➗ 分式方程：分母里藏着未知数➗ Rational Equations: the Unknown in the Denominator

上一片的世界里，答案是一整段区间。今天回到等号，可未知数玩起了躲猫猫——它藏进了分母。机器人的这份糖，还能分得匀吗？ Last leaf, an answer was a whole stretch of numbers. Today we return to the equals sign — but the unknown plays hide-and-seek, tucked inside a denominator. Can the robot still share the candies evenly?

第 1 步：未知数躲进分母Step 1: The unknown hides in the denominator

6 颗糖要分匀，x 该站在哪？Sharing 6 candies evenly — where does x stand?

6 颗糖分给 x 个机器人，每人 6 ÷ x 颗 — 机器人把 6 块糖平均分给 x 个机器人，每人拿到 6 ÷ x 块。今天它想让每人正好拿 3 块——写下方程 6 ÷ x = 3 才发现：x 躲在分母上。像这样分母里含有未知数的方程，就叫**分式方程**。 The robot shares 6 candies equally among x robots, so each gets 6 ÷ x. Today everyone should get exactly 3 — and the equation 6 ÷ x = 3 reveals that x is hiding in the denominator. An equation with an unknown in the denominator is called a **rational equation**.

怎么把 x 从分母里请出来？How do we coax x out of the denominator?

第 2 步：两边同乘公分母Step 2: Multiply both sides by the common denominator

天平两边同乘 x：6 ÷ x = 3 变成 6 = 3x — 等式像一架天平：两边同时乘上同一个数，平衡不变。给 6 ÷ x = 3 的两边都乘 x：左边 6 ÷ x 再乘 x 正好剩下 6，右边 3 乘 x 变成 3x，分母被请走了——得到 6 = 3x，解得 x = 2。这一步叫**去分母**，两边乘的那个式子叫公分母。 An equation is a balance: multiply both sides by the same number and it stays level. Multiply 6 ÷ x = 3 by x on both sides: on the left, 6 ÷ x times x leaves just 6; on the right, 3 becomes 3x. The denominator is gone — 6 = 3x, so x = 2. This step is called **clearing denominators**, and the multiplier is the common denominator.

两边乘的式子里含 x，会不会请进假解？We multiplied by an expression containing x — could a fake root sneak in?

第 3 步：验根——把 x 代回去Step 3: Check — put x back

验根质检闸机：x = 2 通过，分母为 0 的根被拦下 — 如果公分母恰好等于 0，“两边乘 0”会凭空造出假解（增根）。所以解完必须验根：把根代回原方程，只查一件事——每个分母都不等于 0。x = 2 代回去，分母是 2，通过；要是算出 x = 0，分母为 0，果断划掉。 If the common denominator happens to equal zero, “multiplying by 0” can invent a fake root (an extraneous solution). So after solving, always **check**: put the root back into the original equation and make sure no denominator equals zero. x = 2 gives a denominator of 2 — it passes; a root like x = 0 would make the denominator zero — strike it out.

🎮 你来分糖（1 分钟）🎮 Your turn: share the candies (1 minute)

拖动滑块调 x，让左右两边一样大；滑到 0 会被分母拦下，每过一关都自动验根。Drag the slider to tune x until both sides match; zero gets blocked by the denominator, and every cleared stage is checked automatically.

一句话记住它：分式方程 = 未知数躲进分母；两边同乘公分母把它请出来，最后一定要验根（分母 ≠ 0）。 Remember it in one line: a rational equation hides the unknown in a denominator; multiply both sides by the common denominator to set it free, then always check that no denominator is zero.

分母里含未知数的方程，就是分式方程An equation with an unknown in a denominator is a rational equation 去分母：两边同乘最简公分母（含 x 的式子，不是只乘一边）Clear denominators: multiply both sides by the common denominator 验根：把根代回原方程，每个分母都不等于 0Check: put the root back — no denominator may equal zero

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