📚 Solving Linear Equations: Core Skills for KS3 | 解一元一次方程：KS3 核心技能
Linear equations are one of the most important algebra topics in KS3. They appear in Cambridge Checkpoint tests, in word problems and in later topics such as graphs and formulae. This article explains how to solve them step by step, with clear examples and common errors to avoid.
一元一次方程是 KS3 代数中最重要的主题之一。它们出现在剑桥 Checkpoint 考试、应用题以及后来的图像和公式等内容中。本文将逐步讲解如何解方程，并给出清晰的例子和需要避免的常见错误。
1. What Is a Linear Equation? | 什么是一元一次方程？
b = c, where a, b and c are constants and the variable is raised to the power 1. The word 'linear' ght line. In KS3, you will mostly work with equations such as 3x + 4 = 19 or 2(x − 5) = 8.
一元一次方程的一般形式是 ax + b = c，其中 a、b、c 是常数，变量的指数为 1。'线性' 一词表示方程的图像是一条直线。在 KS3 中，你主要会接触到像 3x + 4 = 19 或 2(x − 5) = 8 这样的方程。
ax + b = c
Linear equations can be simple, two-step, contain brackets, fractions or variables on both sides. The goal is always the same: find the value of the unknown that makes the equation true.
一元一次方程可以是简单的、两步的，也可以含括号、分数或两边都有未知数。目标始终相同：求出使方程成立的未知数的值。
2. The Balance Method | 天平法
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. The inverse operation is the key: addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication.
把方程想象成一台天平。你对一边做的任何操作，必须对另一边做同样的操作，才能保持平衡。逆运算是关键：加法用减法抵消，减法用加法抵消，乘法用除法抵消，除法用乘法抵消。
| Operation 运算 | Inverse operation 逆运算 |
| Addition 加法 (+) | Subtraction 减法 (−) |
| Subtraction 减法 (−) | Addition 加法 (+) |
| Multiplication 乘法 (×) | Division 除法 (÷) |
| Division 除法 (÷) | Multiplication 乘法 (×) |
When you solve an equation, always apply the same operation to both sides. This keeps the equation true and leads you to the correct solution.
解方程时，一定要对等号两边同时施加相同的运算。这样能保持等式成立，并帮助你得到正确的解。
3. Solving One-Step Equations | 解一步方程
A one-step equation requires only one inverse operation to isolate the variable. Look at the operation attached to x and undo it.
一步方程只需要一个逆运算就能把未知数单独留在等号一边。观察与 x 相连接的运算，然后使用它的逆运算。
Example 1: x + 7 = 15
例 1：x + 7 = 15
x + 7 − 7 = 15 − 7
x = 8
Example 2: x − 3 = 10
例 2：x − 3 = 10
x − 3 + 3 = 10 + 3
x = 13
Example 3: 4x = 28
例 3：4x = 28
4x ÷ 4 = 28 ÷ 4
x = 7
Example 4: x ÷ 5 = 6
例 4：x ÷ 5 = 6
x ÷ 5 × 5 = 6 × 5
x = 30
Always write the operation on both sides clearly. This will help you avoid mistakes when equations become longer.
一定要在等号两边清楚地写出相同的运算。当方程变长时，这能帮助你避免错误。
4. Solving Two-Step Equations | 解两步方程
For a two-step equation, undo addition or subtraction first, then undo multiplication or division. This order is the reverse of the order of operations.
对于两步方程，先处理加法或减法，再处理乘法或除法。这个顺序与运算顺序相反。
Example 1: 2x + 3 = 11
例 1：2x + 3 = 11
2x + 3 − 3 = 11 − 3
2x = 8
2x ÷ 2 = 8 ÷ 2
x = 4
Example 2: 5x − 4 = 21
例 2：5x − 4 = 21
5x − 4 + 4 = 21 + 4
5x = 25
5x ÷ 5 = 25 ÷ 5
x = 5
Example 3: x ÷ 3 + 2 = 7
例 3：x ÷ 3 + 2 = 7
x ÷ 3 + 2 − 2 = 7 − 2
x ÷ 3 = 5
x ÷ 3 × 3 = 5 × 3
x = 15
Remember: addition and subtraction are undone before multiplication and division when isolating the variable.
记住：在分离未知数时，加法和减法要先于乘法和除法处理。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive law. Then solve the resulting equation as usual. If there is a common factor on both sides, you may divide first.
当方程含有括号时，先用乘法分配律展开括号。然后像平常一样解得到的方程。如果两边有公因数，也可以先相除。
Example 1: 3(x + 4) = 27
例 1：3(x + 4) = 27
3x + 12 = 27
3x + 12 − 12 = 27 − 12
3x = 15
x = 5
Example 2: 2(x − 5) = 14
例 2：2(x − 5) = 14
2x − 10 = 14
2x − 10 + 10 = 14 + 10
2x = 24
x = 12
Be careful with negative signs when expanding brackets. Multiplying two negatives gives a positive.
展开括号时要小心负号。两个负数相乘得到正数。
6. Equations with Unknowns on Both Sides | 两边都含未知数的方程
When x appears on both sides, use inverse operations to collect all x terms on one side and all number terms on the other side. A good strategy is to eliminate the smaller x term first.
当 x 同时出现在等号两边时，使用逆运算把所有含 x 的项移到一边，把所有数字项移到另一边。一个常用策略是先消去较小的 x 项。
Example 1: 5x + 2 = 2x + 14
例 1：5x + 2 = 2x + 14
5x + 2 − 2x = 2x + 14 − 2x
3x + 2 = 14
3x + 2 − 2 = 14 − 2
3x = 12
x = 4
Example 2: 7x − 3 = 3x + 9
例 2：7x − 3 = 3x + 9
7x − 3 − 3x = 3x + 9 − 3x
4x − 3 = 9
4x − 3 + 3 = 9 + 3
4x = 12
x = 3
Always check that you have moved every term correctly. A missing sign is one of the most common errors in this step.
一定要检查每一项是否都移动正确。漏掉符号是这一步最常见的错误之一。
7. Equations with Fraction Coefficients | 分数系数方程
If x has a fractional coefficient, multiply both sides by the denominator to clear the fraction. Then solve the resulting equation.
如果 x 的系数是分数，将等号两边同乘以分母，消去分数。然后解得到的方程。
Example 1: (2/3)x = 8
例 1：(2/3)x = 8
(2/3)x × 3 = 8 × 3
2x = 24
x = 12
Example 2: x/4 + 3 = 9
例 2：x/4 + 3 = 9
x/4 + 3 − 3 = 9 − 3
x/4 = 6
x/4 × 4 = 6 × 4
x = 24
ly both sides by 5 first, then divide by 3. Alternatively, multiply by the reciprocal 5/3.
对于像 3x/5 = 6 这样的方程，先将两边同乘以 5，再除以 3。也可以直接乘以倒数 5/3。
8. Checking Your Solution | 检验你的解
After finding a solution, substitute it back into the original equation. The left-hand side and right-hand side must be equal. This takes a few seconds and prevents many lost marks.
求出解后，把它代回原方程。等号左边和右边必须相等。这只需几秒钟，却能避免很多失分。
Example: Solve 2x + 3 = 11. We found x = 4.
例：解 2x + 3 = 11。我们得到 x = 4。
Left side: 2(4) + 3 = 8 + 3 = 11
Right side: 11
Left side = Right side ✓
If the two sides are not equal, go back and check each step. A small arithmetic error is usually the cause.
如果两边不相等，返回去检查每一步。通常原因是某个算术小错误。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Here are the most common errors students make when solving linear equations. Recognising them will help you avoid losing marks.
以下是学生在解一元一次方程时最常见的错误。认识这些错误能帮助你避免失分。
- Forgetting to apply an operation to both sides. 忘记对等号两边同时施加运算。
- Adding instead of subtracting the constant term. 处理常数项时该减却加了。
- Losing a negative sign when moving terms across the equals sign. 移项时丢掉了负号。
- Expanding brackets incorrectly, especially with negative multipliers. 展开括号时出错，尤其是乘以负数时。
- Dividing before undoing addition or subtraction. 在消去加法或减法之前就进行了除法。
To avoid these, write one operation on each line and check your signs twice before moving on.
为了避免这些错误，每行只写一个运算，在继续之前检查两遍符号。
10. Word Problems Leading to Linear Equations | 由应用题建立方程
Many KS3 questions ask you to form an equation from a written problem. Identify the unknown, write it as x, then translate the words into algebraic expressions and an equation.
许多 KS3 题目要求你根据文字问题列出方程。先找出未知数，设为 x，然后把文字翻译成代数表达式和方程。
Example: The sum of three consecutive numbers is 42. Find the numbers.
例：三个连续整数的和是 42。求这三个数。
Let the numbers be x, x + 1, x + 2.
x + x + 1 + x + 2 = 42
3x + 3 = 42
3x = 39
x = 13
The three numbers are 13, 14 and 15. Always answer the question in context, not just give x.
这三个数是 13、14 和 15。一定要根据题目情境作答，不要只给出 x。
11. Practice Questions with Answers | 练习题与答案
Try these questions before looking at the answers. Show every step using the balance method.
在查看答案之前先尝试以下题目。使用天平法写出每一步。
| Question 题目 | Answer 答案 |
| x + 9 = 20 | x = 11 |
| 4x − 7 = 13 | x = 5 |
| 3(x + 2) = 21 | x = 5 |
| 6x + 1 = 2x + 17 | x = 4 |
| (2/3)x = 10 | x = 15 |
| x/4 + 3 = 9 | x = 24 |
| 5x − 2 = 3x + 8 | x = 5 |
| 2(x − 4) = 3x − 10 | x = 2 |
After solving, substitute your answer back into the original equation to check it.
解完后，把你的答案代回原方程进行检验。
12. Summary and Exam Tips | 总结与考试技巧
To solve a linear equation, use inverse operations in the correct order and keep the equation balanced. Expand brackets first, clear fractions early, and collect x terms on one side when necessary. Then substitute your answer to check.
解一元一次方程时，按正确顺序使用逆运算并保持等式平衡。先展开括号，尽早消去分数，必要时把含 x 的项移到一边。最后代入答案进行检验。
In the exam, show all working. Even if your final answer is wrong, a correct method can earn method marks. Write one step per line, keep signs clear, and never erase your working.
在考试中，要展示所有解题步骤。即使最终答案错误，正确的方法也能得到方法分。每行只写一步，保持符号清晰，绝不要擦掉解题过程。
Balance, inverse operations, check. 平衡、逆运算、检验。
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