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Tutoring abilities

Your navigation app uses the simple formula distance equals rate times time ($d = rt$), but to tell you your estimated arrival time, it needs to solve for time instead.

A **literal equation** is an equation that contains two or more variables, such as geometric formulas or physics laws.

You solve literal equations using the exact same inverse operations as standard linear equations—you simply treat all other variables as if they were known numbers.

<!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>Numeric vs Literal Equations</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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    color: #1e2945;
    background: transparent;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: flex-start;
    padding: 12px;
    min-height: 100%;
  }

  .nav-bar {
    display: flex;
    gap: 6px;
    margin-bottom: 12px;
    background: #f1f5f9;
    padding: 4px;
    border-radius: 8px;
  }

  .nav-btn {
    border: none;
    background: transparent;
    color: #64748b;
    font-size: 13px;
    font-weight: 600;
    padding: 6px 12px;
    border-radius: 6px;
    cursor: pointer;
    transition: all 0.2s ease;
  }

  .nav-btn.active {
    background: #ffffff;
    color: #1e2945;
    box-shadow: 0 1px 3px rgba(0,0,0,0.1);
  }

  .grid-container {
    display: grid;
    grid-template-columns: 1fr 1fr;
    gap: 12px;
    width: 100%;
    max-width: 580px;
  }

  .column-card {
    background: #ffffff;
    border: 1px solid #e6e6e6;
    border-radius: 10px;
    padding: 12px;
    display: flex;
    flex-direction: column;
    gap: 8px;
  }

  .col-header {
    font-size: 13px;
    font-weight: 700;
    text-transform: uppercase;
    letter-spacing: 0.5px;
    color: #64748b;
    text-align: center;
    padding-bottom: 4px;
    border-bottom: 1px solid #f1f5f9;
  }

  .col-header span.target {
    color: #0284c7;
  }

  .step-box {
    background: #f8fafc;
    border: 1px solid #f1f5f9;
    border-radius: 8px;
    padding: 8px;
    text-align: center;
    font-size: 16px;
    transition: opacity 0.3s ease, transform 0.3s ease, border-color 0.3s ease;
  }

  .op-badge {
    align-self: center;
    font-size: 13px;
    font-weight: 700;
    padding: 4px 10px;
    border-radius: 20px;
    display: inline-flex;
    align-items: center;
    gap: 4px;
    transition: opacity 0.3s ease, transform 0.3s ease;
  }

  .op-sub {
    background: #dcfce7;
    border: 1px solid #86efac;
    color: #15803d;
  }

  .op-div {
    background: #ffedd5;
    border: 1px solid #fdba74;
    color: #c2410c;
  }

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

  .step-final {
    background: #f0fdf4;
    border-color: #bbf7d0;
  }

  .dimmed {
    opacity: 0.2;
    transform: scale(0.97);
  }

  .highlight {
    box-shadow: 0 0 0 2px #0284c7;
    background: #ffffff;
  }

  .katex {
    font-size: 1.05em;
  }
</style>
</head>
<body>

<div class="nav-bar">
  <button class="nav-btn" onclick="setStep(0)">Given</button>
  <button class="nav-btn" onclick="setStep(1)">− Subtract</button>
  <button class="nav-btn" onclick="setStep(2)">÷ Divide</button>
  <button class="nav-btn active" onclick="setStep(3)">All</button>
</div>

<div class="grid-container">
  
  <div class="column-card">
    <div class="col-header">Numeric</div>
    
    <div class="step-box" id="num-0">
      <span class="math">14 = 3x + 2</span>
    </div>

    <div class="op-badge op-sub" id="num-op-1">
      <span>↓</span> <span class="math">- 2</span>
    </div>

    <div class="step-box step-result" id="num-1">
      <span class="math">12 = 3x</span>
    </div>

    <div class="op-badge op-div" id="num-op-2">
      <span>↓</span> <span class="math">\div 3</span>
    </div>

    <div class="step-box step-result step-final" id="num-2">
      <span class="math">4 = x</span>
    </div>
  </div>

  
  <div class="column-card">
    <div class="col-header">Literal <span class="target">(solve x)</span></div>
    
    <div class="step-box" id="lit-0">
      <span class="math">y = mx + b</span>
    </div>

    <div class="op-badge op-sub" id="lit-op-1">
      <span>↓</span> <span class="math">- b</span>
    </div>

    <div class="step-box step-result" id="lit-1">
      <span class="math">y - b = mx</span>
    </div>

    <div class="op-badge op-div" id="lit-op-2">
      <span>↓</span> <span class="math">\div m</span>
    </div>

    <div class="step-box step-result step-final" id="lit-2">
      <span class="math">\frac{y - b}{m} = x</span>
    </div>
  </div>
</div>

<script>
  function renderAllMath() {
    document.querySelectorAll('.math').forEach(el => {
      try {
        katex.render(el.textContent, el, { throwOnError: false, displayMode: false });
      } catch (e) {}
    });
  }

  window.addEventListener('DOMContentLoaded', renderAllMath);

  const stepElements = [
    { items: ['num-0', 'lit-0'] },
    { items: ['num-op-1', 'num-1', 'lit-op-1', 'lit-1'] },
    { items: ['num-op-2', 'num-2', 'lit-op-2', 'lit-2'] }
  ];

  const allItemIds = [
    'num-0', 'lit-0',
    'num-op-1', 'lit-op-1', 'num-1', 'lit-1',
    'num-op-2', 'lit-op-2', 'num-2', 'lit-2'
  ];

  function setStep(stepIndex) {
    const buttons = document.querySelectorAll('.nav-btn');
    buttons.forEach((btn, idx) => {
      btn.classList.toggle('active', idx === stepIndex);
    });

    if (stepIndex === 3) {
      // Show all
      allItemIds.forEach(id => {
        const el = document.getElementById(id);
        if (el) {
          el.classList.remove('dimmed', 'highlight');
        }
      });
      return;
    }

    allItemIds.forEach(id => {
      const el = document.getElementById(id);
      if (!el) return;
      el.classList.remove('highlight');
      el.classList.add('dimmed');
    });

    // Reveal up to current step
    for (let s = 0; s <= stepIndex; s++) {
      stepElements[s].items.forEach(id => {
        const el = document.getElementById(id);
        if (!el) return;
        el.classList.remove('dimmed');
        if (s === stepIndex && stepIndex > 0) {
          el.classList.add('highlight');
        }
      });
    }
  }
</script>

</body>
</html>

How do these same inverse operations let us isolate a specific dimension in a geometric formula?

Consider the perimeter of a rectangle, $P = 2l + 2w$, where we want to isolate the width, $w$.

First, subtract $2l$ from both sides to isolate the term with $w$: $$P - 2l = 2w$$

Next, divide the entire left side by $2$ to completely isolate $w$: $$w = \frac{P - 2l}{2}$$

A common mistake is dividing only the $P$ or only the $2l$ by 2, or trying to combine $P - 2l$ into a single term. Remember that unlike terms cannot be combined, and a division step must apply to the whole side.

What happens when our target variable is trapped inside a fraction or parentheses?

The formula converting temperature from Celsius to Fahrenheit is $F = \frac{9}{5}C + 32$.

To solve for $C$, we reverse operations in reverse order: first subtract $32$ from both sides, then multiply both sides by the reciprocal $\frac{5}{9}$.

<!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>Solving Literal Equations</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .stepper-nav {
      display: flex;
      align-items: center;
      gap: 6px;
      margin-bottom: 12px;
      flex-wrap: wrap;
      justify-content: center;
    }

    .step-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: background-color 0.2s ease, color 0.2s ease, border-color 0.2s ease;
    }

    .step-btn:hover {
      background: #e2e8f0;
      color: #1e2945;
    }

    .step-btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
    }

    .play-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 50%;
      width: 32px;
      height: 32px;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 13px;
      color: #1e2945;
      transition: background-color 0.2s;
    }

    .play-btn:hover {
      background: #f1f5f9;
    }

    .stage-container {
      width: 100%;
      max-width: 480px;
      height: 250px;
      position: relative;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      overflow: hidden;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .step-panel {
      position: absolute;
      width: 100%;
      height: 100%;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.3s ease;
    }

    .step-panel.active {
      opacity: 1;
      pointer-events: auto;
    }

    .action-tag {
      font-size: 12px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      padding: 3px 10px;
      border-radius: 12px;
      margin-bottom: 14px;
    }

    .tag-blue { background: #e0f2fe; color: #0369a1; }
    .tag-red { background: #fee2e2; color: #b91c1c; }
    .tag-green { background: #dcfce7; color: #15803d; }
    .tag-result { background: #eff6ff; color: #1d4ed8; border: 1px solid #bfdbfe; }

    .equation-row {
      font-size: 22px;
      margin: 6px 0;
      display: flex;
      align-items: center;
      justify-content: center;
      text-align: center;
    }

    .arrow-indicator {
      font-size: 16px;
      color: #94a3b8;
      margin: 4px 0;
    }

    .result-box {
      border: 2px solid #0284c7;
      background: #f0f9ff;
      border-radius: 8px;
      padding: 10px 20px;
      margin-top: 4px;
    }

    .cancel-bar {
      position: relative;
      display: inline-block;
    }
  </style>
</head>
<body>

  
  <div class="stepper-nav">
    <button class="step-btn active" data-step="0">1. Target</button>
    <button class="step-btn" data-step="1">2. Subtract 32</button>
    <button class="step-btn" data-step="2">3. Multiply ⅖</button>
    <button class="step-btn" data-step="3">4. Solved</button>
    <button class="play-btn" id="playBtn" title="Play/Pause">⏸</button>
  </div>

  
  <div class="stage-container">

    
    <div class="step-panel active" id="step0">
      <div class="action-tag tag-blue">Target: Isolate C</div>
      <div class="equation-row" id="math0"></div>
    </div>

    
    <div class="step-panel" id="step1">
      <div class="action-tag tag-red">Subtract 32 from both sides</div>
      <div class="equation-row" id="math1a"></div>
      <div class="arrow-indicator">↓</div>
      <div class="equation-row" id="math1b"></div>
    </div>

    
    <div class="step-panel" id="step2">
      <div class="action-tag tag-green">Multiply both sides by ⁵⁄₉</div>
      <div class="equation-row" id="math2a"></div>
      <div class="arrow-indicator">↓</div>
      <div class="equation-row" id="math2b"></div>
    </div>

    
    <div class="step-panel" id="step3">
      <div class="action-tag tag-result">Isolated Formula</div>
      <div class="result-box">
        <div class="equation-row" id="math3"></div>
      </div>
    </div>

  </div>

  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script>
    const stepsData = [
      {
        id: "math0",
        tex: "F = \\frac{9}{5}{\\mathbf{\\color{#0284c7}C}} + 32"
      },
      {
        id1: "math1a",
        tex1: "F {\\color{#dc2626}-\\mathbf{32}} = \\frac{9}{5}C + 32 {\\color{#dc2626}-\\mathbf{32}}",
        id2: "math1b",
        tex2: "F - 32 = \\frac{9}{5}{\\mathbf{\\color{#0284c7}C}}"
      },
      {
        id1: "math2a",
        tex1: "{\\mathbf{\\color{#16a34a}\\frac{5}{9}}}(F - 32) = {\\mathbf{\\color{#16a34a}\\frac{5}{9}}} \\cdot \\frac{9}{5}C",
        id2: "math2b",
        tex2: "\\frac{5}{9}(F - 32) = {\\mathbf{\\color{#0284c7}C}}"
      },
      {
        id: "math3",
        tex: "{\\mathbf{\\color{#0284c7}C}} = \\frac{5}{9}(F - 32)"
      }
    ];

    function renderMath() {
      if (typeof katex !== 'undefined') {
        katex.render(stepsData[0].tex, document.getElementById(stepsData[0].id), { displayMode: true });
        katex.render(stepsData[1].tex1, document.getElementById(stepsData[1].id1), { displayMode: true });
        katex.render(stepsData[1].tex2, document.getElementById(stepsData[1].id2), { displayMode: true });
        katex.render(stepsData[2].tex1, document.getElementById(stepsData[2].id1), { displayMode: true });
        katex.render(stepsData[2].tex2, document.getElementById(stepsData[2].id2), { displayMode: true });
        katex.render(stepsData[3].tex, document.getElementById(stepsData[3].id), { displayMode: true });
      } else {
        // Fallback if KaTeX fails to load
        document.getElementById("math0").innerHTML = "F = (9/5)<b style='color:#0284c7'>C</b> + 32";
        document.getElementById("math1a").innerHTML = "F <b style='color:#dc2626'>- 32</b> = (9/5)C + 32 <b style='color:#dc2626'>- 32</b>";
        document.getElementById("math1b").innerHTML = "F - 32 = (9/5)<b style='color:#0284c7'>C</b>";
        document.getElementById("math2a").innerHTML = "<b style='color:#16a34a'>(5/9)</b>(F - 32) = <b style='color:#16a34a'>(5/9)</b> · (9/5)C";
        document.getElementById("math2b").innerHTML = "(5/9)(F - 32) = <b style='color:#0284c7'>C</b>";
        document.getElementById("math3").innerHTML = "<b style='color:#0284c7'>C = (5/9)(F - 32)</b>";
      }
    }

    renderMath();

    let currentStep = 0;
    const totalSteps = 4;
    let isPlaying = true;
    let timer = null;

    const panels = document.querySelectorAll('.step-panel');
    const buttons = document.querySelectorAll('.step-btn');
    const playBtn = document.getElementById('playBtn');

    function goToStep(idx) {
      currentStep = (idx + totalSteps) % totalSteps;
      panels.forEach((p, i) => {
        p.classList.toggle('active', i === currentStep);
      });
      buttons.forEach((b, i) => {
        b.classList.toggle('active', i === currentStep);
      });
    }

    function tick() {
      goToStep(currentStep + 1);
    }

    function startAutoPlay() {
      isPlaying = true;
      playBtn.textContent = '⏸';
      clearInterval(timer);
      timer = setInterval(tick, 3500);
    }

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

    buttons.forEach((btn, idx) => {
      btn.addEventListener('click', () => {
        goToStep(idx);
        pauseAutoPlay();
      });
    });

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

    startAutoPlay();
  </script>
</body>
</html>

What should you do if the variable you want appears in more than one term?

In finance, total account balance with simple interest is $A = P + Prt$, where $P$ is the starting principal. If we need to find $P$, it appears in two separate terms.

We apply the **distributive property** in reverse by **factoring out** the common variable $P$: $$A = P(1 + rt)$$

Because $P$ is now multiplied by the single grouped expression $(1 + rt)$, divide both sides by $(1 + rt)$ to isolate it: $$P = \frac{A}{1 + rt}$$

<!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">
  <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;
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }
    .main-stage {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      margin-top: 4px;
    }
    .step-card {
      width: 100%;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      padding: 18px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 170px;
      position: relative;
    }
    .math-row {
      font-size: 26px;
      line-height: 1.4;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      transition: all 0.3s ease;
    }
    .p-var {
      color: #0066cc;
      font-weight: 700;
    }
    .bin-group {
      color: #7c3aed;
      font-weight: 700;
    }
    .cancel-slash {
      position: relative;
      display: inline-block;
    }
    .cancel-slash::after {
      content: "";
      position: absolute;
      left: -2px;
      top: 50%;
      width: calc(100% + 4px);
      height: 2px;
      background-color: #ef4444;
      transform: rotate(-25deg);
      border-radius: 1px;
    }
    .annotation {
      margin-top: 14px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-size: 14px;
      font-weight: 600;
      color: #475569;
    }
    .badge {
      display: inline-block;
      padding: 3px 8px;
      border-radius: 6px;
      font-size: 13px;
    }
    .badge-p {
      background: #e0f0ff;
      color: #0066cc;
    }
    .badge-purple {
      background: #ede9fe;
      color: #7c3aed;
    }
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
      max-width: 480px;
      margin-top: 10px;
    }
    .nav-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #1e2945;
      padding: 6px 14px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      user-select: none;
    }
    .nav-btn:hover {
      background: #e2e8f0;
    }
    .nav-btn:active {
      background: #cbd5e1;
    }
    .step-pills {
      display: flex;
      gap: 6px;
    }
    .pill {
      border: none;
      background: #e2e8f0;
      color: #475569;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 12px;
      border-radius: 6px;
      cursor: pointer;
      user-select: none;
      transition: background 0.2s, color 0.2s;
    }
    .pill.active {
      background: #0066cc;
      color: #ffffff;
    }
  </style>
</head>
<body>

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How can you be certain your rearranged formula is physically correct?

Units follow algebraic rules just like variables do, meaning you can multiply, divide, and cancel units to check your final formula.

When rearranging $d = rt$ to find speed $r = \frac{d}{t}$, substituting distance in miles ($\text{mi}$) and time in hours ($\text{hr}$) yields $\frac{\text{mi}}{\text{hr}}$, confirming that the isolated variable produces a rate.

math

Solving Literal Equations and Formulas. Rearrange formulas and literal equations to solve for a specified variable.

Rearrange formulas to isolate specific variables

Rearrange multi-variable equations (Algebra 1 Practice)

Isolating quantities | Lesson & Manipulating Formulas

To achieve high student mastery in rearranging literal equations, use the 'Side-by-Side Parallel Method' where numeric equations are solved directly alongside their corresponding literal abstractions. Emphasize highlighting the target variable with highlighters/colors to maintain visual focus. Explicitly integrate dimensional analysis (HSN.Q.A.1) so students verify their algebraic manipulations by checking that unit dimensions cancel appropriately.

Solving Literal Equations and Formulas

Isolating with Inverse Operations

Handling Fractions and Groupings