class 9 ganit manjari part 1 ncert solution exercise 2.6

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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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