📚 Mastering Linear Equations and Inequalities for IGCSE Maths | IGCSE 数学：线性方程与不等式完全掌握
Linear equations and inequalities are the backbone of IGCSE algebra. They appear in almost every paper, both as standalone questions and as embedded steps within larger problems on geometry, measures, and real-life contexts. This revision guide walks through essential methods, common pitfalls, and exam-style worked examples.
线性方程和不等式是 IGCSE 代数的基础。它们在几乎所有试卷中都会出现，既可能作为独立题目，也可能作为几何、测量和实际情境题中的解题步骤。本复习指南将带你梳理核心方法、常见易错点以及考试风格的例题。
1. What Is a Linear Equation? | 什么是线性方程？
t can usually be written in the form ax + b = c, where a, b and c are constants and a ≠ 0.
线性方程是指变量只出现一次幂的代数等式。它的图像总是一条直线，通常可以写成 ax + b = c 的形式，其中 a、b、c 为常数且 a ≠ 0。
rm. In IGCSE exams, linear equations often appear as core questions, and they are also embedded in topics such as angles, perimeter, speed, and currency conversion.
例如，3x + 7 = 19 是线性方程，因为 x 的指数为 1；但 x² + 2x = 5 不是线性方程，因为它含有 x² 项。在 IGCSE 考试中，线性方程经常作为核心题出现，也常嵌入到角度、周长、速度和货币兑换等主题中。
2. Solving One-Step Equations | 解一步方程
The simplest linear equations can be solved in one step by applying the inverse operation. Whatever you do to one side of the equation, you must do to the other side. This balancing method keeps the equation true.
最简单的一步线性方程可以通过使用逆运算一步求解。无论对方程的一边做什么，都必须对另一边做同样的操作。这种平衡方法可以保持等式成立。
Addition and subtraction are inverse operations, as are multiplication and division. The table below shows common inverse pairs used in IGCSE solving.
加法和减法互为逆运算，乘法和除法也互为逆运算。下表列出了 IGCSE 解题中常见的逆运算配对。
| Operation | Inverse Operation |
|---|---|
| + a | − a |
| − a | + a |
| × a | ÷ a |
| ÷ a | × a |
Example 1: Solve x + 7 = 15.
例 1：解方程 x + 7 = 15。
x + 7 = 15 ⇒ x = 15 − 7 ⇒ x = 8
Example 2: Solve 5y = 20.
例 2：解方程 5y = 20。
5y = 20 ⇒ y = 20 ÷ 5 ⇒ y = 4
3. Solving Two-Step and Multi-Step Equations | 解两步和多步方程
When an equation has more than one operation, always undo addition or subtraction before multiplication or division. This is the reverse of the normal order of operations and it keeps the working clear.
当方程包含多个运算时，一定要先消去加法或减法，再消去乘法或除法。这是正常运算顺序的逆序，可以使解题过程更清晰。
Example: Solve 2x − 3 = 11.
例题：解方程 2x − 3 = 11。
Add 3 to both sides: 2x = 14.
两边同时加 3：2x = 14。
Divide both sides by 2: x = 7.
两边同时除以 2：x = 7。
5x + 2 = 3x + 12 becomes 2x = 10 after subtracting 3x and 2 from both sides, giving x = 5.
对于较长的方程，在分离变量之前先合并同类项。例如，5x + 2 = 3x + 12 可以在两边同时减去 3x 和 2 后化为 2x = 10，从而得到 x = 5。
4. Equations with Brackets | 含括号的方程
Expand brackets first using the distributive law, then collect like terms before isolating the variable. A common mistake is to forget to multiply every term inside the bracket by the number outside.
先用分配律展开括号，然后合并同类项，最后再分离变量。常见错误是忘记用括号外的数乘括号内的每一项。
Example: Solve 3(x + 2) = 21.
例题：解方程 3(x + 2) = 21。
Expand the bracket: 3x + 6 = 21.
展开括号：3x + 6 = 21。
Subtract 6 from both sides: 3x = 15.
两边同时减 6：3x = 15。
Divide both sides by 3: x = 5.
两边同时除以 3：x = 5。
3(x − 1) = 20 expands to 2x + 8 + 3x − 3 = 20, which simplifies to 5x + 5 = 20, so x = 3.
如果有两个括号，先展开再化简。例如，2(x + 4) + 3(x − 1) = 20 展开后得到 2x + 8 + 3x − 3 = 20，合并后为 5x + 5 = 20，所以 x = 3。
5. Equations with Fractions | 含分数的方程
Multiply every term by the lowest common denominator (LCD) to clear fractions. This often turns a fraction equation into a simple integer equation and reduces arithmetic errors.
将每一项都乘以最小公分母（LCD）以消去分数。这通常可以将含分数的方程化为简单的整数方程，并减少计算错误。
Example 1: Solve (x + 1)/3 = 2.
例 1：解方程 (x + 1)/3 = 2。
Multiply both sides by 3: x + 1 = 6.
两边同时乘以 3：x + 1 = 6。
Subtract 1 from both sides: x = 5.
两边同时减 1：x = 5。
Example 2: Solve (2x)/3 + 1/2 = 5/6. The LCD of 3, 2 and 6 is 6.
例 2：解方程 (2x)/3 + 1/2 = 5/6。3、2 和 6 的最小公分母为 6。
Multiply every term by 6: 4x + 3 = 5.
将每一项乘以 6：4x + 3 = 5。
Subtract 3 from both sides: 4x = 2.
两边同时减 3：4x = 2。
Divide both sides by 4: x = 1/2.
两边同时除以 4：x = 1/2。
6. Forming Equations from Word Problems | 从文字题建立方程
Read the problem carefully, identify the unknown, and assign it a variable. Translate phrases such as "more than", "less than", "total" and "difference" into mathematical symbols, then solve the resulting equation.
仔细阅读题目，找出未知量并用变量表示。将“比……多”、“比……少”、“总计”和“差”等短语转化为数学符号，然后解出所得方程。
Example: A number doubled and then increased by 9 equals 25. Find the number.
例题：一个数翻倍后再加 9 等于 25。求这个数。
Let x be the unknown number. The equation is 2x + 9 = 25.
设未知数为 x。方程为 2x + 9 = 25。
Subtract 9 from both sides: 2x = 16.
两边同时减 9：2x = 16。
Divide both sides by 2: x = 8. The number is 8.
两边同时除以 2：x = 8。这个数是 8。
Always define your variable clearly at the start and check whether the final answer makes sense in the original context.
一定要在开始时清晰地定义变量，并检验最终答案在原题情境中是否合理。
7. Introduction to Linear Inequalities | 线性不等式入门
. Instead of a single solution, an inequality usually produces a range of possible values.
不等式使用 <、>、≤ 或 ≥ 来比较两个表达式。与方程不同，不等式通常会产生一个取值范围的解。
For example, x > 3 means "x is greater than 3", while x ≤ 5 means "x is less than or equal to 5". The table below compares equations and inequalities.
例如，x > 3 表示“x 大于 3”，而 x ≤ 5 表示“x 小于或等于 5”。下表对等式和不等式进行了比较。
| Equation | Inequality |
|---|---|
| Uses = | Uses <, >, ≤, ≥ |
| One specific solution | A range of solutions |
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