📚 Solving Linear Equations: A Complete KS3 Guide | 解一元一次方程：KS3 完整复习指南
Linear equations are the foundation of algebra at KS3 and the starting point of almost every algebra question in Cambridge Checkpoint, IGCSE and A-Level. They appear in almost every Checkpoint paper, and being able to solve them confidently will help you with graphs, sequences, formulae and word problems. This revision guide explains each method step by step, from one-step equations to equations with unknowns on both sides, and finishes with exam-style practice questions.
线性方程是 KS3 代数的基础，也是剑桥 Checkpoint、IGCSE 乃至 A-Level 中几乎所有代数题目的起点。它们几乎出现在每一份 Checkpoint 试卷中，熟练解方程能帮助你应对图像、数列、公式和应用题。本复习指南从一步方程到未知数在两侧的方程，逐步讲解每种方法，并在最后附上考试风格的练习题。
Every method in this guide is based on one simple idea: whatever you do to one side of an equation, you must do to the other side. Once you understand that idea, solving equations becomes a reliable procedure rather than a guessing game.
本指南中的每一个方法都建立在一个简单的理念之上：你对等式一边做什么运算，就必须对另一边做同样的运算。一旦理解了这个理念，解方程就变成一套可靠的步骤，而不是靠猜。
1. What is a Linear Equation? | 什么是线性方程？
t cubed and not placed under a square root, and it never appears in a denominator such as 1/x. For example, 2x + 3 = 11 is linear, while x² + 2 = 6 is not linear.
线性方程是指未知数（通常写成 x）只出现一次幂的方程。这意味着未知数没有被平方、没有被立方，也没有放在平方根之下，更不会出现在 1/x 这样的分母中。例如 2x + 3 = 11 是线性的，而 x² + 2 = 6 不是线性的。
x + 5 = 12, 3x − 4 = 14, 2(x + 1) = x + 7
These are all linear because the highest power of x is 1. The word "linear" comes from "line" t satisfy a linear equation in two variables, such as y = 2x + 3, you get a straight line.
这些都是线性方程，因为 x 的最高次数是 1。"线性"这个词来自"直线"：如果你把满足二元线性方程（例如 y = 2x + 3）的所有点画出来，你会得到一条直线。
An equation always contains an equals sign, which tells us that the left-hand side and the right-hand side have exactly the same value. An expression such as 3x − 4 has no equals sign, so it cannot be "solved"; it can only be simplified, expanded or factorised.
方程一定含有等号，等号告诉我们左边的值与右边的值完全相等。而像 3x − 4 这样的式子没有等号，因此它不能被"解"，只能被化简、展开或因式分解。
2. Key Vocabulary | 关键术语
Before solving equations, it helps to know the names of the parts. The table below shows the most important words you will meet in Cambridge Checkpoint questions, together with a short explanation and an example.
在解方程之前，了解各部分的名称会很有帮助。下表列出了你在剑桥 Checkpoint 题目中最常遇到的重要词汇，并配有简短解释和例子。
| Variable (变量) | A letter that stands for an unknown number. In 3x + 2 = 14, the variable is x. |
| Coefficient (系数) | The number multiplying the variable. In 3x + 2 = 14, the coefficient of x is 3. |
| Constant (常数项) | number on its own, with no variable attached. In 3x + 2 = 14, the constants are 2 and 14. |
| Term (项) | A part of an expression separated by + or − signs, such as 3x, 2 or 14. |
| Expression (代数式) | A collection of terms with no equals sign, for example 5x − 7. |
| Equation (方程) | A statement that two expressions are equal, for example 5x − 7 = 18. |
| Solution / root (解) | he value of the variable that makes the equation true. If x = 5, then 5x − 7 = 18 is true. |
| LHS and RHS (左边与右边) | The left-hand side and right-hand side of the equals sign. |
| Inverse operation (逆运算) | The operation that undoes another: addition undoes subtraction, division undoes multiplication. |
| Expand (展开) | To multiply out brackets, so 2(x + 3) becomes 2x + 6. |
Learning these words carefully matters, because exam questions often say things like "write an expression for the perimeter" or "state the coefficient of x". If you misread the word "expression" as "equation", you will lose easy marks.
认真掌握这些词汇很重要，因为考试题目常常会说"写出周长的代数式"或"写出 x 的系数"。如果你把"代数式"误读成"方程"，就会白白丢分。
3. The Golden Rule: Keep the Balance | 黄金法则：保持平衡
Think of an equation as a balance scale. The two sides are in perfect balance because they have the same value. If you add 5 to the left pan, you must add 5 to the right pan, otherwise the scale tips. The same is true for subtracting, multiplying and dividing.
把方程想象成一台天平。因为两边数值相等，所以天平完全平衡。如果你在左边的托盘上加 5，就必须在右边的托盘上也加 5，否则天平就会倾斜。减法、乘法和除法也是如此。
This gives us the golden rule of equation solving: whatever operation you perform on one side, you must perform exactly the same operation on the other side. This rule is why we are allowed to "do the same thing to both sides" again and again until the variable stands alone.
这就引出了解方程的黄金法则：你对等式一边做的运算，必须对另一边做完全相同的运算。正因如此，我们才可以不断地"对两边做同样的事"，直到未知数单独留在一边。
ame: to finish with the variable alone on one side, in the form x = a number. We call this "isolating the variable" or "making x the subject".
每一次求解的目标都是一样的：最终让未知数单独留在一边，写成 x = 某数 的形式。我们称之为"把未知数孤立出来"或"把 x 变成主项"。
- Adding the same number to both sides keeps the equation balanced.
- Subtracting the same number from both sides keeps the equation balanced.
- Multiplying or dividing both sides by the same non-zero number keeps the equation balanced.
- 在等式两边同时加上同一个数，等式仍然成立。
- 在等式两边同时减去同一个数，等式仍然成立。
- 在等式两边同时乘以或除以同一个非零数，等式仍然成立。
4. One-Step Equations: Addition and Subtraction | 一步方程：加法与减法
ariable x has had 5 added to it, so we undo the addition by subtracting 5 from both sides.
最简单的方程只需要一步运算就能解出。如果方程是 x + 5 = 12，未知数 x 被加上了 5，所以我们在两边同时减去 5，把加法撤销掉。
x + 5 = 12 → x + 5 − 5 = 12 − 5 → x = 7
e has had 4 subtracted from it, so we undo the subtraction by adding 4 to both sides. Notice that the operation we use is always the inverse (opposite) of the operation in the equation.
如果方程是 x − 4 = 9，未知数被减去了 4，所以我们在两边同时加上 4，把减法撤销掉。注意，我们使用的运算永远是方程中运算的逆运算（相反运算）。
x − 4 = 9 → x − 4 + 4 = 9 + 4 → x = 13
eft-hand side is 13 − 4 = 9, which equals the right-hand side, so the solution is correct.
一定要把答案代回原方程进行检验。对于 x = 13，左边是 13 − 4 = 9，与右边相等，所以解是正确的。
5. One-Step Equations: Multiplication and Division | 一步方程：乘法与除法
g both sides by that number. In 4x = 20, the 4x means 4 × x, so we divide both sides by 4.
当未知数被某个数相乘时，我们在两边同时除以这个数，把乘法撤销掉。在 4x = 20 中，4x 表示 4 × x，所以我们在两边同时除以 4。
4x = 20 → 4x ÷ 4 = 20 ÷ 4 → x = 5
number. In x/3 = 7, x has been divided by 3, so we multiply both sides by 3 to get x = 21.
当未知数被某个数除时，我们在两边同时乘以这个数，把除法撤销掉。在 x/3 = 7 中，x 被 3 除，所以我们在两边同时乘以 3，得到 x = 21。
x/3 = 7 → (x/3) × 3 = 7 × 3 → x = 21
et the negative sign and answer x = 5, which fails the check because −2 × 5 = −10, not 10.
带负系数时要小心。在 −2x = 10 中，两边同时除以 −2，得到 x = −5。一个非常常见的错误是漏掉负号，写成 x = 5，但代回检验会发现 −2 × 5 = −10，并不等于 10。
6. Two-Step Equations | 两步方程
Most Checkpoint equations require two operations. In these equations, the variable has been multiplied (or divided) and then had something added or subtracted. The safe order is: undo the addition or subtraction first, then undo the multiplication or division.
大多数 Checkpoint 方程需要两步运算。在这类方程中，未知数先被乘（或除），然后又被加上或减去某个数。安全的顺序是：先撤销加减法，再撤销乘除法。
First add 4 to both sides to obtain 3x = 18. Then divide both sides by 3 to obtain x = 6.
以 3x − 4 = 14 为例。先在两边加上 4，得到 3x = 18；再在两边除以 3，得到 x = 6。
3x − 4 = 14 → 3x = 18 → x = 6
The same logic works in reverse order if you prefer to think in terms of "functions": the equation 3x − 4 = 14 means "multiply by 3, then subtract 4". To undo it, reverse the steps: "add 4, then divide by 3".
如果你习惯用"函数机器"的思路来思考，也可以反过来理解：方程 3x − 4 = 14 表示"先乘 3，再减 4"。要撤销它，就把步骤反过来："先加 4，再除以 3"。
6. Subtract 2 from both sides to get x/5 = 4, then multiply both sides by 5 to get x = 20.
再看一个先除法的例子：x/5 + 2 = 6。两边先减去 2，得到 x/5 = 4；再两边乘以 5，得到 x = 20。
x/5 + 2 = 6 → x/5 = 4 → x = 20
7. Equations Containing Brackets | 含括号的方程
When an equation contains brackets, you have two reliable options. The first is to expand the brackets and then solve the resulting two-step equation. The second is to divide both sides by the number in front of the bracket first, if that division is exact.
当方程含有括号时，你有两种可靠的方法。第一种是先展开括号，再解得到的两步方程；第二种是如果除以括号前的数能整除，就先在两边同时除以这个数。
t-hand side gives 2x + 2 = x + 7. Subtracting x from both sides gives x + 2 = 7, so x = 5.
以 2(x + 1) = x + 7 为例。展开左边得到 2x + 2 = x + 7；两边同时减去 x 得到 x + 2 = 7，所以 x = 5。
2(x + 1) = x + 7 → 2x + 2 = x + 7 → x = 5
− 2 = 5, so x = 7. Expanding gives the same answer: 4x − 8 = 20, then 4x = 28, then x = 7.
以 4(x − 2) = 20 为例。这里两边同时除以 4 能整除，更快：x − 2 = 5，所以 x = 7。展开也能得到同样的答案：4x − 8 = 20，然后 4x = 28，然后 x = 7。
n error is to write 3(x + 4) = 3x + 4 instead of 3x + 12, which leads to a wrong solution.
展开时请记住，括号外的数要乘括号内的每一项。一个非常常见的错误是把 3(x + 4) 写成 3x + 4，而不是 3x + 12，从而导致解错。
8. Unknowns on Both Sides | 未知数在两侧的方程
ategy is to collect all the variable terms on one side and all the number terms on the other side. Choose the side that keeps the coefficient of x positive, as this usually avoids negative numbers.
有些方程两边都有未知数，例如 5x − 3 = 2x + 9。策略是把所有含未知数的项集中到一边，把所有常数项集中到另一边。选择让 x 的系数保持正数的那一边，这样通常可以避免负数的出现。
from both sides: 3x − 3 = 9. Add 3 to both sides: 3x = 12. Divide both sides by 3: x = 4.
两边同时减去 2x：3x − 3 = 9。两边同时加上 3：3x = 12。两边同时除以 3：x = 4。
→ 3x − 3 = 9 → 3x = 12 → x = 4
side is 5 × 4 − 3 = 17 and the right-hand side is 2 × 4 + 9 = 17. Both sides match, so x = 4 is correct.
做完之后一定要用原方程检验，因为这类题目最容易出现符号错误。当 x = 4 时：左边是 5 × 4 − 3 = 17，右边是 2 × 4 + 9 = 17。两边相等，所以 x = 4 是正确的。
ect terms. For example, 3(x + 2) = 5x − 4 becomes 3x + 6 = 5x − 4, then 10 = 2x, so x = 5.
如果题目既有括号又有未知数在两侧，就先展开括号，再合并同类项。例如 3(x + 2) = 5x − 4 变成 3x + 6 = 5x − 4，然后 10 = 2x，所以 x = 5。
9. Equations with Fractions | 含分数的方程
Fractions look frightening but are easy to handle. The quickest method is to multiply every term by the denominator (or by the lowest common multiple of all denominators) to clear the fractions completely.
分数看起来吓人，其实很容易处理。最快的方法是把每一项都乘以分母（或所有分母的最小公倍数），把分数彻底消掉。
uation x/4 + 1 = 3, subtract 1 to get x/4 = 2, then multiply both sides by 4 to get x = 8.
对于方程 x/4 + 1 = 3，先减去 1 得到 x/4 = 2，再两边同时乘以 4，得到 x = 8。
x/4 + 1 = 3 → x/4 = 2 → x = 8
es by 3 to get 2x + 1 = 15, then subtract 1 to get 2x = 14, then divide by 2 to get x = 7.
当分子整体是一个代数式时，例如 (2x + 1)/3 = 5，两边同时乘以 3 得到 2x + 1 = 15，再减去 1 得到 2x = 14，再除以 2 得到 x = 7。
(2x + 1)/3 = 5 → 2x + 1 = 15 → x = 7
es by 10 to obtain 5x = 2(x + 3), then expand to get 5x = 2x + 6, then solve to get x = 2.
如果两边出现不同的分数，例如 x/2 = (x + 3)/5，两边同时乘以 10，得到 5x = 2(x + 3)，展开得 5x = 2x + 6，再解得 x = 2。
10. Forming Equations from Word Problems | 根据应用题列方程
In Cambridge Checkpoint, many equation questions are word problems. The marks are usually split between forming the equation and solving it, so always write down the equation clearly before you start solving.
在剑桥 Checkpoint 中，很多方程题是应用题。分数通常分为"列方程"和"解方程"两部分，所以在动手求解之前，一定要把方程清楚地写出来。
Example 1: "I think of a number, multiply it by 3 and add 4. The result is 19." Let the number be x, then 3x + 4 = 19. Subtract 4: 3x = 15, so x = 5. The number is 5.
例 1："我想一个数，把它乘以 3 再加上 4，结果是 19。"设这个数为 x，则 3x + 4 = 19。减去 4：3x = 15，所以 x = 5。这个数是 5。
× (length + width), so 2(x + 3 + 5) = 32, giving 2(x + 8) = 32, then x + 8 = 16, so x = 8.
例 2：一个长方形的长是 (x + 3) 厘米，宽是 5 厘米，周长是 32 厘米。周长等于 2 ×（长 + 宽），所以 2(x + 3 + 5) = 32，即 2(x + 8) = 32，于是 x + 8 = 16，所以 x = 8。
48. This simplifies to 3n + 3 = 48, so 3n = 45 and n = 15. The integers are 15, 16 and 17.
例 3：三个连续整数之和是 48。设它们为 n、n + 1 和 n + 2，则 n + (n + 1) + (n + 2) = 48。化简得 3n + 3 = 48，所以 3n = 45，n = 15。这三个整数是 15、16 和 17。
A useful tip: always define your variable in words before writing the equation, for example "Let x be the number of sweets in a small bag." Examiners look for this definition and often award a mark for it.
一个实用技巧：在写方程之前，先用文字定义你的变量，例如"设 x 为一小袋糖果的数量"。阅卷老师会寻找这个定义，并常常为此给分。
11. Checking Answers and Common Mistakes | 检验答案与常见错误
Checking your answer takes less than thirty seconds and can save several marks. Substitute your value back into the original equation and work out both sides separately. If they are equal, your solution is correct; if not, look for a sign error or an arithmetic slip.
检验答案用不到三十秒，却能挽救好几分。把你的答案代回原方程，分别计算两边的值。如果相等，说明解是正确的；如果不相等，请检查是否有符号错误或计算失误。
Here are the mistakes that appear most often in KS3 and Checkpoint scripts:
以下是 KS3 和 Checkpoint 试卷中最常见的错误：
- the equals sign without changing its sign, for example writing 3x + 4 = 19 as 3x = 19 + 4.
- Multiplying only the first term inside a bracket, for example writing 2(x + 3) = 2x + 3.
- Dividing only one side of the equation, which destroys the balance.
- n when dividing by a negative coefficient, such as −2x = 8 giving x = 4 instead of x = −4.
- 把一项移到等号另一边时忘记变号，例如把 3x + 4 = 19 写成 3x = 19 + 4。
- 只乘括号内的第一项，例如把 2(x + 3) 写成 2x + 3。
- 只对等式的一边做除法，破坏了平衡。
- 除以负系数时漏掉负号，例如 −2x = 8 写成 x = 4，而不是 x = −4。
Another good habit is to use a rough estimate. If x is roughly 5, then 3x − 4 should be roughly 11. If your answer gives 60, something has gone wrong long before the final line.
另一个好习惯是做粗略估算。如果 x 大约是 5，那么 3x − 4 应该大约是 11。如果你的答案算出 60，那么问题早在最后一步之前就出现了。
12. Where Linear Equations Lead Next | 线性方程之后的延伸
mx + c shows that y and x are connected by a linear relationship, and the gradient m and intercept c can be found by solving equations.
解线性方程不是一项孤立的技能。它是你在直线图像中要使用的工具：y = mx + c 表示 y 与 x 之间存在线性关系，而斜率 m 和截距 c 都可以通过解方程求得。
Linear equations also appear in sequences. If a sequence has a constant difference d and first term a, the nth term is a + (n − 1)d, and finding which term equals a particular value means solving a linear equation in n.
线性方程也出现在数列中。
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