Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

Imagine you have $10 to spend entirely on $2 drinks and $1 snacks. You could buy 3 drinks and 4 snacks, or 5 drinks and no snacks — both combinations hit your total exactly.

In algebra, we write this relationship as a **linear equation in two variables**: $2x + y = 10$. Unlike single-variable equations with just one answer, equations with two variables have infinitely many paired solutions.

A solution is written as an **ordered pair** $(x, y)$, where the first number replaces $x$ and the second replaces $y$ to make the statement true.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Ordered Pair Substitution</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background-color: #f8fafc;
      color: #1e293b;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .main-layout {
      width: 100%;
      max-width: 620px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* Pair Selector Tabs */
    .selector-bar {
      display: flex;
      justify-content: center;
      gap: 8px;
    }

    .pair-btn {
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      border-radius: 8px;
      border: 1.5px solid #cbd5e1;
      background: #ffffff;
      color: #475569;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .pair-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }

    /* Diagram Grid */
    .diagram-grid {
      display: grid;
      grid-template-columns: 1fr auto 1.3fr;
      gap: 12px;
      align-items: stretch;
    }

    @media (max-width: 520px) {
      .diagram-grid {
        grid-template-columns: 1fr;
      }
      .arrow-col {
        transform: rotate(90deg);
        margin: -4px 0;
      }
    }

    .panel {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 12px;
      padding: 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 10px;
    }

    /* Anatomy Side */
    .pair-display {
      font-size: 26px;
      font-weight: 700;
      letter-spacing: 1px;
    }

    .x-val {
      color: #2563eb;
      background: #eff6ff;
      padding: 2px 8px;
      border-radius: 6px;
      border: 1px solid #bfdbfe;
    }

    .y-val {
      color: #7c3aed;
      background: #f5f3ff;
      padding: 2px 8px;
      border-radius: 6px;
      border: 1px solid #ddd6fe;
    }

    .callout-group {
      display: flex;
      flex-direction: column;
      gap: 6px;
      width: 100%;
    }

    .callout {
      display: flex;
      align-items: center;
      gap: 6px;
      font-size: 13px;
      font-weight: 600;
      padding: 4px 8px;
      border-radius: 6px;
    }

    .callout.x-call {
      color: #1d4ed8;
      background: #eff6ff;
    }

    .callout.y-call {
      color: #6d28d9;
      background: #f5f3ff;
    }

    /* Arrow */
    .arrow-col {
      display: flex;
      align-items: center;
      justify-content: center;
      color: #94a3b8;
      font-size: 22px;
      font-weight: bold;
    }

    /* Substitution Side */
    .sub-steps {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 8px;
      font-size: 15px;
      font-weight: 600;
    }

    .step-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 6px 10px;
      background: #f8fafc;
      border-radius: 6px;
      border: 1px solid #f1f5f9;
    }

    .step-label {
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }

    /* Result Card */
    .result-card {
      width: 100%;
      padding: 10px;
      border-radius: 8px;
      text-align: center;
      font-size: 14px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.2s ease;
    }

    .result-true {
      background: #ecfdf5;
      color: #059669;
      border: 1.5px solid #a7f3d0;
    }

    .result-false {
      background: #fef2f2;
      color: #dc2626;
      border: 1.5px solid #fecaca;
    }
  </style>
</head>
<body>

  <div class="main-layout">
    
    <div class="selector-bar">
      <button class="pair-btn active" id="btn-pair1" onclick="selectPair(0)">Test Pair (3, 4)</button>
      <button class="pair-btn" id="btn-pair2" onclick="selectPair(1)">Test Pair (2, 5)</button>
    </div>

    
    <div class="diagram-grid">
      
      <div class="panel">
        <div class="pair-display">
          (<span class="x-val" id="disp-x">3</span>, <span class="y-val" id="disp-y">4</span>)
        </div>
        <div class="callout-group">
          <div class="callout x-call">
            <span>x = 1st Value (Input)</span>
          </div>
          <div class="callout y-call">
            <span>y = 2nd Value (Output)</span>
          </div>
        </div>
      </div>

      
      <div class="arrow-col">➔</div>

      
      <div class="panel">
        <div class="sub-steps">
          <div class="step-row">
            <span class="step-label">Equation</span>
            <span>2<span style="color:#2563eb">x</span> + <span style="color:#7c3aed">y</span> = 10</span>
          </div>
          <div class="step-row">
            <span class="step-label">Substitute</span>
            <span id="step-sub">2(<span class="x-val" style="padding:0 3px">3</span>) + <span class="y-val" style="padding:0 3px">4</span> = 10</span>
          </div>
          <div class="step-row">
            <span class="step-label">Evaluate</span>
            <span id="step-eval">6 + 4 = 10</span>
          </div>
        </div>

        
        <div class="result-card result-true" id="result-box">
          <span id="result-icon">✓</span>
          <span id="result-text">True Statement ➔ Solution!</span>
        </div>
      </div>
    </div>
  </div>

  <script>
    const data = [
      {
        x: 3,
        y: 4,
        sub: '2(<span class="x-val" style="padding:0 3px">3</span>) + <span class="y-val" style="padding:0 3px">4</span> = 10',
        eval: '6 + 4 = 10 ➔ 10 = 10',
        isSolution: true,
        text: 'True Statement ➔ Solution!'
      },
      {
        x: 2,
        y: 5,
        sub: '2(<span class="x-val" style="padding:0 3px">2</span>) + <span class="y-val" style="padding:0 3px">5</span> = 10',
        eval: '4 + 5 = 9 ≠ 10',
        isSolution: false,
        text: 'False Statement ➔ Not a Solution'
      }
    ];

    function selectPair(idx) {
      document.getElementById('btn-pair1').classList.toggle('active', idx === 0);
      document.getElementById('btn-pair2').classList.toggle('active', idx === 1);

      const item = data[idx];
      document.getElementById('disp-x').textContent = item.x;
      document.getElementById('disp-y').textContent = item.y;
      document.getElementById('step-sub').innerHTML = item.sub;
      document.getElementById('step-eval').innerHTML = item.eval;

      const resBox = document.getElementById('result-box');
      const resIcon = document.getElementById('result-icon');
      const resText = document.getElementById('result-text');

      if (item.isSolution) {
        resBox.className = 'result-card result-true';
        resIcon.textContent = '✓';
        resText.textContent = item.text;
      } else {
        resBox.className = 'result-card result-false';
        resIcon.textContent = '✗';
        resText.textContent = item.text;
      }
    }
  </script>
</body>
</html>

How can you tell if a specific mystery pair actually works for an equation?

To check if an ordered pair is a solution, substitute the $x$-value and $y$-value into the equation and simplify both sides.

Let's test if $(4, 1)$ solves $3x - 2y = 10$. Replace $x$ with $4$ and $y$ with $1$: $$3(4) - 2(1) = 12 - 2 = 10$$ Since $10 = 10$, the pair $(4, 1)$ is a valid solution.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Testing Solutions of a Linear Equation</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    .header-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 15px;
      font-weight: 600;
      color: #475569;
      background: #f1f5f9;
      padding: 6px 14px;
      border-radius: 20px;
    }
    .x-var { color: #2563eb; font-weight: 700; }
    .y-var { color: #d97706; font-weight: 700; }

    /* Comparison grid */
    .cards-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      max-width: 600px;
    }

    @media (max-width: 480px) {
      .cards-grid {
        grid-template-columns: 1fr;
        gap: 10px;
      }
    }

    .test-card {
      background: #ffffff;
      border-radius: 12px;
      border: 1.5px solid #e2e8f0;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.04);
      padding: 14px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      transition: border-color 0.2s ease, box-shadow 0.2s ease;
    }

    .test-card.valid-card {
      border-color: #86efac;
      background: linear-gradient(180deg, #ffffff 0%, #f0fdf4 100%);
    }

    .test-card.invalid-card {
      border-color: #fca5a5;
      background: linear-gradient(180deg, #ffffff 0%, #fef2f2 100%);
    }

    .card-top {
      display: flex;
      justify-content: space-between;
      align-items: center;
      border-bottom: 1px dashed #cbd5e1;
      padding-bottom: 8px;
    }

    .point-badge {
      font-size: 15px;
      font-weight: 700;
      background: #ffffff;
      padding: 3px 8px;
      border-radius: 6px;
      border: 1px solid #cbd5e1;
    }

    .status-badge {
      font-size: 12px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 12px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .status-badge.valid {
      background: #dcfce7;
      color: #15803d;
      border: 1px solid #86efac;
    }

    .status-badge.invalid {
      background: #fee2e2;
      color: #b91c1c;
      border: 1px solid #fca5a5;
    }

    /* Math steps container */
    .steps-list {
      display: flex;
      flex-direction: column;
      gap: 8px;
      font-size: 15px;
    }

    .step-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      background: #ffffff;
      padding: 6px 10px;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
    }

    .step-tag {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    .step-math {
      font-weight: 600;
    }

    .conclusion-row {
      display: flex;
      align-items: center;
      justify-content: center;
      padding: 8px;
      border-radius: 8px;
      font-weight: 700;
      font-size: 15px;
      gap: 6px;
    }

    .valid-card .conclusion-row {
      background: #dcfce7;
      color: #166534;
    }

    .invalid-card .conclusion-row {
      background: #fee2e2;
      color: #991b1b;
    }

    /* Interactive Stepper Bar */
    .stepper-nav {
      display: flex;
      align-items: center;
      gap: 6px;
      margin-top: 2px;
    }

    .step-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #475569;
      padding: 5px 12px;
      border-radius: 16px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      transition: background 0.15s ease, color 0.15s ease;
    }

    .step-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
    }

    .step-reveal {
      transition: opacity 0.25s ease;
    }
    .hidden-step {
      opacity: 0.25;
      filter: blur(0.5px);
    }
  </style>
</head>
<body>

  
  <div class="header-bar">
    <span>Equation:</span>
    <span id="eq-main"></span>
  </div>

  
  <div class="cards-grid">
    
    <div class="test-card valid-card">
      <div class="card-top">
        <span class="point-badge" id="p1-label"></span>
        <span class="status-badge valid" id="badge-1">✓ Valid Solution</span>
      </div>

      <div class="steps-list">
        <div class="step-row step-reveal" id="p1-s1">
          <span class="step-tag">Substitute</span>
          <span class="step-math" id="p1-sub"></span>
        </div>
        <div class="step-row step-reveal" id="p1-s2">
          <span class="step-tag">Simplify</span>
          <span class="step-math" id="p1-calc"></span>
        </div>
        <div class="conclusion-row step-reveal" id="p1-s3">
          <span id="p1-eval"></span>
          <span>✓ True</span>
        </div>
      </div>
    </div>

    
    <div class="test-card invalid-card">
      <div class="card-top">
        <span class="point-badge" id="p2-label"></span>
        <span class="status-badge invalid" id="badge-2">✗ Not a Solution</span>
      </div>

      <div class="steps-list">
        <div class="step-row step-reveal" id="p2-s1">
          <span class="step-tag">Substitute</span>
          <span class="step-math" id="p2-sub"></span>
        </div>
        <div class="step-row step-reveal" id="p2-s2">
          <span class="step-tag">Simplify</span>
          <span class="step-math" id="p2-calc"></span>
        </div>
        <div class="conclusion-row step-reveal" id="p2-s3">
          <span id="p2-eval"></span>
          <span>✗ False</span>
        </div>
      </div>
    </div>
  </div>

  
  <div class="stepper-nav">
    <button class="step-btn active" id="btn-all" onclick="showStep(3)">All Steps</button>
    <button class="step-btn" id="btn-1" onclick="showStep(1)">1. Substitute</button>
    <button class="step-btn" id="btn-2" onclick="showStep(2)">2. Simplify</button>
    <button class="step-btn" id="btn-3" onclick="showStep(3)">3. Verify</button>
  </div>

  <script>
    function renderMath() {
      if (typeof katex === 'undefined') return;

      katex.render("3{\\color{#2563eb}x} - 2{\\color{#d97706}y} = 10", document.getElementById('eq-main'), { throwOnError: false });

      // Point 1 (4, 1)
      katex.render("({\\color{#2563eb}4},\\; {\\color{#d97706}1})", document.getElementById('p1-label'), { throwOnError: false });
      katex.render("3({\\color{#2563eb}4}) - 2({\\color{#d97706}1})", document.getElementById('p1-sub'), { throwOnError: false });
      katex.render("12 - 2 = 10", document.getElementById('p1-calc'), { throwOnError: false });
      katex.render("10 = 10", document.getElementById('p1-eval'), { throwOnError: false });

      // Point 2 (2, 3)
      katex.render("({\\color{#2563eb}2},\\; {\\color{#d97706}3})", document.getElementById('p2-label'), { throwOnError: false });
      katex.render("3({\\color{#2563eb}2}) - 2({\\color{#d97706}3})", document.getElementById('p2-sub'), { throwOnError: false });
      katex.render("6 - 6 = 0", document.getElementById('p2-calc'), { throwOnError: false });
      katex.render("0 \\neq 10", document.getElementById('p2-eval'), { throwOnError: false });
    }

    function showStep(stepNum) {
      const s1 = [document.getElementById('p1-s1'), document.getElementById('p2-s1')];
      const s2 = [document.getElementById('p1-s2'), document.getElementById('p2-s2')];
      const s3 = [document.getElementById('p1-s3'), document.getElementById('p2-s3')];
      const b1 = document.getElementById('badge-1');
      const b2 = document.getElementById('badge-2');

      const btns = [
        document.getElementById('btn-all'),
        document.getElementById('btn-1'),
        document.getElementById('btn-2'),
        document.getElementById('btn-3')
      ];

      btns.forEach((btn, idx) => {
        btn.classList.toggle('active', (stepNum === 3 && idx === 0) || idx === stepNum);
      });

      s1.forEach(el => el.classList.toggle('hidden-step', stepNum < 1));
      s2.forEach(el => el.classList.toggle('hidden-step', stepNum < 2));
      s3.forEach(el => el.classList.toggle('hidden-step', stepNum < 3));

      b1.style.opacity = stepNum >= 3 ? '1' : '0.2';
      b2.style.opacity = stepNum >= 3 ? '1' : '0.2';
    }

    window.addEventListener('DOMContentLoaded', renderMath);
  </script>
</body>
</html>

Testing given points is easy, but what if you need to create your own solutions from scratch?

Because every input $x$ links to an output $y$, you can pick **any** convenient value for $x$ and solve the remaining single-variable equation for $y$.

Consider $y = 2x - 1$. If we choose $x = 0$, then $y = 2(0) - 1 = -1$, giving the pair $(0, -1)$. If we pick $x = 3$, then $y = 2(3) - 1 = 5$, giving $(3, 5)$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Value Generator: y = 2x - 1</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      gap: 16px;
    }
    .flow-container {
      display: flex;
      flex-direction: row;
      align-items: stretch;
      justify-content: center;
      width: 100%;
      max-width: 620px;
      gap: 8px;
    }
    .stage {
      flex: 1;
      background: #f8fafc;
      border: 2px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 220px;
    }
    .stage-title {
      font-size: 13px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
      margin-bottom: 8px;
      text-align: center;
    }
    .badge-list {
      display: flex;
      flex-direction: column;
      gap: 6px;
      width: 100%;
      align-items: center;
    }
    .x-btn {
      width: 100%;
      max-width: 90px;
      padding: 6px 0;
      border-radius: 20px;
      border: 1.5px solid #cbd5e1;
      background: #ffffff;
      color: #1e2945;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      text-align: center;
      transition: background-color 0.2s ease, border-color 0.2s ease, color 0.2s ease;
      user-select: none;
    }
    .x-btn.active {
      background: #22b7ff;
      border-color: #0284c7;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.4);
    }
    .arrow-divider {
      display: flex;
      align-items: center;
      justify-content: center;
      color: #94a3b8;
      font-size: 20px;
      font-weight: bold;
    }
    .machine-box {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 10px;
      width: 100%;
      height: 100%;
    }
    .gear-icon {
      font-size: 26px;
      display: inline-block;
      transition: transform 0.4s cubic-bezier(0.4, 0, 0.2, 1);
    }
    .gear-spin {
      transform: rotate(90deg);
    }
    .formula-badge {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 4px 8px;
      font-size: 14px;
    }
    .math-calc {
      background: #ffffff;
      border: 1.5px solid #22b7ff;
      border-radius: 8px;
      padding: 8px 6px;
      font-size: 14px;
      font-weight: 600;
      text-align: center;
      width: 100%;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .pair-badge {
      width: 100%;
      max-width: 100px;
      padding: 6px 0;
      border-radius: 20px;
      border: 1.5px solid #cbd5e1;
      background: #ffffff;
      color: #64748b;
      font-size: 14px;
      font-weight: 600;
      text-align: center;
      transition: background-color 0.2s ease, border-color 0.2s ease, color 0.2s ease;
    }
    .pair-badge.active {
      background: #22b7ff;
      border-color: #0284c7;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.4);
    }
    .controls {
      display: flex;
      align-items: center;
      gap: 12px;
    }
    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    @media (max-width: 480px) {
      .flow-container {
        flex-direction: column;
      }
      .arrow-divider {
        transform: rotate(90deg);
        height: 16px;
      }
      .badge-list {
        flex-direction: row;
        justify-content: center;
        flex-wrap: wrap;
      }
      .stage {
        min-height: auto;
        padding: 10px;
      }
      .x-btn, .pair-badge {
        max-width: 70px;
        font-size: 13px;
      }
    }
  </style>
</head>
<body>

  <div class="flow-container">
    
    <div class="stage">
      <div class="stage-title">1. Pick x</div>
      <div class="badge-list" id="xList">
        <button class="x-btn active" data-index="0">x = -1</button>
        <button class="x-btn" data-index="1">x = 0</button>
        <button class="x-btn" data-index="2">x = 2</button>
        <button class="x-btn" data-index="3">x = 3</button>
      </div>
    </div>

    <div class="arrow-divider">→</div>

    
    <div class="stage">
      <div class="stage-title">2. Compute</div>
      <div class="machine-box">
        <span class="gear-icon" id="gear">⚙️</span>
        <div class="formula-badge" id="formulaKatex"></div>
        <div class="math-calc" id="calcKatex"></div>
      </div>
    </div>

    <div class="arrow-divider">→</div>

    
    <div class="stage">
      <div class="stage-title">3. Pair (x, y)</div>
      <div class="badge-list" id="pairList">
        <div class="pair-badge active">(-1, -3)</div>
        <div class="pair-badge">(0, -1)</div>
        <div class="pair-badge">(2, 3)</div>
        <div class="pair-badge">(3, 5)</div>
      </div>
    </div>
  </div>

  <div class="controls">
    <button class="ctrl-btn" id="playBtn">⏸ Pause</button>
  </div>

  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <script>
    const data = [
      { x: -1, y: -3, expr: "2(-1) - 1 = -3" },
      { x: 0,  y: -1, expr: "2(0) - 1 = -1" },
      { x: 2,  y: 3,  expr: "2(2) - 1 = 3" },
      { x: 3,  y: 5,  expr: "2(3) - 1 = 5" }
    ];

    let currentIndex = 0;
    let isPlaying = true;
    let timer = null;

    const xButtons = document.querySelectorAll('.x-btn');
    const pairBadges = document.querySelectorAll('.pair-badge');
    const formulaEl = document.getElementById('formulaKatex');
    const calcEl = document.getElementById('calcKatex');
    const gearEl = document.getElementById('gear');
    const playBtn = document.getElementById('playBtn');

    function renderFormula() {
      if (window.katex) {
        katex.render("y = 2x - 1", formulaEl, { throwOnError: false });
      } else {
        formulaEl.textContent = "y = 2x - 1";
      }
    }

    function setStep(idx) {
      currentIndex = idx;

      // Update X Buttons
      xButtons.forEach((btn, i) => {
        btn.classList.toggle('active', i === idx);
      });

      // Update Pair Badges
      pairBadges.forEach((badge, i) => {
        badge.classList.toggle('active', i === idx);
      });

      // Animate Gear
      gearEl.classList.toggle('gear-spin');

      // Update KaTeX Expression
      const item = data[idx];
      if (window.katex) {
        katex.render(`y = ${item.expr}`, calcEl, { throwOnError: false });
      } else {
        calcEl.textContent = `y = ${item.expr}`;
      }
    }

    function nextStep() {
      const nextIdx = (currentIndex + 1) % data.length;
      setStep(nextIdx);
    }

    function startAutoPlay() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3200);
      isPlaying = true;
      playBtn.textContent = '⏸ Pause';
    }

    function pauseAutoPlay() {
      if (timer) clearInterval(timer);
      isPlaying = false;
      playBtn.textContent = '▶ Play';
    }

    // User interactions
    xButtons.forEach((btn) => {
      btn.addEventListener('click', () => {
        const idx = parseInt(btn.getAttribute('data-index'), 10);
        setStep(idx);
        pauseAutoPlay();
      });
    });

    playBtn.addEventListener('click', () => {
      if (isPlaying) {
        pauseAutoPlay();
      } else {
        nextStep();
        startAutoPlay();
      }
    });

    // Init
    renderFormula();
    setStep(0);
    startAutoPlay();
  </script>
</body>
</html>

What are the two absolute fastest numbers you can plug in to find clean pairs?

Setting a variable to **zero** eliminates one term entirely, making arithmetic lightning fast. These special pairs are the **intercepts**, where the line crosses each axis.

Take $4x + 3y = 12$. Setting $x = 0$ gives $3y = 12$, so $y = 4$, yielding $(0, 4)$. Setting $y = 0$ gives $4x = 12$, so $x = 3$, yielding $(3, 0)$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Line Intercepts: 4x + 3y = 12</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      min-height: 100%;
    }

    .diagram-content {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    .equation-badge {
      font-size: 17px;
      font-weight: 700;
      color: #0f172a;
      letter-spacing: 0.5px;
      padding: 4px 14px;
      background: #f1f5f9;
      border-radius: 20px;
      border: 1px solid #e2e8f0;
    }

    svg {
      width: 100%;
      height: auto;
      max-height: 270px;
      overflow: visible;
      user-select: none;
    }

    /* Grid & Axis styles */
    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
    }

    .axis-line {
      stroke: #64748b;
      stroke-width: 2;
    }

    .axis-arrow {
      fill: #64748b;
    }

    .tick-label {
      font-size: 11px;
      fill: #94a3b8;
      font-weight: 500;
      text-anchor: middle;
      dominant-baseline: central;
    }

    .axis-title {
      font-size: 13px;
      font-weight: 700;
      fill: #475569;
    }

    /* Plot styles */
    .plot-line {
      stroke: #3b82f6;
      stroke-width: 3.5;
      stroke-linecap: round;
    }

    .proj-line {
      stroke: #94a3b8;
      stroke-width: 1.5;
      stroke-dasharray: 4, 4;
      opacity: 0.8;
    }

    /* Markers */
    .glow-y {
      fill: #0284c7;
      filter: drop-shadow(0 0 6px rgba(2, 132, 199, 0.6));
      cursor: pointer;
      transition: r 0.2s ease, opacity 0.2s ease;
    }

    .glow-x {
      fill: #10b981;
      filter: drop-shadow(0 0 6px rgba(16, 185, 129, 0.6));
      cursor: pointer;
      transition: r 0.2s ease, opacity 0.2s ease;
    }

    .marker-label-bg {
      fill: #ffffff;
      opacity: 0.92;
      rx: 4;
    }

    .marker-label-y {
      font-size: 12px;
      font-weight: 700;
      fill: #0369a1;
    }

    .marker-sub-y {
      font-size: 10px;
      font-weight: 500;
      fill: #64748b;
    }

    .marker-label-x {
      font-size: 12px;
      font-weight: 700;
      fill: #047857;
    }

    .marker-sub-x {
      font-size: 10px;
      font-weight: 500;
      fill: #64748b;
    }

    /* Chips container */
    .chips-row {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
      flex-wrap: wrap;
    }

    .chip {
      flex: 1 1 180px;
      max-width: 220px;
      padding: 8px 12px;
      border-radius: 8px;
      border: 1.5px solid transparent;
      background: #f8fafc;
      display: flex;
      flex-direction: column;
      gap: 2px;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
      transition: background 0.2s ease, border-color 0.2s ease, box-shadow 0.2s ease;
    }

    .chip.active-y {
      background: #f0f9ff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.15);
    }

    .chip.active-x {
      background: #ecfdf5;
      border-color: #10b981;
      box-shadow: 0 2px 6px rgba(16, 185, 129, 0.15);
    }

    .chip-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.4px;
    }

    .chip-y .chip-title { color: #0284c7; }
    .chip-x .chip-title { color: #10b981; }

    .chip-math {
      font-size: 14px;
      font-weight: 600;
      color: #1e2945;
    }

    .dimmed {
      opacity: 0.25 !important;
    }
  </style>
</head>
<body>

  <div class="diagram-content">
    <div class="equation-badge">4x + 3y = 12</div>

    <svg viewBox="0 0 380 230" aria-label="Coordinate graph showing x and y intercepts for 4x + 3y = 12">
      <defs>
        <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1.5 L 8 5 L 0 8.5 z" class="axis-arrow" />
        </marker>
      </defs>

      
      
      <g>
        
        <line x1="32" y1="180" x2="330" y2="180" class="grid-line" />
        <line x1="32" y1="142" x2="330" y2="142" class="grid-line" />
        <line x1="32" y1="104" x2="330" y2="104" class="grid-line" />
        <line x1="32" y1="66"  x2="330" y2="66"  class="grid-line" />
        <line x1="32" y1="28"  x2="330" y2="28"  class="grid-line" />

        
        <line x1="70"  y1="205" x2="70"  y2="15" class="grid-line" />
        <line x1="108" y1="205" x2="108" y2="15" class="grid-line" />
        <line x1="146" y1="205" x2="146" y2="15" class="grid-line" />
        <line x1="184" y1="205" x2="184" y2="15" class="grid-line" />
        <line x1="222" y1="205" x2="222" y2="15" class="grid-line" />
        <line x1="260" y1="205" x2="260" y2="15" class="grid-line" />
      </g>

      
      
      <line x1="36" y1="180" x2="340" y2="180" class="axis-line" marker-end="url(#arrow)" />
      
      <line x1="70" y1="200" x2="70" y2="10" class="axis-line" marker-end="url(#arrow)" />

      
      <text x="348" y="184" class="axis-title">x</text>
      <text x="70" y="5" class="axis-title" text-anchor="middle">y</text>

      
      <text x="70"  y="194" class="tick-label">0</text>
      <text x="108" y="194" class="tick-label">1</text>
      <text x="146" y="194" class="tick-label">2</text>
      <text x="184" y="194" class="tick-label" style="font-weight: 700; fill: #10b981;">3</text>
      <text x="222" y="194" class="tick-label">4</text>
      <text x="260" y="194" class="tick-label">5</text>

      <text x="56" y="142" class="tick-label">1</text>
      <text x="56" y="104" class="tick-label">2</text>
      <text x="56" y="66"  class="tick-label">3</text>
      <text x="56" y="28"  class="tick-label" style="font-weight: 700; fill: #0284c7;">4</text>

      
      
      
      <line x1="50" y1="1.33" x2="210.5" y2="215.33" class="plot-line" />

      
      <g id="group-y" onclick="selectIntercept('y')">
        <circle id="dot-y" cx="70" cy="28" r="6" class="glow-y" />
        <rect x="82" y="15" width="138" height="28" class="marker-label-bg" />
        <text x="86" y="27" class="marker-label-y">(0, 4) y-intercept</text>
        <text x="86" y="39" class="marker-sub-y">set x = 0</text>
      </g>

      
      <g id="group-x" onclick="selectIntercept('x')">
        <circle id="dot-x" cx="184" cy="180" r="6" class="glow-x" />
        <rect x="194" y="148" width="138" height="28" class="marker-label-bg" />
        <text x="198" y="160" class="marker-label-x">(3, 0) x-intercept</text>
        <text x="198" y="172" class="marker-sub-x">set y = 0</text>
      </g>
    </svg>

    
    <div class="chips-row">
      <div id="chip-y" class="chip chip-y active-y" onclick="selectIntercept('y')">
        <span class="chip-title">y-intercept</span>
        <span class="chip-math">When x = 0 → y = 4</span>
      </div>

      <div id="chip-x" class="chip chip-x active-x" onclick="selectIntercept('x')">
        <span class="chip-title">x-intercept</span>
        <span class="chip-math">When y = 0 → x = 3</span>
      </div>
    </div>
  </div>

  <script>
    let activeMode = 'both';

    function selectIntercept(mode) {
      if (activeMode === mode) {
        activeMode = 'both';
      } else {
        activeMode = mode;
      }
      updateUI();
    }

    function updateUI() {
      const groupY = document.getElementById('group-y');
      const groupX = document.getElementById('group-x');
      const chipY = document.getElementById('chip-y');
      const chipX = document.getElementById('chip-x');
      const dotY = document.getElementById('dot-y');
      const dotX = document.getElementById('dot-x');

      if (activeMode === 'y') {
        groupY.classList.remove('dimmed');
        groupX.classList.add('dimmed');
        chipY.classList.add('active-y');
        chipX.classList.remove('active-x');
        dotY.setAttribute('r', '8');
        dotX.setAttribute('r', '5');
      } else if (activeMode === 'x') {
        groupY.classList.add('dimmed');
        groupX.classList.remove('dimmed');
        chipY.classList.remove('active-y');
        chipX.classList.add('active-x');
        dotY.setAttribute('r', '5');
        dotX.setAttribute('r', '8');
      } else {
        groupY.classList.remove('dimmed');
        groupX.classList.remove('dimmed');
        chipY.classList.add('active-y');
        chipX.classList.add('active-x');
        dotY.setAttribute('r', '6');
        dotX.setAttribute('r', '6');
      }
    }
  </script>
</body>
</html>

Even when the calculations are simple, there is one common trap students fall into when writing their answers.

When you choose a $y$-value first (like $y = 0$), make sure you still write the solution as $(x, y)$, putting the calculated $x$-value first. Writing $(0, 3)$ instead of $(3, 0)$ flips the axes and gives an incorrect point.

Try picking $x = -2$ for $5x - 2y = 6$. What $y$-value balances the equation, and how will you write the complete ordered pair?

math

Solutions of a Linear Equation: Finding Pairs of Values. Find ordered pairs that satisfy a linear equation in two variables.

Solutions to Linear Equations in Two Variables

Does (x, y) Satisfy the Linear Equation?

Solutions to Linear Equations

When teaching 9th-10th grade students, bridge single-variable equation solving (HSA.REI.B.3) to two-variable relations by framing substitution as fixing one dimension to solve the other. Emphasize multiple representations: algebraic verification ($LHS = RHS$), tabular generation (T-charts), and graphical visualization (the continuous line as the infinite set of solutions). Explicitly address the structure of fractions to help students strategically select domain values that produce integer outputs.

Solutions of a Linear Equation: Finding Pairs of Values

Testing a Candidate Pair

Generating Ordered Pairs