12 Linear equations

Finding the unknown

You and three friends want to split a bill evenly, but one person already paid for something earlier. How much does each person owe now?

You earn £8.50 an hour at a weekend job. You need £120 for a gig ticket. How many shifts do you need?

You’re buying snacks for a group. Each bag costs the same. You spend £14.40 in total. How much was each bag?

These are all the same question underneath: you know the relationship, you’re missing one of the values. A linear equation is just that relationship written down, with a placeholder for the missing part.

12.1 What the notation is saying

\[3x + 7 = 22\]

Read it as: three of something, plus seven, gives twenty-two. The \(x\) is the placeholder — a name for the gap in your knowledge. The equals sign means both sides describe the same amount.

Peel away what’s around the unknown until it stands alone. Three of something plus seven is twenty-two, so three of something is fifteen, so the something is five. Check: \(3 \times 5 + 7 = 22\). Yes.

That move — isolating the unknown — is what this chapter is about.

The concept stays this simple. What gets harder is the notation accumulating, and equations appearing inside larger problems where it isn’t obvious that’s what they are.

Interactive: Equation balance. The scale below shows both sides of an equation. Choose an operation and a value. Apply it to both sides and watch the equation update — the balance holds because you’re changing both sides equally.

Code

{
  // Starting equation: 3x + 7 = 22
  // We track the numeric sides as they stand after one operation.
  // Left side: 3x + 7  →  represented symbolically
  // Right side: 22
  // For display we show the symbolic equation text and what each side becomes.

  const op  = balanceOp;
  const val = Number.isFinite(balanceVal) ? Math.round(balanceVal) : 3;

  // Guard against division by zero
  const divByZero = op === "÷" && val === 0;

  // Represent the original equation symbolically
  const origLeft  = "3x + 7";
  const origRight = "22";

  // Compute what happens to the right-hand side (a pure number)
  let newRight;
  if (op === "+")       newRight = 22 + val;
  else if (op === "−")  newRight = 22 - val;
  else if (op === "×")  newRight = 22 * val;
  else if (op === "÷")  newRight = divByZero ? null : 22 / val;

  // Build a symbolic description of the new left side
  let newLeftDesc;
  const valStr = val < 0 ? `(${val})` : `${val}`;
  if (op === "+")       newLeftDesc = `3x + 7 + ${valStr}`;
  else if (op === "−")  newLeftDesc = `3x + 7 − ${valStr}`;
  else if (op === "×")  newLeftDesc = `${valStr}(3x + 7)`;
  else if (op === "÷")  newLeftDesc = divByZero ? "3x + 7 ÷ 0" : `(3x + 7) ÷ ${valStr}`;

  // Display
  const W = 560;
  const H = 260;
  const MID_X = W / 2;
  const BEAM_Y = 120;
  const PIVOT_Y = 80;
  const PAN_Y   = 170;
  const PAN_W   = 150;
  const PAN_H   = 44;
  const ROPE_H  = 50;

  const svg = d3.create("svg")
    .attr("viewBox", `0 0 ${W} ${H}`)
    .attr("width", "100%")
    .attr("style", `max-width:${W}px; font-family: inherit;`);

  // ── Pivot post ──
  svg.append("line")
    .attr("x1", MID_X).attr("y1", PIVOT_Y - 10)
    .attr("x2", MID_X).attr("y2", BEAM_Y)
    .attr("stroke", "#9ca3af").attr("stroke-width", 3);

  svg.append("circle")
    .attr("cx", MID_X).attr("cy", PIVOT_Y - 10)
    .attr("r", 6)
    .attr("fill", "#6b7280");

  // ── Beam ──
  svg.append("line")
    .attr("x1", 80).attr("y1", BEAM_Y)
    .attr("x2", W - 80).attr("y2", BEAM_Y)
    .attr("stroke", "#374151").attr("stroke-width", 4)
    .attr("stroke-linecap", "round");

  // ── Ropes ──
  const leftPanCX  = 130;
  const rightPanCX = W - 130;

  svg.append("line")
    .attr("x1", leftPanCX).attr("y1", BEAM_Y)
    .attr("x2", leftPanCX).attr("y2", PAN_Y - PAN_H / 2)
    .attr("stroke", "#9ca3af").attr("stroke-width", 2);

  svg.append("line")
    .attr("x1", rightPanCX).attr("y1", BEAM_Y)
    .attr("x2", rightPanCX).attr("y2", PAN_Y - PAN_H / 2)
    .attr("stroke", "#9ca3af").attr("stroke-width", 2);

  // ── Pans ──
  const panFill = "#f3f4f6";
  const panStroke = "#d1d5db";

  svg.append("rect")
    .attr("x", leftPanCX - PAN_W / 2).attr("y", PAN_Y - PAN_H / 2)
    .attr("width", PAN_W).attr("height", PAN_H)
    .attr("rx", 6)
    .attr("fill", panFill).attr("stroke", panStroke).attr("stroke-width", 1.5);

  svg.append("rect")
    .attr("x", rightPanCX - PAN_W / 2).attr("y", PAN_Y - PAN_H / 2)
    .attr("width", PAN_W).attr("height", PAN_H)
    .attr("rx", 6)
    .attr("fill", panFill).attr("stroke", panStroke).attr("stroke-width", 1.5);

  // ── Pan labels: new expressions ──
  const leftLabel  = divByZero ? "÷ 0 not defined" : newLeftDesc;
  const rightLabel = divByZero ? "÷ 0 not defined" :
    (Number.isInteger(newRight) ? `${newRight}` : newRight.toFixed(2));

  svg.append("text")
    .attr("x", leftPanCX).attr("y", PAN_Y + 5)
    .attr("text-anchor", "middle")
    .attr("fill", divByZero ? "#dc2626" : "#1f2937")
    .attr("font-size", divByZero ? "10px" : "12px")
    .attr("font-weight", "700")
    .text(leftLabel);

  svg.append("text")
    .attr("x", rightPanCX).attr("y", PAN_Y + 5)
    .attr("text-anchor", "middle")
    .attr("fill", divByZero ? "#dc2626" : "#1f2937")
    .attr("font-size", "14px")
    .attr("font-weight", "700")
    .text(rightLabel);

  // ── Equation line above balance ──
  const origEq = `Original: ${origLeft} = ${origRight}`;
  svg.append("text")
    .attr("x", MID_X).attr("y", 32)
    .attr("text-anchor", "middle")
    .attr("fill", "#374151").attr("font-size", "13px").attr("font-weight", "600")
    .text(origEq);

  // ── Operation applied ──
  const opSymbol = op === "÷" ? "÷" : op === "×" ? "×" : op;
  const opDesc = divByZero
    ? "Cannot divide by zero — the balance breaks"
    : `Apply  ${opSymbol} ${val}  to both sides`;

  svg.append("text")
    .attr("x", MID_X).attr("y", 52)
    .attr("text-anchor", "middle")
    .attr("fill", divByZero ? "#dc2626" : "#0d9488")
    .attr("font-size", "12px")
    .text(opDesc);

  // ── "Both sides equal" label ──
  if (!divByZero) {
    svg.append("text")
      .attr("x", MID_X).attr("y", H - 12)
      .attr("text-anchor", "middle")
      .attr("fill", "#6b7280").attr("font-size", "11px")
      .text("Both sides still describe the same amount — the balance holds.");
  }

  // ── "=" centred between pans ──
  svg.append("text")
    .attr("x", MID_X).attr("y", PAN_Y + 6)
    .attr("text-anchor", "middle")
    .attr("fill", "#6b7280").attr("font-size", "18px").attr("font-weight", "700")
    .text("=");

  return svg.node();
}

12.2 The method

The rule is: whatever you do to one side of the equals sign, do to the other. The equation is a balance. You can add weight to both sides, remove weight from both sides, or multiply both sides by the same amount — as long as you do the same thing to both, the balance holds.

\[3x + 7 = 22\]

Subtract 7 from both sides — this removes the 7 from the left, leaving just the term with \(x\):

\[3x = 15\]

Divide both sides by 3 — this peels away the coefficient, leaving \(x\) alone:

\[x = 5\]

That’s it. Two operations, one unknown, done.

Why this works

The equals sign is a statement that two things are the same amount. If you change one side without changing the other, they’re no longer the same amount — you’ve broken the equation. Doing the same thing to both sides preserves the equality. That’s the whole method.

Interactive: Equation solver. Set the coefficients \(a\), \(b\), and \(c\) in the equation \(ax + b = c\). The solution steps update live with your numbers.

Code

{
  const a = Number.isFinite(solveA) ? Math.round(solveA) : 3;
  const b = Number.isFinite(solveB) ? Math.round(solveB) : 7;
  const c = Number.isFinite(solveC) ? Math.round(solveC) : 22;

  const noSolution = a === 0;
  const cMinusB    = c - b;
  const x          = noSolution ? null : cMinusB / a;

  // Format a coefficient: omit "1x" → "x", handle negatives neatly
  const coeffStr = (n) => {
    if (n === 1)  return "x";
    if (n === -1) return "−x";
    if (n < 0)    return `${n}x`;
    return `${n}x`;
  };

  // Format the left side "ax + b" nicely
  const leftSide = () => {
    const aPart = coeffStr(a);
    if (b === 0) return `${aPart}`;
    if (b > 0)   return `${aPart} + ${b}`;
    return `${aPart} − ${Math.abs(b)}`;
  };

  const W = 560;
  const H = noSolution ? 130 : 200;
  const ROW_H = 38;
  const LABEL_X = 20;
  const EQ_X = 180;

  const svg = d3.create("svg")
    .attr("viewBox", `0 0 ${W} ${H}`)
    .attr("width", "100%")
    .attr("style", `max-width:${W}px; font-family: inherit;`);

  // Header: equation as given
  svg.append("text")
    .attr("x", LABEL_X).attr("y", 28)
    .attr("fill", "#1f2937").attr("font-size", "15px").attr("font-weight", "700")
    .text(`Solve:  ${leftSide()} = ${c}`);

  if (noSolution) {
    svg.append("text")
      .attr("x", LABEL_X).attr("y", 70)
      .attr("fill", "#dc2626").attr("font-size", "13px").attr("font-weight", "600")
      .text("a = 0: no unique solution (x drops out of the equation).");

    svg.append("text")
      .attr("x", LABEL_X).attr("y", 96)
      .attr("fill", "#6b7280").attr("font-size", "12px")
      .text(b === c
        ? `With a = 0 the equation reads ${b} = ${c}, which is always true — any x works.`
        : `With a = 0 the equation reads ${b} = ${c}, which is never true — no x works.`
      );
  } else {
    // Step 1
    const step1Y = 64;
    svg.append("text")
      .attr("x", LABEL_X).attr("y", step1Y)
      .attr("fill", "#6b7280").attr("font-size", "12px")
      .text(`Step 1 — subtract ${b < 0 ? `(${b})` : b} from both sides`
            + (b < 0 ? "  [adding a negative]" : "")
            + ":");

    const bStr = b < 0 ? `+ (${b})` : `− ${b}`;
    svg.append("text")
      .attr("x", EQ_X).attr("y", step1Y + 20)
      .attr("fill", "#1f2937").attr("font-size", "14px").attr("font-weight", "600")
      .text(`${coeffStr(a)} = ${c} ${bStr}  =  ${cMinusB}`);

    // Step 2
    const step2Y = step1Y + 56;
    svg.append("text")
      .attr("x", LABEL_X).attr("y", step2Y)
      .attr("fill", "#6b7280").attr("font-size", "12px")
      .text(`Step 2 — divide both sides by ${a < 0 ? `(${a})` : a}:`);

    const xDisplay = Number.isInteger(x)
      ? `${x}`
      : x.toFixed(4).replace(/\.?0+$/, "");

    svg.append("text")
      .attr("x", EQ_X).attr("y", step2Y + 20)
      .attr("fill", "#1f2937").attr("font-size", "14px").attr("font-weight", "600")
      .text(`x = ${cMinusB} ÷ ${a}  =  ${xDisplay}`);

    // Answer box
    const ansY = step2Y + 46;
    svg.append("rect")
      .attr("x", EQ_X - 8).attr("y", ansY - 20)
      .attr("width", 220).attr("height", 32)
      .attr("rx", 6)
      .attr("fill", "#0d9488").attr("opacity", 0.12)
      .attr("stroke", "#0d9488").attr("stroke-width", 1.5);

    svg.append("text")
      .attr("x", EQ_X + 2).attr("y", ansY)
      .attr("fill", "#0d9488").attr("font-size", "15px").attr("font-weight", "700")
      .text(`Answer:  x = ${xDisplay}`);

    // Check
    const checkY = ansY + 28;
    const checkLeft  = a * (Number.isInteger(x) ? x : parseFloat(xDisplay)) + b;
    const checkRight = c;
    const checkOk    = Math.abs(checkLeft - checkRight) < 1e-9;
    svg.append("text")
      .attr("x", LABEL_X).attr("y", checkY)
      .attr("fill", checkOk ? "#6b7280" : "#dc2626")
      .attr("font-size", "11px")
      .text(checkOk
        ? `Check: ${a}×${xDisplay} + ${b} = ${checkLeft} ✓`
        : `Check: ${a}×${xDisplay} + ${b} = ${checkLeft}  (expected ${c}) — rounding`
      );
  }

  return svg.node();
}

12.3 Worked examples

Example 1. You’re saving up for a festival wristband. The wristband costs £85. You already have £20 saved and earn £13 per hour doing odd jobs. How many hours do you need to work?

Let \(h\) = hours worked.

\[20 + 13h = 85\]

Subtract 20 from both sides — to clear the fixed amount already saved:

\[13h = 65\]

Divide both sides by 13 — to get from “13 of something” to “one of something”:

\[h = 5\]

Five hours. Check: \(20 + 13 \times 5 = 20 + 65 = 85\). Yes.

Example 2. Two streaming services. Service A costs £8 per month plus £2 per film you rent. Service B costs £18 per month with unlimited films. How many film rentals per month makes the two services equal in cost?

Let \(f\) = films rented per month.

Service A costs: \(8 + 2f\)

Service B costs: \(18\)

They’re equal when:

\[8 + 2f = 18\]

\[2f = 10\]

\[f = 5\]

At 5 films per month they cost the same. Above 5 films, Service B is cheaper.

Example 3. You’re making a fruit drink by mixing juice concentrate with water. The concentrate is 60% real juice. You want the final drink to be 20% real juice. You start with 1 litre of concentrate. How much water do you add?

The amount of real juice stays fixed at \(0.60 \times 1 = 0.6\) litres. After adding \(w\) litres of water, the total volume is \(1 + w\) litres, and you want the juice fraction to be 20%:

\[\frac{0.6}{1 + w} = 0.2\]

Multiply both sides by \((1 + w)\) — to clear the fraction:

\[0.6 = 0.2(1 + w)\]

Expand the right side:

\[0.6 = 0.2 + 0.2w\]

Subtract 0.2 from both sides:

\[0.4 = 0.2w\]

Divide both sides by 0.2:

\[w = 2\]

Add 2 litres of water.

12.4 Where this goes

This chapter covered one unknown on one side. The next two chapters in this volume — inequalities and graphing — extend the same idea in different directions: inequalities ask which values satisfy a relationship rather than pinning down one exact answer, and graphing shows what happens when you let the unknown vary continuously. The further horizon worth knowing about is systems of equations (Volume 4): two unknowns, two equations, solved together. The method is the same balancing act you’ve already learned, applied twice at once.

Where this shows up

An electrical engineer writing Kirchhoff’s voltage law is solving a linear equation for current or voltage.
A nutritionist calculating how much of an ingredient hits a target percentage is solving a linear equation.
A financial analyst finding a break-even point is solving a linear equation.
A data scientist fitting a regression line is solving a system of linear equations.

The tool is the same. The domain changes.

12.5 Exercises

These are puzzles, not drills. Each one has a clean answer, but the interesting part is setting up the equation — translating the situation into the notation before you solve it.

Code

function makeStepperHTML(exerciseNum, steps) {
  let current = 0;
  const totalSteps = steps.length;
  const container = document.createElement("div");
  container.style.cssText = "border:1px solid #e5e7eb; border-radius:8px; padding:1rem 1.25rem; margin:0.75rem 0; font-family:inherit;";
  const stepsDiv = document.createElement("div");
  stepsDiv.style.marginBottom = "0.75rem";

  function renderSteps() {
    stepsDiv.innerHTML = "";
    for (let i = 0; i < current; i++) {
      const s = steps[i];
      const row = document.createElement("div");
      row.style.cssText = "display:grid; grid-template-columns:200px 1fr; gap:0.35rem 1rem; align-items:baseline; padding:0.35rem 0; border-top:1px solid #f3f4f6;";
      const opCell = document.createElement("span");
      opCell.style.cssText = "font-size:0.85em; color:#6b7280; font-style:italic;";
      opCell.textContent = s.op;
      const eqCell = document.createElement("span");
      eqCell.innerHTML = katex.renderToString(s.eq, { throwOnError: false, displayMode: false });
      row.appendChild(opCell);
      row.appendChild(eqCell);
      if (s.note) {
        const noteRow = document.createElement("div");
        noteRow.style.cssText = "grid-column:2; font-size:0.82em; color:#6b7280; padding-bottom:0.2rem;";
        noteRow.textContent = s.note;
        row.appendChild(noteRow);
      }
      stepsDiv.appendChild(row);
    }
    if (current === totalSteps) {
      const done = document.createElement("div");
      done.style.cssText = "margin-top:0.5rem; font-size:0.9em; color:#059669; font-weight:500;";
      done.textContent = "✓ Solution complete";
      stepsDiv.appendChild(done);
    }
  }

  const controls = document.createElement("div");
  controls.style.cssText = "display:flex; gap:0.5rem; align-items:center;";
  const nextBtn = document.createElement("button");
  nextBtn.textContent = "Next step →";
  nextBtn.style.cssText = "padding:0.35rem 0.85rem; border:1px solid #d1d5db; border-radius:4px; background:#fff; cursor:pointer; font-size:0.9em;";
  const resetBtn = document.createElement("button");
  resetBtn.textContent = "Reset";
  resetBtn.style.cssText = "padding:0.35rem 0.75rem; border:1px solid #e5e7eb; border-radius:4px; background:#f9fafb; cursor:pointer; font-size:0.9em; color:#6b7280;";
  const counter = document.createElement("span");
  counter.style.cssText = "font-size:0.82em; color:#9ca3af; margin-left:0.25rem;";

  function updateButtons() {
    nextBtn.disabled = current === totalSteps;
    nextBtn.style.opacity = current === totalSteps ? "0.4" : "1";
    counter.textContent = current === 0 ? "Click to reveal steps" : `Step ${current} of ${totalSteps}`;
  }

  nextBtn.onclick = () => { if (current < totalSteps) { current++; renderSteps(); updateButtons(); } };
  resetBtn.onclick = () => { current = 0; renderSteps(); updateButtons(); };
  controls.appendChild(nextBtn);
  controls.appendChild(resetBtn);
  controls.appendChild(counter);
  container.appendChild(stepsDiv);
  container.appendChild(controls);
  renderSteps();
  updateButtons();
  return container;
}

A bus pass costs £1.50 to buy, then £0.90 per journey. You spend £9.60 in total. How many journeys did you take?

Code

makeStepperHTML(1, [
  { op: "Write as an equation",           eq: "1.50 + 0.90j = 9.60" },
  { op: "Subtract 1.50 from both sides",  eq: "0.90j = 8.10" },
  { op: "Divide both sides by 0.90",      eq: "j = 9" },
  { op: "Check",                           eq: "1.50 + 0.90 \\times 9 = 1.50 + 8.10 = 9.60 \\checkmark" }
])

A recipe makes 12 cookies using 180g of flour. You have 480g of flour. How many cookies can you make? (Set up an equation first. Don’t just divide.)

Code

makeStepperHTML(2, [
  { op: "Write as an equation",          eq: "\\frac{180}{12} = \\frac{480}{c}", note: "Flour per cookie is constant, so set up equal ratios." },
  { op: "Simplify left side",             eq: "15 = \\frac{480}{c}" },
  { op: "Multiply both sides by c",       eq: "15c = 480" },
  { op: "Divide both sides by 15",        eq: "c = 32" },
  { op: "Check",                          eq: "\\frac{180}{12} = 15 \\text{ g/cookie},\\quad 15 \\times 32 = 480 \\checkmark" }
])

Two cyclists start 120km apart and ride toward each other. One rides at 20km/h, the other at 25km/h. After how many hours do they meet?

Code

makeStepperHTML(3, [
  { op: "Write as an equation",           eq: "20t + 25t = 120", note: "Together they close 45 km each hour, starting 120 km apart." },
  { op: "Collect like terms",             eq: "45t = 120" },
  { op: "Divide both sides by 45",        eq: "t = \\frac{120}{45} = \\frac{8}{3} \\approx 2.67 \\text{ h}" },
  { op: "Check",                          eq: "20 \\times \\tfrac{8}{3} + 25 \\times \\tfrac{8}{3} = \\tfrac{160}{3} + \\tfrac{200}{3} = \\tfrac{360}{3} = 120 \\checkmark" }
])

When two quantities are changing in opposite directions and you want to know when they’re equal, the unknown ends up on both sides of the equation. The fix is simple: collect all the terms with the unknown on one side, and all the plain numbers on the other.

For example, if one thing is shrinking and another is growing, you might get something like \(500 - 35t = 200 + 15t\). Subtract \(15t\) from both sides to collect the variable terms: \(500 - 50t = 200\). Then subtract 200: \(300 = 50t\). Divide by 50: \(t = 6\). The same balance rule as before — it just takes one extra step to gather the unknowns first.

A water tank holds 800 litres and drains at 35 litres per minute. A second tank holds 200 litres and fills at 15 litres per minute. After how many minutes do they contain the same amount?

Code

makeStepperHTML(4, [
  { op: "Write as an equation",           eq: "800 - 35t = 200 + 15t", note: "Tank 1 decreases; tank 2 increases. Set them equal." },
  { op: "Subtract 15t from both sides",   eq: "800 - 50t = 200" },
  { op: "Subtract 800 from both sides",   eq: "-50t = -600" },
  { op: "Divide both sides by −50",       eq: "t = 12" },
  { op: "Check",                          eq: "800 - 35 \\times 12 = 800 - 420 = 380;\\quad 200 + 15 \\times 12 = 200 + 180 = 380 \\checkmark" }
])

You and two friends are splitting the cost of a group gift. The total cost is £96, but one friend already put in £18 as a deposit. How much does each of the three of you pay now?

Code

makeStepperHTML(5, [
  { op: "Write as an equation",           eq: "3p + 18 = 96", note: "Three people each pay p, plus the £18 already paid, equals £96." },
  { op: "Subtract 18 from both sides",    eq: "3p = 78" },
  { op: "Divide both sides by 3",         eq: "p = 26" },
  { op: "Check",                          eq: "3 \\times 26 + 18 = 78 + 18 = 96 \\checkmark" }
])