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+ <!DOCTYPE html>
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+ <html lang="hi-IN">
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+ <head>
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+ <meta charset="UTF-8">
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+ <meta name="viewport" content="width=device-width, initial-scale=1.0">
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+ <title>Gauss Elimination Method Se Equations Solve Karna</title>
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+ <style>
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+ body {
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+ font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
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+ line-height: 1.8;
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+ margin: 0;
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+ padding: 20px;
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+ background-color: #e6e6fa; /* Lavender background */
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+ color: #333;
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+ }
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+ .container {
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+ max-width: 800px;
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+ margin: auto;
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+ background: #fff;
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+ padding: 25px;
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+ border-radius: 8px;
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+ box-shadow: 0 0 15px rgba(0,0,0,0.1);
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+ }
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+ h1, h2, h3 {
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+ color: #483d8b; /* Dark Slate Blue */
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+ border-bottom: 2px solid #9370db; /* Medium Purple border */
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+ padding-bottom: 5px;
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+ }
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+ h1 {
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+ text-align: center;
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+ font-size: 2em;
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+ }
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+ h2 {
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+ font-size: 1.5em;
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+ margin-top: 30px;
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+ }
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+ h3 {
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+ font-size: 1.2em;
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+ margin-top: 20px;
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+ color: #9370db; /* Medium Purple */
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+ }
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+ p {
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+ margin-bottom: 15px;
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+ }
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+ .equations, .matrix-display {
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+ background-color: #f8f8ff; /* Ghost White */
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+ border: 1px solid #d8bfd8; /* Thistle border */
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+ padding: 15px;
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+ border-radius: 5px;
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+ margin-bottom: 20px;
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+ font-family: 'Courier New', Courier, monospace;
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+ font-size: 1.1em;
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+ overflow-x: auto;
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+ white-space: pre;
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+ }
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+ .matrix-display code {
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+ display: block;
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+ }
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+ .solution {
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+ background-color: #f0fff0; /* Honeydew */
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+ border: 1px solid #98fb98; /* Pale Green border */
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+ padding: 15px;
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+ border-radius: 5px;
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+ font-size: 1.1em;
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+ font-weight: bold;
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+ color: #3cb371; /* Medium Sea Green */
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+ }
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+ .operation {
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+ font-style: italic;
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+ color: #6a5acd; /* Slate Blue */
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+ }
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+ .highlight {
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+ color: #db7093; /* Pale Violet Red for pivot */
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+ font-weight: bold;
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+ }
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+ .comment {
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+ color: #2e8b57; /* Sea Green for comments */
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+ font-style: italic;
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+ }
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+ .back-substitution {
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+ background-color: #fffacd; /* Lemon Chiffon */
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+ padding: 10px;
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+ border-left: 4px solid #ffd700; /* Gold */
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+ margin-top: 15px;
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+ }
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+ </style>
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+ </head>
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+ <body>
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+ <div class="container">
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+ <h1>Gauss Elimination Method</h1>
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+ <h2>(a) Sawaal (Problem Statement)</h2>
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+ <p>Gauss Elimination method ka istemal karke yeh system of equations solve karo:</p>
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+ <div class="equations">
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+ 6x₁ + 3x₂ + 2x₃ = 6
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+ 6x₁ + 4x₂ + 3x₃ = 0
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+ 20x₁ + 15x₂ + 12x₃ = 0
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+ </div>
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+
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+ <h2>Gauss Elimination Ke Steps</h2>
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+ <p>Pehle, augmented matrix banate hain:</p>
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+ <div class="matrix-display"><code>[ 6 3 2 | 6 ]
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+ [ 6 4 3 | 0 ]
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+ [ 20 15 12 | 0 ]</code></div>
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+
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+ <h3>Step 1: Pehla pivot (R1,C1) ko 1 banana</h3>
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+ <p>Pehla element (R1,C1) 6 hai, isko 1 banana hai.</p>
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+ <p class="operation">R1 → R1 / 6</p>
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+ <div class="matrix-display"><code>[ <span class="highlight">1</span> 1/2 1/3 | 1 ] <span class="comment"><-- (6/6=1, 3/6=1/2, 2/6=1/3, 6/6=1)</span>
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+ [ 6 4 3 | 0 ]
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+ [ 20 15 12 | 0 ]</code></div>
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+
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+ <h3>Step 2: Pehle pivot ke neeche zeros banana</h3>
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+ <p>Ab R1,C1 wale pivot (1) ke neeche ke elements ko zero karenge.</p>
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+ <p class="operation">R2 → R2 - 6*R1</p>
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+ <p class="operation">R3 → R3 - 20*R1</p>
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+ <div class="matrix-display"><code>[ 1 1/2 1/3 | 1 ]
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+ [ 0 1 1 | -6 ] <span class="comment"><-- R2: [6-6*1, 4-6*(1/2), 3-6*(1/3) | 0-6*1] = [0, 1, 1 | -6]</span>
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+ [ 0 5 16/3 | -20] <span class="comment"><-- R3: [20-20*1, 15-20*(1/2), 12-20*(1/3) | 0-20*1] = [0, 5, (36-20)/3 | -20] = [0, 5, 16/3 | -20]</span></code></div>
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+ <p>Yahaan R2,C2 mein already 1 aa gaya, toh accha hai!</p>
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+
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+ <h3>Step 3: Dusre pivot ke neeche zero banana</h3>
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+ <p>Ab R2,C2 wala pivot 1 hai. Iske neeche (R3,C2) zero banana hai.</p>
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+ <p class="operation">R3 → R3 - 5*R2</p>
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+ <div class="matrix-display"><code>[ 1 1/2 1/3 | 1 ]
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+ [ 0 <span class="highlight">1</span> 1 | -6 ]
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+ [ 0 0 1/3 | 10] <span class="comment"><-- R3: [0-5*0, 5-5*1, 16/3-5*1 | -20-5*(-6)] = [0, 0, (16-15)/3 | -20+30] = [0, 0, 1/3 | 10]</span></code></div>
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+ <p>Matrix ab Row Echelon Form (REF) mein hai. Ab pivot elements ko 1 banana hai (teesre ko).</p>
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+
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+ <h3>Step 4: Teesra pivot (R3,C3) ko 1 banana</h3>
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+ <p class="operation">R3 → R3 * 3</p>
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+ <div class="matrix-display"><code>[ 1 1/2 1/3 | 1 ]
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+ [ 0 1 1 | -6 ]
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+ [ 0 0 <span class="highlight">1</span> | 30 ]</code></div>
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+ <p>Matrix ab Row Echelon Form (REF) mein hai aur pivots 1 hain. Gauss Elimination yahan tak hota hai. Ab hum back-substitution karenge.</p>
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+
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+ <h2>Back-Substitution (Peeche se values nikalna)</h2>
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+ <p>Matrix se equations wapas likhte hain:</p>
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+ <div class="back-substitution">
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+ Equation 1: x₁ + (1/2)x₂ + (1/3)x₃ = 1 <br>
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+ Equation 2: x₂ + x₃ = -6 <br>
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+ Equation 3: x₃ = 30
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+ </div>
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+
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+ <p><strong>Equation 3 se:</strong></p>
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+ <p>x₃ = <strong>30</strong></p>
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+
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+ <p><strong>x₃ ki value Equation 2 mein daalo:</strong></p>
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+ <p>x₂ + (30) = -6</p>
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+ <p>x₂ = -6 - 30</p>
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+ <p>x₂ = <strong>-36</strong></p>
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+
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+ <p><strong>x₂ aur x₃ ki values Equation 1 mein daalo:</strong></p>
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+ <p>x₁ + (1/2)(-36) + (1/3)(30) = 1</p>
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+ <p>x₁ - 18 + 10 = 1</p>
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+ <p>x₁ - 8 = 1</p>
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+ <p>x₁ = 1 + 8</p>
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+ <p>x₁ = <strong>9</strong></p>
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+
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+ <h2>Hal (Solution)</h2>
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+ <div class="solution">
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+ x₁ = 9 <br>
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+ x₂ = -36 <br>
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+ x₃ = 30
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+ </div>
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+
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+ <h2>Jaanch (Verification)</h2>
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+ <p>Values ko original equations mein daal kar check karte hain:</p>
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+
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+ <h3>Original Equation 1: 6x₁ + 3x₂ + 2x₃ = 6</h3>
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+ <p>6(9) + 3(-36) + 2(30) = 54 - 108 + 60 = 114 - 108 = <strong>6</strong> (Sahi hai!)</p>
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+
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+ <h3>Original Equation 2: 6x₁ + 4x₂ + 3x₃ = 0</h3>
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+ <p>6(9) + 4(-36) + 3(30) = 54 - 144 + 90 = 144 - 144 = <strong>0</strong> (Sahi hai!)</p>
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+
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+ <h3>Original Equation 3: 20x₁ + 15x₂ + 12x₃ = 0</h3>
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+ <p>20(9) + 15(-36) + 12(30) = 180 - 540 + 360 = 540 - 540 = <strong>0</strong> (Sahi hai!)</p>
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+
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+ <p>Solution bilkul sahi hai!</p>
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+ </div>
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+ </body>
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+ </html>