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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>Dusra Sawaal: Equations Solve Karna Gauss-Jordan Method Se</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: #f0f8ff; /* AliceBlue 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: #2c3e50; /* Dark blue */
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+ border-bottom: 2px solid #5dade2; /* Lighter blue 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: #5dade2; /* Lighter blue */
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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: #eaf2f8; /* Light blue-gray */
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+ border: 1px solid #aed6f1; /* Soft blue 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: #e8f8f5; /* Light cyan */
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+ border: 1px solid #76d7c4; /* Mint 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: #1abc9c; /* Turquoise */
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+ }
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+ .operation {
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+ font-style: italic;
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+ color: #7f8c8d; /* Gray */
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+ }
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+ .highlight {
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+ color: #e74c3c; /* Red for pivot */
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+ font-weight: bold;
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+ }
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+ .comment {
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+ color: #27ae60; /* Green for comments */
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+ font-style: italic;
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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>Linear Equations Ko Solve Karna (Part 2)</h1>
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+ <h2>(a) Sawaal (Problem Statement)</h2>
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+ <p>Gauss-Jordan method ka istemal karke yeh equations solve karo:</p>
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+ <div class="equations">
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+ 2x - 6y + 8z = 24
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+ 5x + 4y - 3z = 2
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+ 3x + y + 2z = 16
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+ </div>
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+
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+ <h2>Gauss-Jordan Elimination Ke Steps</h2>
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+ <p>Sabse pehle, in equations ka augmented matrix banayenge:</p>
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+ <div class="matrix-display"><code>[ 2 -6 8 | 24 ]
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+ [ 5 4 -3 | 2 ]
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+ [ 3 1 2 | 16 ]</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) abhi 2 hai, isko 1 banana hai.</p>
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+ <p class="operation">R1 → R1 / 2 (Row 1 ko 2 se divide karo)</p>
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+ <div class="matrix-display"><code>[ <span class="highlight">1</span> -3 4 | 12 ]
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+ [ 5 4 -3 | 2 ]
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+ [ 3 1 2 | 16 ]</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 (R2,C1 aur R3,C1) ko zero karenge.</p>
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+ <p class="operation">R2 → R2 - 5*R1</p>
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+ <p class="operation">R3 → R3 - 3*R1</p>
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+ <div class="matrix-display"><code>[ 1 -3 4 | 12 ]
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+ [ 0 19 -23 | -58 ] <span class="comment"><-- R2: [5-5*1, 4-5*(-3), -3-5*4 | 2-5*12] = [0, 19, -23 | -58]</span>
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+ [ 0 10 -10 | -20 ] <span class="comment"><-- R3: [3-3*1, 1-3*(-3), 2-3*4 | 16-3*12] = [0, 10, -10 | -20]</span></code></div>
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+
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+ <h3>Step 3: Dusra pivot (R2,C2) ko 1 banana (Thoda Smart Work)</h3>
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+ <p>Dekho, Row 3 (R3) ko 10 se divide karke simplify kar sakte hain:</p>
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+ <p class="operation">R3 → R3 / 10</p>
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+ <div class="matrix-display"><code>[ 1 -3 4 | 12 ]
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+ [ 0 19 -23 | -58 ]
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+ [ 0 1 -1 | -2 ] <span class="comment"><-- Simplified R3</span></code></div>
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+ <p>Ab R2 aur R3 ko swap (badal) kar lete hain taaki R2,C2 mein 1 aa jaaye.</p>
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+ <p class="operation">R2 ↔ R3</p>
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+ <div class="matrix-display"><code>[ 1 -3 4 | 12 ]
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+ [ 0 <span class="highlight">1</span> -1 | -2 ]
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+ [ 0 19 -23 | -58 ]</code></div>
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+ <p>Ab R2,C2 wala pivot 1 ho gaya!</p>
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+
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+ <h3>Step 4: Dusre pivot ke upar aur neeche zeros banana</h3>
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+ <p>Ab R2,C2 wale pivot (1) ke upar (R1,C2) aur neeche (R3,C2) zero banana hai.</p>
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+ <p class="operation">R1 → R1 + 3*R2</p>
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+ <p class="operation">R3 → R3 - 19*R2</p>
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+ <div class="matrix-display"><code>[ 1 0 1 | 6 ] <span class="comment"><-- R1: [1, -3+3*1, 4+3*(-1) | 12+3*(-2)] = [1, 0, 1 | 6]</span>
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+ [ 0 1 -1 | -2 ]
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+ [ 0 0 -4 | -20 ] <span class="comment"><-- R3: [0, 19-19*1, -23-19*(-1) | -58-19*(-2)] = [0, 0, -4 | -20]</span></code></div>
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+
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+ <h3>Step 5: Teesra pivot (R3,C3) ko 1 banana</h3>
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+ <p>Ab R3,C3 wale element (-4) ko 1 banana hai.</p>
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+ <p class="operation">R3 → R3 / (-4)</p>
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+ <div class="matrix-display"><code>[ 1 0 1 | 6 ]
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+ [ 0 1 -1 | -2 ]
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+ [ 0 0 <span class="highlight">1</span> | 5 ]</code></div>
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+
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+ <h3>Step 6: Teesre pivot ke upar zeros banana</h3>
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+ <p>Ab R3,C3 wale pivot (1) ke upar (R1,C3 aur R2,C3) zero banana hai.</p>
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+ <p class="operation">R1 → R1 - R3</p>
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+ <p class="operation">R2 → R2 + R3</p>
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+ <div class="matrix-display"><code>[ 1 0 0 | 1 ] <span class="comment"><-- R1: [1-0, 0-0, 1-1 | 6-5] = [1, 0, 0 | 1]</span>
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+ [ 0 1 0 | 3 ] <span class="comment"><-- R2: [0+0, 1+0, -1+1 | -2+5] = [0, 1, 0 | 3]</span>
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+ [ 0 0 1 | 5 ]</code></div>
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+ <p>Yeh matrix ab Reduced Row Echelon Form (RREF) mein hai.</p>
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+
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+ <h2>Hal (Solution)</h2>
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+ <p>RREF matrix se humein solution milta hai:</p>
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+ <div class="solution">
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+ x = 1 <br>
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+ y = 3 <br>
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+ z = 5
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+ </div>
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+
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+ <h2>Jaanch (Verification)</h2>
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+ <p>Ab x, y, aur z ki values ko original equations mein daal kar check karte hain:</p>
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+
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+ <h3>Equation 1: 2x - 6y + 8z = 24</h3>
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+ <p>2(1) - 6(3) + 8(5) = 2 - 18 + 40 = -16 + 40 = <strong>24</strong> (Sahi hai!)</p>
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+
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+ <h3>Equation 2: 5x + 4y - 3z = 2</h3>
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+ <p>5(1) + 4(3) - 3(5) = 5 + 12 - 15 = 17 - 15 = <strong>2</strong> (Sahi hai!)</p>
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+
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+ <h3>Equation 3: 3x + y + 2z = 16</h3>
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+ <p>3(1) + (3) + 2(5) = 3 + 3 + 10 = 6 + 10 = <strong>16</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>