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234 lines
11 KiB
HTML
234 lines
11 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Problems that need two unknowns at once are ancient. The Chinese text
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<em>Nine Chapters on the Mathematical Art</em> (around 200 BCE, with later commentary)
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includes <strong>fangcheng</strong> methods — arranging coefficients in a table and eliminating
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unknowns, much like modern elimination.
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</p>
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<p>
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In the 9th century, <strong>al-Khwarizmi</strong> organised linear and quadratic problems in
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systematic rules that spread to Europe. By the 17th century, <strong>Descartes</strong> linked
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algebra with geometry: an equation in \(x\) and \(y\) describes a curve, so solving a system is
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finding where two curves meet. Today, computers solve huge systems, but the same ideas — replace,
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combine, or intersect — remain the core of school algebra.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation</h2>
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<h3 id="why-elimination" class="book-prose__heading">Why elimination preserves solutions</h3>
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<p>
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If \((x_0, y_0)\) satisfies both \(ax + by = c\) and \(dx + ey = f\), then any
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<em>linear combination</em> of the two equations is still true at \((x_0, y_0)\). For example,
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adding the left sides and adding the right sides gives
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\[ (a+d)x + (b+e)y = c + f, \]
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which \((x_0, y_0)\) also satisfies. Elimination chooses combinations so one unknown disappears,
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leaving a single equation in one unknown. Multiplying an equation by a non-zero constant before
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adding does not change the solution set either.
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</p>
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<h3 id="why-intersection" class="book-prose__heading">Why the graph solution is the intersection</h3>
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<p>
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The graph of \(ax + by = c\) is all points \((x, y)\) that make the equation true. A solution of
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the <em>system</em> must lie on <em>both</em> graphs, so it must be a point common to both lines.
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If the lines cross once, that point is the unique solution; if they never meet, there is no
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solution; if they are the same line, every point on the line works.
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<p>
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Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
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</p>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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In the graph explorer, use the “Parallel” preset. What does the status panel say about
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solutions? Does the canvas show an intersection point?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Solve \(y = 3x - 2\) and \(2x + y = 8\) by substitution without a calculator. Does your pair
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\((x, y)\) satisfy both originals?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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For \(x + 2y = 8\) and \(2x + 4y = 16\), are the two equations independent, or is one a
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multiple of the other? How many solutions do you expect?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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When eliminating, why do we sometimes <em>subtract</em> equations instead of add them? Give an
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example where subtracting cancels \(x\).
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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A shop sells notebooks for \$4 and pens for \$2. You buy 6 items for \$18. What two equations
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describe this? Which method would you use first?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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In the step-by-step workbench, run elimination on the ticket-style preset. At which step does
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\(y\) (or your chosen variable) disappear?
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</p>
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</div>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<h3 id="ticket-pricing" class="book-prose__heading">Ticket pricing</h3>
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<p>
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Cinemas and transport systems often use different prices for adults and children. If you know the
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total number of tickets and the total cost, you have two equations in two unknowns — the counts
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of each type. The same structure appears in fundraising dinners (plate vs concession) and sports
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seating.
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</p>
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<h3 id="mixtures" class="book-prose__heading">Mixtures and concentrations</h3>
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<p>
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A chemist may mix a \(20\%\) solution with a \(50\%\) solution to obtain 2 L of
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\(35\%\) solution. Let \(x\) and \(y\) be the volumes (in litres) of the weaker and stronger
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solutions. Then \(x + y = 2\) (total volume) and \(0.2x + 0.5y = 0.35 \times 2\) (total amount
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of pure solute). Solving the system gives the recipe.
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</p>
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<h3 id="break-even" class="book-prose__heading">Two constraints in business</h3>
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<p>
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A small business might need at least 100 units to cover fixed costs and also stay within a
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production cap of 150 units while hitting a revenue target. Each constraint is linear; the
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feasible plan is where the constraints overlap — a preview of ideas used later in linear
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programming.
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</p>
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<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
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<p>
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Work each question, then open the solution to check your reasoning. Difficulty increases through
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the sets.
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</p>
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<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Solve by substitution: \(y = x + 2\) and \(x + y = 8\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((3, 5)\).</p>
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<p><strong>Steps:</strong> Substitute \(y = x + 2\) into \(x + y = 8\): \(x + (x+2) = 8\), so \(2x = 6\), \(x = 3\), \(y = 5\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Solve by elimination: \(x + y = 6\) and \(x - y = 2\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((4, 2)\).</p>
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<p><strong>Steps:</strong> Add: \(2x = 8\), \(x = 4\). Then \(y = 6 - 4 = 2\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Is \((1, 2)\) a solution of \(2x + y = 4\) and \(x - y = -1\)?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Yes.</p>
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<p><strong>Check:</strong> \(2(1)+2 = 4\) and \(1 - 2 = -1\).</p>
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</div>
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</details>
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<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Solve: \(2x + y = 7\) and \(x - y = 2\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((3, 1)\).</p>
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<p><strong>Steps:</strong> Add equations: \(3x = 9\), \(x = 3\). Then \(3 - y = 2\), \(y = 1\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Solve: \(3x + 2y = 12\) and \(x + y = 5\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((2, 3)\).</p>
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<p><strong>Steps:</strong> From \(x + y = 5\), \(y = 5 - x\). Substitute: \(3x + 2(5-x) = 12\), \(3x + 10 - 2x = 12\), \(x = 2\), \(y = 3\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Two lines have equations \(y = 2x + 1\) and \(y = 2x - 3\). How many solutions does the system have?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> No solution (parallel lines).</p>
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<p><strong>Explanation:</strong> Same slope \(2\), different intercepts — the lines never meet.</p>
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</div>
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</details>
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<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Solve: \(2x + 3y = 13\) and \(4x - y = 5\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((2, 3)\).</p>
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<p><strong>Steps:</strong> From \(4x - y = 5\), \(y = 4x - 5\). Substitute into \(2x + 3y = 13\): \(2x + 3(4x-5) = 13\), \(2x + 12x - 15 = 13\), \(14x = 28\), \(x = 2\), \(y = 3\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Eliminate \(x\): \(3x + 2y = 11\) and \(5x - 2y = 13\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((3, 1)\).</p>
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<p><strong>Steps:</strong> Add equations: \(8x = 24\), \(x = 3\). Then \(3(3) + 2y = 11\), \(2y = 2\), \(y = 1\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A purse has 12 coins, all 5¢ or 10¢, with total value 95¢. How many of each?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> 5 coins at 5¢ and 7 coins at 10¢.</p>
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<p><strong>Steps:</strong> Let \(n\) be 5¢ coins and \(m\) be 10¢ coins. Then \(n + m = 12\) and \(5n + 10m = 95\). From the first, \(n = 12 - m\). Substitute: \(5(12-m) + 10m = 95\), \(60 - 5m + 10m = 95\), \(5m = 35\), \(m = 7\), \(n = 5\).</p>
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</div>
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</details>
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<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
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<details class="book-question">
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<summary class="book-question__prompt">For what value of \(k\) does \(x + ky = 4\) and \(2x + 2ky = 8\) have infinitely many solutions?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Every \(k\) (the second equation is \(2\times\) the first).</p>
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<p><strong>Explanation:</strong> The lines are coincident for all \(k\) where the second is a multiple of the first; here they are always multiples, so infinitely many solutions.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Solve: \(\dfrac{x}{2} + y = 5\) and \(x - y = 1\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((4, 3)\).</p>
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<p><strong>Steps:</strong> Multiply the first equation by 2: \(x + 2y = 10\). With \(x - y = 1\), substitute \(x = 1 + y\) into \(x + 2y = 10\): \(1 + y + 2y = 10\), \(3y = 9\), \(y = 3\), \(x = 4\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A rectangle has perimeter 26 cm. The length is 4 cm more than the width. Find the dimensions.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> width \(4.5\) cm, length \(8.5\) cm.</p>
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<p><strong>Steps:</strong> \(2\ell + 2w = 26\), \(\ell = w + 4\). Substitute: \(2(w+4)+2w=26\), \(4w+8=26\), \(w=4.5\), \(\ell=8.5\).</p>
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</div>
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</details>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li><em>The Nine Chapters on the Mathematical Art</em> — early elimination methods (fangcheng).</li>
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<li>al-Khwarizmi, <em>Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala</em> (c. 820 CE).</li>
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<li>René Descartes, <em>La Géométrie</em> (1637) — coordinates and curves.</li>
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<li>HKDSE Compulsory Part curriculum — Number and Algebra strand (equations in two unknowns).</li>
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</ul>
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</div>
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