📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra and appear in almost every area of mathematics. Understanding how to solve them is a key skill at the KS3 level. This article walks you through the essential techniques, from balancing both sides of an equation to handling brackets and fractions. You’ll also see how linear equations model real-world problems, making them not just an abstract exercise but a practical tool.
一元一次方程是代数的基础，在数学的几乎每个领域都会出现。掌握解方程的方法是KS3阶段的一项关键技能。本文带你学习从等式两边平衡到处理括号和分数的各种基本技巧。你还会看到一元一次方程如何用于模拟现实世界的问题，这让它不只是一个抽象的练习，而是一个实用的工具。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown variable, usually represented by x, is only raised to the power of 1. It can be written in the form ax + b = c, where a, b and c are numbers and a ≠ 0. The term ‘linear’ comes from the fact that the graph of such an equation is always a straight line.
一元一次方程是指未知数（通常用 x 表示）的指数仅为一次的方程。它可以写成 ax + b = c 的形式，其中 a、b 和 c 是数，且 a ≠ 0。“一次”这个名称来源于这种方程的图像总是一条直线。
2. The Balancing Method | 天平法
Think of an equation as a perfectly balanced set of scales. Whatever you do to one side of the equation, you must do exactly the same to the other side to keep it balanced. This idea is the backbone of solving any linear equation. For example, to solve x + 3 = 7, you subtract 3 from both sides, leaving x = 4.
把一个方程想象成一个完美平衡的天平。无论你对方程的一边做了什么操作，都必须对另一边做完全相同的操作以保持平衡。这个思想是解任何一元一次方程的核心。例如，解 x + 3 = 7 时，你需要在两边同时减去 3，从而得到 x = 4。
3. Solving Equations with Addition and Subtraction | 通过加减法解方程
When a number is added to or subtracted from the variable, you can isolate the variable by doing the inverse operation. If the number is added, subtract it from both sides. If the number is subtracted, add it to both sides.
当未知数加上或减去一个数时，你可以通过逆运算来隔离未知数。如果是加上的数，那就从两边同时减去它；如果是减去的数，那就给两边同时加上它。
- x + 5 = 12 → Subtract 5 from both sides: x = 7
- x – 4 = 9 → Add 4 to both sides: x = 13
- x + 5 = 12 → 两边同时减5：x = 7
- x – 4 = 9 → 两边同时加4：x = 13
4. Multiplication and Division in Equations | 方程中的乘法和除法
If the variable is multiplied by a number, divide both sides by that number. If the variable is divided by a number, multiply both sides by that number. These steps undo the operation and leave the variable by itself.
如果未知数乘以了一个数，那就两边同时除以那个数；如果未知数被除以了一个数，那就两边同时乘以那个数。这些步骤会撤销运算，让未知数单独留在一边。
- 3x = 21 → Divide both sides by 3: x = 7
- x / 4 = 5 → Multiply both sides by 4: x = 20
- 3x = 21 → 两边同时除以3：x = 7
- x / 4 = 5 → 两边同时乘以4：x = 20
5. Combining Operations: Two-Step Equations | 组合运算：两步方程
In many equations, the variable has two operations acting on it, such as multiplication and then addition. To solve, first undo the addition or subtraction, and then undo the multiplication or division. Always reverse the order of operations.
在许多方程中，未知数受到两种运算的影响，例如先乘再加。解这类方程时，首先要撤销加法或减法，然后再撤销乘法或除法。一定要按照运算顺序的逆顺序来操作。
For example: 2x + 3 = 11. First, subtract 3 from both sides to get 2x = 8. Then divide both sides by 2 to get x = 4.
例如：2x + 3 = 11。首先，两边同时减去3，得到 2x = 8。然后两边同时除以2，得到 x = 4。
6. Variables on Both Sides | 未知数在等式两边
Sometimes the unknown appears on both the left and right sides of the equation. In such cases, you must first collect all the variable terms on one side and all the number terms on the other. This is done by adding or subtracting the same variable term from both sides.
有时候未知数会同时出现在等式的左边和右边。在这种情况下，你必须先把所有含未知数的项移到一边，把所有数字项移到另一边。这可以通过在两边同时加上或减去相同的未知数项来实现。
For example: 5x – 4 = 3x + 6. Subtract 3x from both sides to get 2x – 4 = 6. Then add 4 to both sides to get 2x = 10, so x = 5.
例如：5x – 4 = 3x + 6。两边同时减去 3x，得到 2x – 4 = 6。然后两边同时加4，得到 2x = 10，所以 x = 5。
7. Equations Involving Brackets | 带括号的方程
When brackets are present, you must expand them using the distributive law before solving. Multiply the term outside the bracket by each term inside the bracket. Then proceed with the usual steps to isolate the variable.
当方程中含有括号时，必须在求解之前先用分配律展开。把括号外的项乘以括号内的每一项。然后再按照通常的步骤隔离未知数。
For example: 3(x + 2) = 15. Expand to 3x + 6 = 15. Subtract 6 from both sides to get 3x = 9, so x = 3.
例如：3(x + 2) = 15。展开得到 3x + 6 = 15。两边同时减6，得到 3x = 9，所以 x = 3。
8. Equations with Fractions | 含分数的方程
Equations become slightly more complex when fractions are involved. The best strategy is to clear the fractions by multiplying both sides of the equation by the lowest common denominator (LCD) of all the fractions. This turns the equation into an integer equation, which is easier to solve.
当方程中含有分数时，问题会变得稍微复杂一些。最好的策略是通过把等式两边同时乘以所有分母的最小公倍数（LCD）来去掉分数。这会把方程变成一个整数系数的方程，解起来就更容易了。
For example: x/2 + 3 = 5. Multiply both sides by 2: x + 6 = 10. Subtract 6: x = 4.
例如：x/2 + 3 = 5。两边同时乘以2：x + 6 = 10。再减6：x = 4。
9. Checking Your Solution | 验证你的解
Always substitute your answer back into the original equation to check that it is correct. This only takes a moment and can catch mistakes in arithmetic or logic. Replace the variable with your solution and simplify both sides to confirm they are equal.
永远要把你的答案代回原方程进行检查，确保它是正确的。这个过程只需要一会儿，却可以查出算术或逻辑错误。用你的解替换未知数，然后计算两边是否相等。
If you solve 4x – 7 = 9 and get x = 4, then the left side becomes 4(4) – 7 = 16 – 7 = 9, which matches the right side. Correct!
如果你解 4x – 7 = 9 得到 x = 4，那么左边变成 4(4) – 7 = 16 – 7 = 9，与右边相等。正确！
10. Common Mistakes to Avoid | 要避免的常见错误
Some errors occur frequently when learning to solve linear equations. These include forgetting to perform an operation on both sides, making sign errors when moving terms, and incorrectly expanding brackets. Being aware of these pitfalls will help you improve your accuracy.
学习解一元一次方程时，有些错误经常出现。这些错误包括忘记对两边都进行相同的操作、移项时弄错正负号，以及展开括号时出错。意识到这些陷阱将有助于你提高正确率。
| Mistake | Example | Correction |
|---|---|---|
| Only dividing one term | 2x + 4 = 10 → x + 4 = 5 | Divide the whole side: x + 2 = 5 |
| Sign error when moving terms | x – 3 = 7 → x = 7 – 3 | Must add 3: x = 7 + 3 |
| Incomplete expansion | 2(x + 3) = 2x + 3 | 2(x + 3) = 2x + 6 |
中文对照表格：
| 错误 | 举例 | 纠正 |
|---|---|---|
| 只除以一项 | 2x + 4 = 10 → x + 4 = 5 | 应该整个边除以2：x + 2 = 5 |
| 移项时符号错误 | x – 3 = 7 → x = 7 – 3 | 应该加3：x = 7 + 3 |
| 展开不完整 | 2(x + 3) = 2x + 3 | 2(x + 3) = 2x + 6 |
11. Linear Equations in Real Life | 现实生活中的一元一次方程
Linear equations model many everyday situations, such as calculating costs, distances, and conversion rates. For instance, a mobile phone plan charges a fixed monthly fee of £10 plus £0.05 per text message. The total monthly cost C for t texts is C = 0.05t + 10. To find how many texts you sent when the total is £15, you solve 0.05t + 10 = 15 → t = 100.
一元一次方程可以用来模拟许多日常情景，比如计算费用、距离和换算率。例如，一个手机套餐每月固定收费 10 英镑，每条短信收费 0.05 英镑。发送 t 条短信的月总费用 C 为 C = 0.05t + 10。要计算当总费用为 15 英镑时发送了多少条短信，就需要解方程 0.05t + 10 = 15，得到 t = 100。
12. Practice and Exam Tips | 练习与考试技巧
To master linear equations, regular practice is essential. Start with simple one-step equations and gradually move to multi-step ones involving brackets and fractions. In exams, always write out each step clearly to avoid mistakes and earn method marks, even if the final answer is wrong. Time management is also key—don’t spend too long on a single equation.
要掌握一元一次方程，坚持练习是必不可少的。从简单的一步方程开始，逐步过渡到含有括号和分数的多步方程。在考试中，一定要把每个步骤清楚地写出来，这样即使最后答案错了，也能得到步骤分。时间管理也很关键——不要在一个方程上花费太长时间。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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