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第 12 週 · 聯立方程與變分 · 第 6 節

聯變與部分變 ＋ 本週關卡

變分的最後兩種組合 — 聯變（一個量隨多個量變）、部分變（固定部分 ＋ 變化部分）。學完用本週 BOSS 把 W12 四大主題一次打通。配滿圖例。

▶ 第一關： 聯變 z = kxy（一個量隨多個量變）
z varies jointly as x and yxyz = k x yone constant kz
joint variation: ONE constant, ONE data setwrite z = k x y(set up form)→sub ONE data set12 = k(2)(3) → k = 2→use z = 2 x yfind any zonly ONE unknown (k) → ONE equation is enough
example: z = k x y , double both x and yz = k x y→k (2x)(2y) = 4 k x y× 4both factors grow 2× → product grows 2 × 2 = 4×
▶ 第二關： 部分變 y = k₁ + k₂x（固定 ＋ 變化）
y = k₁ + k₂ xfixed partvarying part (∝ x)
taxi fare C = 20 + 8ddCstart fare 20 (d=0)fixed part k₁slope 8 per kmline does NOT pass origin → partial variation
partial variation: TWO constants → need TWO data setswrite y = k₁ + k₂ xtwo unknowns→sub TWO data setsget 2 equations→solve thesimultaneouscontrast: joint variation needs only ONE data set (one k)
▶ 第三關： 四類變分總覽 ＋ 本週知識地圖
direct y = k xthru origininverse y = k / xx↑ y↓joint z = k x yz follows x AND yONE constant kneed 1 data setpartial y = k₁ + k₂ xfixed + varyingTWO constantsneed 2 data sets
W12 map① simultaneousgraph & algebra② fractioneqn + check③ wordproblems④ variation4 typesdirect · inverse · joint · partialBOSS draws one question from each block
⚔ BOSS： 本週總關卡（W12 四大主題）
BOSS — four checkpoints of W12simultaneousx = 6add to kill yfraction eqnx = 2check rootword + inversed = 15w·d = 60joint varz = 24k = 4clear all four → W12 complete!next stop: W13 · indices & logarithms