📚 IGCSE Maths Core: Linear Equations and Inequalities | IGCSE 数学核心：线性方程与不等式
Linear equations and inequalities are the backbone of IGCSE algebra. This article reviews the key skills you need: solving one-step and two-step equations, expanding brackets, factorising common factors, dealing with fractions, and representing inequalities on a number line. Work through the examples and then test yourself with exam-style questions.
线性方程和不等式是 IGCSE 代数的基础。本文梳理你需要掌握的关键技能：解一步和两步方程、展开括号、提取公因式、处理分数方程，以及在数轴上表示不等式。请认真学习例题，再用考试风格的问题进行自测。
1. Solving One-Step Equations | 解一步方程
A one-step equation is solved by performing the opposite operation to both sides. If a number is added, subtract it. If a number is multiplied, divide by it. The golden rule is: whatever you do to one side, you must do to the other side.
一步方程通过对方程两边进行相反的运算来求解。如果加上一个数，就减去它；如果乘以一个数，就除以它。黄金法则是：对一边做什么运算，对另一边也必须做同样的运算。
x + 5 = 12
x = 12 − 5
x = 7
ubstituting x = 7 back into the original equation: 7 + 5 = 12, so the solution is correct.
把 x = 7 代回原方程检验：7 + 5 = 12，因此解是正确的。
2. Solving Two-Step Equations | 解两步方程
ion or subtraction first, then multiplication or division. For example, solve 2x + 3 = 11.
解两步方程时，按相反顺序消去运算：先处理加减，再处理乘除。例如，解 2x + 3 = 11。
2x + 3 = 11
2x = 11 − 3
2x = 8
x = 8 ÷ 2
x = 4
Always perform addition or subtraction before multiplication or division when reversing the equation. This order is the reverse of the order of operations.
逆推方程时，总是先做加减，再做乘除。这个顺序与运算顺序相反。
Try another: 3x − 4 = 14. First add 4 to both sides, then divide by 3.
再试一个：3x − 4 = 14。先两边加 4，再除以 3。
3x − 4 = 14
3x = 18
x = 6
3. Expanding Brackets | 展开括号
Expanding brackets means multiplying each term inside the bracket by the term outside. For example, expand 3(x + 4).
展开括号就是把括号内的每一项都乘以括号外的项。例如，展开 3(x + 4)。
3(x + 4) = 3 × x + 3 × 4 = 3x + 12
With negative coefficients, be careful with signs: −2(x − 5) = −2x + 10.
涉及负系数时，要特别注意符号：−2(x − 5) = −2x + 10。
You also need to expand double brackets such as (x + 2)(x + 5). Multiply every term in the first bracket by every term in the second bracket, then collect like terms.
你还需要展开双括号，例如 (x + 2)(x + 5)。把第一个括号中的每一项分别乘以第二个括号中的每一项，然后合并同类项。
(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10
4. Factorising by Common Factor | 提取公因式
Factorising is the reverse of expanding. Look for the greatest common factor of all terms and write it outside a bracket. For example, factorise 6x + 9.
因式分解是展开括号的逆运算。找出所有项的最大公因式，把它写在括号外。例如，分解 6x + 9。
6x + 9 = 3(2x + 3)
by expanding your answer. Here, 3(2x + 3) = 6x + 9, which matches the original expression.
一定要通过展开检验答案。这里 3(2x + 3) = 6x + 9，与原式一致。
For 8x² + 12x, the highest common factor is 4x, so 8x² + 12x = 4x(2x + 3).
对于 8x² + 12x，最大公因式是 4x，所以 8x² + 12x = 4x(2x + 3)。
5. Equations with Fractions | 含分数方程
ons, multiply every term by the lowest common denominator. For example, solve x/3 + 2 = 5.
解含分数的方程时，把每一项都乘以最小公分母。例如，解 x/3 + 2 = 5。
x/3 + 2 = 5
x/3 = 5 − 2
x/3 = 3
x = 3 × 3
x = 9
ly all terms by 3 at the start: x + 6 = 15, then x = 9. Both methods give the same result.
也可以在开始时把所有项乘以 3：x + 6 = 15，然后得到 x = 9。两种方法结果相同。
If the equation has two fractions, find the lowest common multiple of the denominators and multiply every term by it.
如果方程中有两个分数，先求出分母的最小公倍数，然后把每一项都乘以这个公倍数。
6. Forming Equations from Word Problems | 文字题列方程
Many IGCSE questions ask you to translate a word problem into an equation. Identify the unknown, give it a letter, and write an equation using the given relationships.
许多 IGCSE 题目要求把文字题转化为方程。设未知数为字母，根据题目给出的关系写出方程。
Example: The sum of three consecutive integers is 42. Find the integers. Let the first integer be x, so the next two are x + 1 and x + 2.
例题：三个连续整数的和是 42，求这些整数。设第一个整数为 x，则后两个为 x + 1 和 x + 2。
x + (x + 1) + (x + 2) = 42
3x + 3 = 42
3x = 39
x = 13
The integers are 13, 14 and 15. Check: 13 + 14 + 15 = 42.
这些整数是 13、14 和 15。检验：13 + 14 + 15 = 42。
Always define your variable clearly at the start. This helps you earn method marks even if the final answer is not fully correct.
一定要在开头清晰地定义变量。即使最终答案不完全正确，这也能帮助你拿到方法分。
7. Introduction to Inequalities | 不等式入门
nd ≥. Unlike equations, inequalities have a range of solutions rather than a single value.
不等式用符号 <、>、≤ 和 ≥ 比较两个表达式。与方程不同，不等式的解是一个范围，而不是一个单一的值。
| Symbol | Meaning | Example |
|---|---|---|
| < | less than | x < 5 |
| > | greater thanPublished by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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