📚 Solving Linear Equations: Step-by-Step | 解一元一次方程：分步指南
Linear equations form the foundation of algebra and appear regularly in KS3 Cambridge Mathematics. An equation tells us that two expressions are equal, and solving it means finding the value of the unknown variable, usually represented by a letter such as x. This article will guide you through the key methods for solving linear equations, from simple one-step operations to problems involving brackets and variables on both sides. Understanding each step will build your confidence and help you avoid common mistakes.
一元一次方程是代数的基础，在 KS3 剑桥数学中经常出现。方程告诉我们两个表达式相等，解方程就是找出未知变量（通常用字母如 x 表示）的值。本文将从简单的一步运算方程开始，一直到包含括号和两边变量的题目，逐步指导你掌握求解线性方程的关键方法。透彻理解每一步将建立你的信心，并帮助你避开常见错误。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation in which the variable appears only to the first power. There are no terms like x² or x³. The general form can be written as ax + b = c, where a, b and c are known numbers and x is the unknown. For example, 2x + 3 = 11 and x − 5 = 7 are both linear equations. The graph of a linear equation in one variable is a point on a number line, but in two variables it forms a straight line – hence the name ‘linear’.
一元一次方程是指变量只出现一次幂的方程，没有 x² 或 x³ 这样的项。其一般形式可写作 ax + b = c，其中 a、b、c 是已知数，x 是未知数。例如，2x + 3 = 11 和 x − 5 = 7 都属于一元一次方程。单变量线性方程在数轴上的图像是一个点，而两变量的图像则是一条直线——因此得名“线性”。
2. The Balance Method | 天平法
Think of an equation as a perfectly balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep it balanced. This idea is called the balance method, and it is the golden rule for solving equations. If you add, subtract, multiply or divide one side by a number, you must apply the same operation to the other side. The aim is to isolate the variable on one side, leaving a number on the other.
把方程想象成一台完全平衡的天平。你对一端所做的任何操作，必须对另一端做完全相同的操作，才能保持平衡。这一理念称为天平法，是解方程的黄金法则。如果你在一端加上、减去、乘以或除以一个数，你必须对另一端进行相同的运算。目标是让变量单独留在等式的一边，另一边是一个数值。
The inverse (opposite) operations you will use repeatedly are:
| Operation | Inverse Operation |
|---|---|
| Addition (+) | Subtraction (−) |
| Subtraction (−) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
你将反复用到的逆运算如下：
| 运算 | 逆运算 |
|---|---|
| 加法 (+) | 减法 (−) |
| 减法 (−) | 加法 (+) |
| 乘法 (×) | 除法 (÷) |
| 除法 (÷) | 乘法 (×) |
3. Solving Basic Equations: x + a = b | 解基本方程：x + a = b
When a number is added to the variable, we reverse the operation by subtracting that number from both sides. Consider the equation:
x + 3 = 7
Subtract 3 from both sides to isolate x:
x + 3 − 3 = 7 − 3
x = 4
The solution is x = 4. Always write the final answer clearly and check your working.
当一个数加上变量时，我们通过在两边同时减去该数来逆转运算。考虑方程：
x + 3 = 7
两边同时减去 3 以分离 x：
x + 3 − 3 = 7 − 3
x = 4
解为 x = 4。要始终清晰书写最终答案，并检查你的解答过程。
4. Equations with Subtraction: x − a = b | 含减法的方程：x − a = b
If a number is subtracted from the variable, we add the same number to both sides. For example:
x − 5 = 2
Add 5 to each side:
x − 5 + 5 = 2 + 5
x = 7
This works because adding 5 cancels out the −5, leaving x alone on the left.
如果从变量中减去一个数，我们就在两边同时加上该数。例如：
x − 5 = 2
每边加上 5：
x − 5 + 5 = 2 + 5
x = 7
之所以能这样，是因为加 5 消掉了 −5，使 x 单独留在左边。
5. Equations with Multiplication: ax = b | 含乘法的方程：ax = b
Here the variable is multiplied by a known number a. The inverse operation is division. If 3x = 15, we divide both sides by 3:
3x = 15
(3x) ÷ 3 = 15 ÷ 3
x = 5
Remember that 3x means 3 × x, so dividing by 3 leaves just x.
这里变量乘以已知数 a，其逆运算是除法。如果 3x = 15，两边同除以 3：
3x = 15
(3x) ÷ 3 = 15 ÷ 3
x = 5
记住 3x 表示 3 × x，所以除以 3 后只剩下 x。
6. Equations with Division: x ÷ a = b | 含除法的方程：x ÷ a = b
When the variable is divided by a number, we multiply both sides by that number. For instance:
x ÷ 4 = 3
Multiply each side by 4:
(x ÷ 4) × 4 = 3 × 4
x = 12
Multiplication and division are inverse operations; applying multiplication cancels the division.
当变量除以一个数时，我们在两边同时乘以该数。例如：
x ÷ 4 = 3
每边乘以 4：
(x ÷ 4) × 4 = 3 × 4
x = 12
乘法和除法互为逆运算；运用乘法可以消去除法。
7. Two-Step Equations | 两步方程
Many equations require two inverse operations. Solve 2x + 3 = 11 by reversing the addition first, then the multiplication. Step 1: subtract 3 from both sides.
2x + 3 − 3 = 11 − 3
2x = 8
Step 2: divide both sides by 2.
2x ÷ 2 = 8 ÷ 2
x = 4
Always undo the addition or subtraction before dealing with the multiplication or division – this follows the reverse order of operations (BIDMAS/BODMAS in reverse).
许多方程需要两步逆运算。解 2x + 3 = 11 时，先逆转加法，再逆转乘法。步骤1：两边减去 3。
2x + 3 − 3 = 11 − 3
2x = 8
步骤2：两边除以 2。
2x ÷ 2 = 8 ÷ 2
x = 4
始终先处理加法或减法，再处理乘法或除法——这遵循逆运算顺序（BIDMAS/BODMAS 的逆向）。
8. Equations with Brackets | 含括号的方程
Brackets tell us that a number outside multiplies everything inside. Expanding the brackets is often the first step. Suppose we have 2(x + 3) = 10. Expand the left side: 2x + 6 = 10. Then follow the two-step method: subtract 6 (→ 2x = 4), and divide by 2 (→ x = 2). Alternatively, you can divide both sides by 2 first, giving x + 3 = 5, then subtract 3 to get x = 2. Both routes are valid, but expanding is usually safer when the equation is more complex.
括号意味着外面的数与括号内的每一项相乘。展开括号通常是第一步。例如 2(x + 3) = 10，先展开左边得 2x + 6 = 10。然后按两步法：减 6 得 2x = 4，再除以 2 得 x = 2。另一种方法是先两边除以 2，得 x + 3 = 5，再减 3 得 x = 2。两种途径均正确，但当方程更复杂时，展开通常更稳妥。
Consider another example: 3(2x − 1) = 15. Expand to get 6x − 3 = 15, add 3 (→ 6x = 18), divide by 6 (→ x = 3).
再举一例：3(2x − 1) = 15。展开得 6x − 3 = 15，加 3 得 6x = 18，除以 6 得 x = 3。
9. Equations with Variables on Both Sides | 含两边变量的方程
When variables appear on both sides, collect all variable terms on one side and constant terms on the other. For example, solve 5x + 2 = 3x + 8. Subtract 3x from both sides to bring variable terms together:
5x − 3x + 2 = 3x − 3x + 8
2x + 2 = 8
Now this is a two-step equation. Subtract 2 (→ 2x = 6) and divide by 2 (→ x = 3).
当变量出现在等式两边时，把所有含变量的项集中到一边，常数项集中到另一边。例如，解 5x + 2 = 3x + 8。两边同时减去 3x，将变量项归拢：
5x − 3x + 2 = 3x − 3x + 8
2x + 2 = 8
现在它变成了两步方程。减 2 得 2x = 6，除以 2 得 x = 3。
Always aim to have fewer variable terms after each step. You can subtract the smaller variable term to keep coefficients positive.
每一步都应使变量项更少。可以减去较小的变量项，这样系数保持为正。
10. Checking Your Solution | 检验你的解
After solving an equation, substitute your answer back into the original equation to verify it. If the left side equals the right side, your solution is correct. For x = 4 in the equation 2x + 3 = 11, we get 2(4) + 3 = 8 + 3 = 11, which matches the right side. This habit catches arithmetic mistakes and reinforces your understanding.
解出方程后，将你的答案代回原方程进行检验。如果左边等于右边，解就是正确的。对于方程 2x + 3 = 11 的解 x = 4，我们有 2(4) + 3 = 8 + 3 = 11，与右边一致。这一习惯能发现计算错误并加深理解。
Checking is especially important when the solution is negative or a fraction. For example, if you solved 3x + 7 = 1 and got x = −2, check: 3(−2) + 7 = −6 + 7 = 1, which is correct.
当解为负数或分数时，检验尤为重要。例如，解 3x + 7 = 1 得 x = −2，检验：3(−2) + 7 = −6 + 7 = 1，正确。
11. Word Problems Involving Equations | 涉及方程的文字题
Many real-life situations can be modelled with linear equations. The key is to translate the written statement into algebraic language. For instance: ‘If I add 5 to a number, the result is 12. Find the number.’ Let the number be x. Then x + 5 = 12. Solve by subtracting 5: x = 7.
现实生活中的许多情境可以用一元一次方程建模。关键是将文字叙述翻译成代数语言。例如：“一个数加上 5 等于 12，求这个数。”设这个数为 x，则 x + 5 = 12，解出 x = 7。
A slightly more complex problem: ‘Three times a number, plus 7, equals 22.’ This gives 3x + 7 = 22. Subtract 7 (→ 3x = 15), then divide by 3 (→ x = 5). Always define your variable clearly at the start.
稍复杂的问题：“一个数的 3 倍再加 7 等于 22。”对应方程为 3x + 7 = 22。先减 7 得 3x = 15，再除以 3 得 x = 5。解题开始时务必明确定义变量。
12. Common Mistakes to Avoid | 常见错误避免
Here is a concise list of typical errors and how to avoid them:
- Unbalanced operations: Adding a number to one side but not the other. Always apply the same change to both sides.
- Sign errors when moving terms: When a term is subtracted from one side, adding it to the other is correct, but many students forget to change its sign properly. Use the balance method step by step.
- Forgetting to multiply all terms inside brackets: In 2(x + 3), multiply both x and 3 by 2, giving 2x + 6, not 2x + 3.
- Incorrect order of inverse operations: Always undo addition/subtraction before multiplication/division unless brackets allow a division shortcut.
- Arithmetic slip-ups: Small mistakes in mental calculation can lead to wrong answers. Take your time and double-check.
以下是一份常见错误清单及避免方法：
- 操作不对称：只在一边加一个数，却忘了另一边。务必对两边同时进行相同操作。
- 移项时符号错误：从一边减去一项，另一边却未能正确改变符号。要严格遵循天平法，一步一检验。
- 忘记乘括号内的所有项：在 2(x + 3) 中，要把 x 和 3 都乘以 2，得到 2x + 6，而不是 2x + 3。
- 逆运算顺序错误：除非括号允许先除，否则总是先做加减逆运算，再做乘除逆运算。
- 计算疏忽：心算时的细小错误会导致错误答案。放慢速度并复查。
Awareness of these pitfalls will make your equation-solving more reliable and accurate.
意识到这些陷阱，将使你做方程题时更可靠、更准确。
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