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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-Jordan Method Se Equations Solve Karna (x,y,z)</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: #fff0f5; /* LavenderBlush 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: #c71585; /* MediumVioletRed */
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+ border-bottom: 2px solid #ff69b4; /* HotPink 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: #ff69b4; /* HotPink */
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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: #ffe4e1; /* MistyRose */
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+ border: 1px solid #ffb6c1; /* LightPink 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: #f5fffa; /* MintCream */
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+ border: 1px solid #90ee90; /* LightGreen 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: #32cd32; /* LimeGreen */
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+ }
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+ .operation {
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+ font-style: italic;
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+ color: #8a2be2; /* BlueViolet */
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+ }
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+ .highlight {
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+ color: #ff4500; /* OrangeRed for pivot */
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+ font-weight: bold;
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+ }
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+ .comment {
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+ color: #20b2aa; /* LightSeaGreen 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>Gauss-Jordan Method (x,y,z Variables)</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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+ x + 2y + z = 3
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+ 2x + 3y + 3z = 10
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+ 3x - y + 2z = 13
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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, augmented matrix banayenge:</p>
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+ <div class="matrix-display"><code>[ 1 2 1 | 3 ]
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+ [ 2 3 3 | 10 ]
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+ [ 3 -1 2 | 13 ]</code></div>
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+ <p>Pehla pivot (R1,C1) already 1 hai, bahut accha!</p>
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+
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+ <h3>Step 1: Pehle pivot ke neeche zeros banana</h3>
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+ <p class="operation">R2 → R2 - 2*R1</p>
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+ <p class="operation">R3 → R3 - 3*R1</p>
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+ <div class="matrix-display"><code>[ <span class="highlight">1</span> 2 1 | 3 ]
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+ [ 0 -1 1 | 4 ] <span class="comment"><-- R2: [2-2*1, 3-2*2, 3-2*1 | 10-2*3] = [0, -1, 1 | 4]</span>
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+ [ 0 -7 -1 | 4 ] <span class="comment"><-- R3: [3-3*1, -1-3*2, 2-3*1 | 13-3*3] = [0, -7, -1 | 4]</span></code></div>
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+
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+ <h3>Step 2: Dusra pivot (R2,C2) ko 1 banana</h3>
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+ <p>Ab R2,C2 wale element (-1) ko 1 banana hai.</p>
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+ <p class="operation">R2 → R2 * (-1)</p>
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+ <div class="matrix-display"><code>[ 1 2 1 | 3 ]
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+ [ 0 <span class="highlight">1</span> -1 | -4 ]
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+ [ 0 -7 -1 | 4 ]</code></div>
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+
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+ <h3>Step 3: Dusre pivot ke upar aur neeche zeros banana</h3>
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+ <p class="operation">R1 → R1 - 2*R2</p>
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+ <p class="operation">R3 → R3 + 7*R2</p>
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+ <div class="matrix-display"><code>[ 1 0 3 | 11 ] <span class="comment"><-- R1: [1-2*0, 2-2*1, 1-2*(-1) | 3-2*(-4)] = [1, 0, 3 | 11]</span>
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+ [ 0 1 -1 | -4 ]
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+ [ 0 0 -8 | -24 ] <span class="comment"><-- R3: [0+7*0, -7+7*1, -1+7*(-1) | 4+7*(-4)] = [0, 0, -8 | -24]</span></code></div>
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+
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+ <h3>Step 4: Teesra pivot (R3,C3) ko 1 banana</h3>
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+ <p>Ab R3,C3 wale element (-8) ko 1 banana hai.</p>
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+ <p class="operation">R3 → R3 / (-8)</p>
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+ <div class="matrix-display"><code>[ 1 0 3 | 11 ]
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+ [ 0 1 -1 | -4 ]
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+ [ 0 0 <span class="highlight">1</span> | 3 ]</code></div>
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+
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+ <h3>Step 5: Teesre pivot ke upar zeros banana</h3>
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+ <p class="operation">R1 → R1 - 3*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 | 2 ] <span class="comment"><-- R1: [1-3*0, 0-3*0, 3-3*1 | 11-3*3] = [1, 0, 0 | 2]</span>
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+ [ 0 1 0 | -1 ] <span class="comment"><-- R2: [0+0, 1+0, -1+1 | -4+3] = [0, 1, 0 | -1]</span>
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+ [ 0 0 1 | 3 ]</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 = 2 <br>
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+ y = -1 <br>
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+ z = 3
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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: x + 2y + z = 3</h3>
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+ <p>(2) + 2(-1) + (3) = 2 - 2 + 3 = 0 + 3 = <strong>3</strong> (Sahi hai!)</p>
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+
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+ <h3>Equation 2: 2x + 3y + 3z = 10</h3>
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+ <p>2(2) + 3(-1) + 3(3) = 4 - 3 + 9 = 1 + 9 = <strong>10</strong> (Sahi hai!)</p>
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+
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+ <h3>Equation 3: 3x - y + 2z = 13</h3>
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+ <p>3(2) - (-1) + 2(3) = 6 + 1 + 6 = 7 + 6 = <strong>13</strong> (Sahi hai!)</p>
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+
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+ <p>Solution bilkul sahi hai! Ekdum mast!</p>
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+ </div>
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+ </body>
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+ </html>