = −2x − 3</span> and <span class="math">y = −2x</span> have the same slope (−2) → parallel lines.</li>
<li><span class="math">y = 2x + 3</span> has a positive slope → different direction.</li>
<li>‘b’ changes the vertical position of the line.</li>
</ul>
<div class="result-box">
<span class="icon">✅</span>
<span><strong>Conclusion:</strong><br>
‘a’ (coefficient of x) controls <strong>slope</strong> (steepness and direction).<br>
‘b’ (constant term) controls <strong>vertical shift</strong> (y-intercept).</span>
</div>
</div>
</div>
</div>
<!-- ============================================================ -->
<!-- SUMMARY -->
<!-- ============================================================ -->
<div class="question-block">
<div class="q-header">
<span class="q-label" style="background:#0b2b4a;">★</span>
<span style="font-weight:600; color:#1e4c7a; font-size:1.05rem;">Summary — Role of ‘a’ and ‘b’ in y = ax + b</span>
</div>
<div class="solution">
<div class="table-wrap">
<table class="summary-table">
<thead>
<tr>
<th>Parameter</th>
<th>Name</th>
<th>Effect on the Line</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>a</strong></td>
<td>Slope</td>
<td>
• Positive → line slopes <strong>upward</strong><br>
• Negative → line slopes <strong>downward</strong><br>
• Larger |a| → <strong>steeper</strong> line<br>
• a = 0 → horizontal line
</td>
</tr>
<tr>
<td><strong>b</strong></td>
<td>y-intercept</td>
<td>
• Determines where the line crosses the <strong>y-axis</strong><br>
• Positive b → line crosses above origin<br>
• Negative b → line crosses below origin<br>
• b = 0 → line passes through origin
</td>
</tr>
</tbody>
</table>
</div>
<div class="result-box info">
<span class="icon">📐</span>
<span>
<strong>Same ‘a’</strong> → parallel lines.<br>
<strong>Different ‘a’</strong> → lines intersect at a point.
</span>
</div>
</div>
</div>
<!-- ===== FOOTER ===== -->
<div class="footer-note">
<span>⚡ Complete step‑by‑step solutions</span>
<span class="dot"></span>
<span>NCERT / Ganita Manjari · Chapter 2</span>
</div>
</div>
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