📚 Mastering Linear Equations at KS3 | KS3 线性方程完全掌握
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in nearly every topic, from number problems to geometry, and they prepare you for more advanced algebra at IGCSE. This article explains how to solve linear equations step by step, with clear examples and common exam-style questions.
线性方程是 KS3 数学中最重要的基础内容之一。它们几乎出现在每一个主题中，从数字问题到几何问题，并且为你 IGCSE 阶段更高级的代数学习做好准备。本文逐步讲解如何解线性方程，配有清晰的示例和常见考试题型。
1. What is a Linear Equation? | 什么是线性方程？
cause the highest power of x is 1. Linear equations can have one solution, no solution, or infinitely many solutions, but at KS3 you will usually work with equations that have exactly one solution.
线性方程是指未知数（通常用 x 表示）的指数只为 1 的方程。例如，2x + 3 = 11 就是一个线性方程，因为 x 的最高次数是 1。线性方程可能有一个解、无解或无穷多个解，但在 KS3 阶段你通常会遇到恰好有一个解的方程。
nd c are constants and a ≠ 0. Solving the equation means finding the value of x that makes both sides equal.
一元线性方程的一般形式是 ax + b = c，其中 a、b 和 c 是常数且 a ≠ 0。解方程就是求出使等号两边相等的 x 的值。
2. The Balance Method | 天平法
Think of an equation as a balanced scale. Both sides must always have the same value. Whatever you do to one side, you must do exactly the same to the other side to keep the balance.
可以把方程想象成一个平衡的天平。两边必须始终保持相同的值。无论你对一边做什么操作，都必须对另一边做完全相同的操作，才能保持平衡。
multiplication and division. For example, to solve x + 4 = 9, subtract 4 from both sides.
天平法使用逆运算来分离 x。加法和减法互为逆运算，乘法和除法也互为逆运算。例如，解 x + 4 = 9 时，两边同时减去 4。
Always write each step on a new line. This shows the examiner your method and helps you avoid mistakes. For example:
每一步要写在新的行上。这样能向阅卷人展示你的解题过程，也有助于避免错误。例如：
x + 4 = 9
x = 9 - 4
x = 5
3. Solving One-Step Equations | 解一步方程
One-step equations require just one inverse operation to solve. They usually involve addition, subtraction, multiplication or division.
一步方程只需要一个逆运算就能解出。它们通常涉及加法、减法、乘法或除法。
- Addition: x + 7 = 15 → x = 15 - 7 → x = 8 | 加法：x + 7 = 15 → x = 15 - 7 → x = 8
- Subtraction: x - 6 = 4 → x = 4 + 6 → x = 10 | 减法：x - 6 = 4 → x = 4 + 6 → x = 10
- Multiplication: 3x = 21 → x = 21 ÷ 3 → x = 7 | 乘法：3x = 21 → x = 21 ÷ 3 → x = 7
- Division: x ÷ 5 = 6 → x = 6 × 5 → x = 30 | 除法：x ÷ 5 = 6 → x = 6 × 5 → x = 30
ck into the original equation. For x = 8 in x + 7 = 15, we have 8 + 7 = 15, which is true.
解完后，一定要把答案代回原方程检验。例如 x = 8 代入 x + 7 = 15，得到 8 + 7 = 15，成立。
4. Solving Two-Step Equations | 解两步方程
volves multiplication by 2 and then subtraction of 3. To solve, reverse the order of operations: undo addition or subtraction first, then undo multiplication or division.
两步方程涉及两个运算。例如，2x - 3 = 7 先乘以 2，再减去 3。解方程时，要按相反顺序进行逆运算：先去加减，再去乘除。
Worked example: Solve 2x - 3 = 7.
解题示范：解 2x - 3 = 7。
2x - 3 = 7
2x = 7 + 3
2x = 10
x = 10 ÷ 2
x = 5
Subtract 2 from both sides first to get 5x = 25, then divide both sides by 5 to get x = 5.
另一个例子：解 5x + 2 = 27。先将两边同时减去 2，得到 5x = 25，然后两边同时除以 5，得到 x = 5。
Notice that the order matters: always undo the constant term before dividing by the coefficient of x. If you divide too early, you create fractions unnecessarily.
注意顺序很重要：一定要先消去常数项，再除以 x 的系数。如果太早除以，会产生不必要的分数。
5. Equations with Brackets | 含括号的方程
ibutive law: a(b + c) = ab + ac. Then solve the resulting two-step or multi-step equation.
当方程含有括号时，要先用分配律展开：
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