📚 Solving Linear Equations: Brackets and Fractions | 解线性方程：括号与分数
This tutorial walks through the types of linear equations found in Cambridge KS3 Maths, page 133, Exercise 2. You will learn to expand brackets, clear fractions, balance both sides, and check solutions using substitution.
本教程梳理剑桥 KS3 数学第133页练习2中的线性方程类型。你将学习展开括号、去分母、平衡方程两边，并用代入法检验解。
1. What Makes an Equation Linear? | 什么是线性方程？
A linear equation has variables raised only to the power 1. There are no x² terms, no square roots, and no products like xy.
线性方程中的变量只出现一次方。没有 x² 项、没有平方根，也没有 xy 这样的乘积项。
ar. The expression 2(x − 1) = x + 4 is also linear because expanding gives 2x − 2 = x + 4.
例如，3x + 5 = 20 是线性方程。表达式 2(x − 1) = x + 4 也是线性方程，因为展开后得到 2x − 2 = x + 4。
+ d, where a, b, c, d are constants. Recognising this structure is the first step to choosing a reliable method.
在第133页练习2中，每道题都可以改写成 ax + b = cx + d 的形式，其中 a、b、c、d 是常数。识别这种结构是选择可靠解法的第一步。
2. Reading Exercise 2 on Page 133 | 阅读第133页练习2
Exercise 2 usually mixes equations with brackets, fractions, and variables on both sides. Read each question carefully to identify the first operation to undo.
练习2通常混合了含括号、含分数以及两边有变量的方程。仔细读题，判断第一步需要逆运算的操作。
A useful habit is to circle the unknown, underline the equals sign, and box any brackets or fraction bars before solving. This makes the structure visible.
一个有用的习惯是：解题前圈出未知数，在等号下画线，并用方框标出括号或分数线。这样可以让结构更清晰。
fraction bar groups 3x + 6 as one block. Treat it as a single package until it is cleared.
例如，在 (3x + 6)/2 = 9 中，分数线把 3x + 6 组合成一个整体。在去分母之前，把它当作一个整体处理。
3. Expanding Brackets Before Solving | 先展开括号再求解
law: a(b + c) = ab + ac. This removes the grouping and reveals the variable term clearly.
如果方程含有括号，首先用分配律展开：a(b + c) = ab + ac。这样可以去掉括号，清楚地显示出变量项。
gives 3x + 12 = 27. Subtract 12 from both sides to get 3x = 15. Divide by 3 to find x = 5.
解方程 3(x + 4) = 27。展开得到 3x + 12 = 27。两边同时减去 12，得到 3x = 15。再除以 3，得到 x = 5。
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
e signs outside brackets. For 4 − 2(x − 1) = 10, expand −2(x − 1) as −2x + 2, not −2x − 2.
注意括号外的负号。对于 4 − 2(x − 1) = 10，应把 −2(x − 1) 展开为 −2x + 2，而不是 −2x − 2。
Once all brackets are expanded, collect like terms before applying inverse operations. This reduces mistakes in later steps.
展开所有括号后，先合并同类项，再进行逆运算。这样可以减少后续步骤中的错误。
4. Clearing Fractions by Multiplication | 通过乘法去分母
When an equation contains a fraction, multiply every term on both sides by the denominator. This clears the fraction and makes the equation easier to solve.
当方程含有分数时，把两边每一项都乘以分母。这样可以去掉分母，使方程更容易求解。
multiply both sides by 3 to get 2x + 4 = 18. Then subtract 4 to obtain 2x = 14, so x = 7.
对于 (2x + 4)/3 = 6，两边同时乘以 3，得到 2x + 4 = 18。然后减去 4，得到 2x = 14，所以 x = 7。
(2x + 4)/3 = 6 → 2x + 4 = 18 → 2x = 14 → x = 7
If there are two fractions, multiply
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