📚 Solving Linear Equations and Inequalities | 一元一次方程与不等式的解法
ce sheet focuses on balancing equations, using inverse operations, expanding brackets, and interpreting inequality symbols on a number line. This revision guide explains each method step by step, with worked examples and common mistakes to avoid.
一元一次方程和不等式是剑桥初中 KS3 数学课程中代数推理的核心内容。p159_2 练习页重点关注方程的平衡、逆运算、括号展开，以及在数轴上表示不等式解集。本复习指南逐步讲解每种方法，配有例题和常见错误提醒。
1. Expressions, Equations and Inequalities | 代数式、方程与不等式
ns, for example 2x − 3 < 7. Knowing the difference helps you decide whether to simplify, solve, or shade a solution set.
代数式（如 3x + 5）不包含等号，因此不能“求解”。方程表示两个代数式相等，例如 3x + 5 = 17。不等式使用 <、>、≤ 或 ≥ 来比较两个代数式，例如 2x − 3 < 7。分清三者有助于判断应当化简、求解还是用数轴表示解集。
3x + 5 = 17 and 2x − 3 < 7
2. The Balance Method | 等式平衡法
ance method. For example, to solve x + 7 = 15, subtract 7 from both sides to obtain x = 8.
方程就像一个平衡的天平。只要对等式两边同时加、减、乘或除以同一个非零数，等式仍然成立，这就是平衡法。例如，解 x + 7 = 15 时，两边同时减去 7，得到 x = 8。
x + 7 = 15 → x = 8
3. Solving Two-Step Linear Equations | 解两步一元一次方程
= 11, subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.
两步方程指的是对变量进行了两次运算。解题关键是逆着运算顺序：先消去加法或减法，再消去乘法或除法。对于 2x + 3 = 11，先两边减 3 得到 2x = 8，再两边除以 2 得到 x = 4。
2x + 3 = 11 → 2x = 8 → x = 4
4. Equations with Brackets | 含括号的方程
d x = 4. You can also divide both sides by 3 first, which is quicker: x + 2 = 6, so x = 4.
当括号前有数字或符号时，要用括号外的因数乘以括号内的每一项。在 3(x + 2) = 18 中，展开左边得到 3x + 6 = 18，然后减 6、除以 3，得到 x = 4。也可以先两边同时除以 3，得到 x + 2 = 6，所以 x = 4，这种方法更快。
3(x + 2) = 18 → 3x + 6 = 18 → x = 4
5. Variables on Both Sides | 含有两侧变量的方程
Always choose the smaller variable term to remove so that the coefficient stays positive.
如果变量同时出现在方程两边，应把变量项集中到一边，常数项集中到另一边。对于 5x = 2x + 9，两边同时减去 2x，得到 3x = 9，再除以 3 得到 x = 3。通常先消去较小的变量项，这样系数可以保持为正。
5x = 2x + 9 → 3x = 9 → x = 3
6. Forming Equations from Word Problems | 由文字题列方程
require you to translate English or everyday statements into algebraic form. For example, 'A number doubled and then increased by 5 gives 21' the unknown, assign a letter, write the operations in order, and then solve the equation.
文字题要求把日常语言转化为代数形式。例如，“一个数乘以 2 后再加 5 得到 21”可以写成 2n + 5 = 21。首先要确定未知数并用字母表示，然后按顺序写出运算过程，最后解方程。
2n + 5 = 21 → 2n = 16 → n = 8
7. Inequality Symbols and the Number Line | 不等式符号与数轴表示
Inequality symbols show the relationship between two quantities. The symbol < means 'less than', > means 'greater than', ≤ means 'less than or equal to', and ≥ means 'greater than or equal to'. On a number line, strict inequalities use an open circle, while ≤ and ≥
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