📚 Solving Linear Equations with Unknowns on Both Sides | 解含两边未知数的一元一次方程
In KS3 Cambridge Mathematics, solving linear equations is a central algebra skill. It builds on your understanding of inverse operations, simplifying expressions, and keeping equations balanced. This article focuses on equations where the unknown appears on both sides, a topic that often appears in Cambridge Checkpoint tests and school assessments.
在 KS3 剑桥数学中，解一元一次方程是一项核心代数技能。它建立在你对逆运算、化简表达式以及保持等式平衡的理解之上。本文重点讲解未知数出现在等式两边的方程，这是剑桥 Checkpoint 考试和校内测评中经常出现的主题。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is an equation in which the unknown, usually written as x or n, appears only to the first power. It can be written in forms such as ax + b = c or ax + b = cx + d. No x², x³, √x or 1/x terms appear. In KS3 you mostly solve equations by reversing operations while keeping the two sides balanced.
线性方程是未知数（通常写作 x 或 n）只出现一次方的方程。它可以写成 ax + b = c 或 ax + b = cx + d 这样的形式。不会出现 x²、x³、√x 或 1/x 这样的项。在 KS3 阶段，你主要通过逆运算并保持等式两边平衡来解方程。
2. The Balancing Method | 天平平衡法
Think of an equation as a balance scale. The equals sign tells you that the left side and the right side have the same value. Whatever you do to one side, you must also do to the other side, otherwise the balance is broken. This is called the balancing method.
把方程想象成一架天平。等号告诉你左边和右边的值相等。你对一边做的任何操作，必须同时对另一边做，否则天平就会失去平衡。这叫做平衡法。
For example, to solve x + 5 = 12, subtract 5 from both sides. This gives x + 5 − 5 = 12 − 5, so x = 7.
例如，解 x + 5 = 12 时，两边同时减去 5。得到 x + 5 − 5 = 12 − 5，所以 x = 7。
3. Moving Terms Across the Equals Sign | 移项法则
When you move a term from one side of an equation to the other, you use the inverse operation. Addition becomes subtraction, subtraction becomes addition, multiplication becomes division, and division becomes multiplication. This is often called transposition.
当你把一项从方程一边移到另一边时，要使用逆运算。加法变减法，减法变加法，乘法变除法，除法变乘法。这通常称为移项。
For instance, in 3x = 18, x is multiplied by 3. To undo this, divide both sides by 3 to get x = 18 ÷ 3 = 6.
例如，在 3x = 18 中，x 被乘以 3。要撤销这个运算，两边同时除以 3，得到 x = 18 ÷ 3 = 6。
4. Unknowns on Both Sides | 两边都有未知数
Many KS3 equations have unknowns on both sides, such as 5x + 3 = 2x + 15. The first goal is to collect all x terms on one side and all constant terms on the other.
很多 KS3 方程两边都有未知数，例如 5x + 3 = 2x + 15。第一步目标是把所有含 x 的项移到一边，把所有常数项移到另一边。
Start by subtracting 2x from both sides to remove the smaller x term. Then subtract 3 from both sides to isolate the x term. Finally divide by the coefficient of x.
首先两边同时减去 2x，去掉较小的 x 项。然后两边同时减去 3，使含 x 的项单独在一边。最后除以 x 的系数。
5x + 3 = 2x + 15
5x − 2x = 15 − 3
3x = 12
x = 4
The solution is x = 4.
解为 x = 4。
5. Expanding Brackets First | 先去括号
If a bracket appears in an equation, expand it first. For example, 2(x + 3) = x + 10. Multiply both terms inside the bracket by 2: 2x + 6 = x + 10. Then subtract x from both sides and subtract 6.
如果方程中有括号，要先去括号。例如 2(x + 3) = x + 10。把括号内的两项都乘以 2：2x + 6 = x + 10。然后两边同时减去 x，再同时减去 6。
2x + 6 = x + 10
x + 6 = 10
x = 4
Always expand before collecting like terms.
一定要先去括号，再合并同类项。
6. Equations with Fractions | 含有分数的方程
When an equation contains a fraction, multiply every term by the denominator to clear it. For example, x/2 + 3 = 7. Multiplying every term by 2 gives x + 6 = 14, so x = 8.
当方程含有分数时，把每一项乘以分母来去掉分数。例如 x/2 + 3 = 7。每一项乘以 2 得到 x + 6 = 14，所以 x = 8。
If the fraction has a numerator with more than one term, such as (2x + 1)/3 = 5, multiply both sides by 3 first. Then solve 2x + 1 = 15.
如果分子有多项，例如 (2x + 1)/3 = 5，先两边同时乘以 3。然后解 2x + 1 = 15。
(2x + 1) ÷ 3 = 5
2x + 1 = 15
x = 7
7. Checking Solutions by Substitution | 代入检验答案
Always check your solution by substituting it back into the original equation. For 5x + 3 = 2x + 15, substitute x = 4. The left side is 5(4) + 3 = 23, and the right side is 2(4) + 15 = 23. Both sides match.
得到答案后，要把它代回原方程检验。对于 5x + 3 = 2x + 15，代入 x = 4。左边是 5(4) + 3 = 23，右边是 2(4) + 15 = 23。两边相等。
| Left side 左边 | 5(4) + 3 = 23 |
| Right side 右边 | 2(4) + 15 = 23 |
If the two sides are not equal, go back and find the error in your working.
如果两边不相等，就回去检查你解题过程中的错误。
8. Word Problems and Modelling | 应用题与建模
Equations often model real problems. For example, the sum of three consecutive numbers is 72. Let the first number be n. The next two are n + 1 and n + 2.
方程经常用来建立实际问题的模型。例如，三个连续整数的和是 72。设第一个数为 n。接下来两个是 n + 1 和 n + 2。
n + (n + 1) + (n + 2) = 72
Combine like terms: 3n + 3 = 72. Subtract 3, then divide by 3: 3n = 69, so n = 23. The three numbers are 23, 24, and 25.
合并同类项：3n + 3 = 72。减去 3，再除以 3：3n = 69，所以 n = 23。三个数为 23、24 和 25。
When reading a word problem, translate the key phrases carefully. The phrase ‘more than’ suggests addition, ‘less than’ suggests subtraction, and ‘times’ suggests multiplication.
在读应用题时，要仔细翻译关键词。‘比……多’表示加法，‘比……少’表示减法，‘倍’表示乘法。
9. Common Errors to Avoid | 常见错误
Students often lose marks because of small errors. Common mistakes include forgetting to apply an operation to every term, changing the sign of only one term when moving it, and multiplying only part of an expression by the denominator.
学生经常因为小错误失分。常见错误包括：忘记对每一项进行相同的运算；移项时只改变一项的符号；去分母时只乘以表达式的一部分。
- Forgetting to subtract 2x from both sides, not just one side. | 忘记从两边同时减去 2x，而只从一边减。
- Writing 3x + 2 as 3x − 2 when moving it. | 把 3x + 2 移项时写成 3x − 2。
- Multiplying only the fraction by the lowest common denominator but not the whole equation. | 只对分数乘以最小公分母，而没有对整条方程每一项都乘。
10. Practice and Exam Tips | 练习与考试技巧
To prepare for Cambridge Checkpoint or school tests, practise solving equations step by step. Write every step neatly so the examiner can follow your method. Check your final answer by substitution and re-read the question to make sure you answered what was asked.
为了备考剑桥 Checkpoint 或校内测试，要分步骤练习解方程。每一步写清楚，让考官能看懂你的方法。用代入法检查最终答案，并重读题目，确保你回答的是题目所问。
A useful routine is: expand brackets, clear fractions, collect x terms, collect constant terms, divide, and check.
一个实用的流程
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