📚 Mastering Linear Equations | 掌握一元一次方程
Linear equations lie at the very heart of algebra and are a key topic in the Cambridge KS3 Mathematics curriculum. Being able to solve them confidently gives you a toolkit that you will use again and again – from tackling word problems to understanding graphs. This article breaks down everything you need to know, from the simplest one-step equations to trickier ones with brackets and variables on both sides. Work through each section carefully and you will soon feel at home with equations.
线性方程是代数的核心，也是剑桥 KS3 数学课程的重点内容。能够自信地解方程，就掌握了一个会反复用到的工具——无论是在解决应用题还是理解图像时都是如此。本文详细讲解你需要知道的所有内容，从最简单的一步方程到带括号和两边有变量的复杂方程。认真完成每个小节，你很快就会对方程得心应手。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement where two expressions are equal, and the unknown (usually a letter like x or y) is raised to the power of 1. For example, 2x + 3 = 11 is a linear equation. There are no squares, cubes or roots of the unknown – just plain ‘x to the power of 1’. Linear equations can always be written in the form ax + b = c, where a, b and c are numbers.
线性方程是表示两个代数式相等的陈述，其中未知数（通常用 x 或 y 这样的字母表示）的指数为 1。例如，2x + 3 = 11 就是一个线性方程。这里没有未知数的平方、立方或根号——只有“x 的一次方”。线性方程总是可以写成 ax + b = c 的形式，其中 a、b 和 c 都是数字。
In KS3, you will mostly see equations with one unknown, such as x + 7 = 10 or 5y − 2 = 18. The goal is always the same: find the value of the variable that makes the equation balance.
在 KS3 阶段，你见到的大多是含一个未知数的方程，例如 x + 7 = 10 或 5y − 2 = 18。目标始终不变：找到使等式成立的变量的值。
2. Understanding Variables, Constants and Terms | 理解变量、常量与项
Before you start solving, it helps to know the names of the parts. In 3x + 4 = 19, the letter x is the variable. The number 3 is the coefficient (it multiplies the variable). The numbers 4 and 19 are constant terms. Expressions like 3x and 4 are called terms, and the equals sign shows that the left side has the same value as the right side.
在开始解方程之前，了解各部分的名称很有帮助。在 3x + 4 = 19 中，字母 x 是变量。数字 3 是系数（它与变量相乘）。数字 4 和 19 是常量项。像 3x 和 4 这样的式子叫做项，等号表示左边与右边的值相等。
Recognising these parts makes it easier to describe what you are doing when you ‘move’ terms from one side to the other. Instead of saying ‘throw the 4 over’, you can say ‘subtract 4 from both sides to eliminate the constant term on the left’.
认识这些部分后，当你把一个项从一边“移”到另一边时，就能更准确地描述你的操作。你不会再说“把 4 扔过去”，而会说“两边都减去 4，以消去左边的常数项”。
3. The Balance Method | 平衡法
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 is the golden rule of solving equations. If you add 5 to the left, add 5 to the right. If you divide the left by 2, divide the right by 2.
可以把方程想象成一个完全平衡的天平。你对一边做的任何操作，都必须对另一边做完全相同的操作，这样才能保持平衡。这是解方程的黄金法则。如果你在左边加 5，右边也要加 5。如果你在左边除以 2，右边也要除以 2。
Many students attempt to ‘just move it to the other side and change the sign’. While this shortcut works, it is safer to always write down the same operation on both sides until you are very fluent. This reduces mistakes and helps you understand why the method works.
很多学生试图“直接把它移到另一边并变号”。这个捷径虽然有效，但在你非常熟练之前，更稳妥的做法是始终在两边写下相同的运算。这样可以减少错误，并帮助你理解方法背后的道理。
4. Solving One-Step Equations | 解一步方程
One-step equations are the simplest type: you only need one operation to find the answer. For an addition equation like x + 6 = 15, subtract 6 from both sides: x + 6 − 6 = 15 − 6, which gives x = 9. For a subtraction equation like y − 4 = 10, add 4 to both sides: y − 4 + 4 = 10 + 4, so y = 14.
一步方程是最简单的一类：你只需要一步运算就能求出答案。像 x + 6 = 15 这样的加法方程，两边都减去 6：x + 6 − 6 = 15 − 6，得到 x = 9。像 y − 4 = 10 这样的减法方程，两边都加上 4：y − 4 + 4 = 10 + 4，所以 y = 14。
For multiplication equations such as 7p = 42, divide both sides by 7: 7p ÷ 7 = 42 ÷ 7, giving p = 6. For division equations such as q ÷ 3 = 9, multiply both sides by 3: q ÷ 3 × 3 = 9 × 3, so q = 27. Always write the operation on both sides clearly.
像 7p = 42 这样的乘法方程，两边都除以 7：7p ÷ 7 = 42 ÷ 7，得到 p = 6。像 q ÷ 3 = 9 这样的除法方程，两边都乘以 3：q ÷ 3 × 3 = 9 × 3，所以 q = 27。一定要清楚地写出两边的运算。
5. Solving Two-Step Equations | 解两步方程
A two-step equation involves two operations, like 2x + 5 = 17. To solve it, you undo the operations in reverse order of BIDMAS. First, get rid of the addition or subtraction, then the multiplication or division. So for 2x + 5 = 17, subtract 5 from both sides: 2x = 12. Then divide both sides by 2: x = 6.
两步方程涉及两种运算，例如 2x + 5 = 17。解这类方程时，要按与运算优先级相反的顺序进行逆运算。首先处理加法或减法，再处理乘法或除法。因此对于 2x + 5 = 17，先两边减 5：2x = 12。然后两边除以 2：x = 6。
If the equation were (x ÷ 4) − 3 = 2, you would first add 3 to both sides to get x ÷ 4 = 5, and then multiply both sides by 4 to find x = 20. Setting out your working in clear steps is essential to avoid errors.
如果方程是 (x ÷ 4) − 3 = 2，要先两边加 3 得到 x ÷ 4 = 5，再两边乘 4 得出 x = 20。清晰地展示解题步骤对避免错误至关重要。
6. Equations with Brackets | 带括号的方程
When an equation contains brackets, such as 3(2y − 1) = 15, you usually need to expand the brackets first. Multiply each term inside the bracket by the number outside: 3 × 2y gives 6y, and 3 × (−1) gives −3, so the equation becomes 6y − 3 = 15. Then solve as a two-step equation: add 3 to get 6y = 18, divide by 6 to get y = 3.
当方程含有括号时，例如 3(2y − 1) = 15，通常需要先展开括号。将括号外的数乘以括号内的每一项：3 × 2y 得 6y，3 × (−1) 得 −3，方程变为 6y − 3 = 15。然后按两步方程求解：加 3 得 6y = 18，除以 6 得 y = 3。
Sometimes you can divide both sides by the number in front of the bracket first if it is a factor of the other side. In the example above, since 15 is divisible by 3, you could divide both sides by 3 to get 2y − 1 = 5, then add 1 and divide by 2. Both approaches give the same answer; choose the one that feels easier.
有时，如果括号前的数是另一边数的因数，你也可以先两边同时除以这个数。在上面的例子中，因为 15 能被 3 整除，你可以先两边除以 3 得到 2y − 1 = 5，然后加 1 再除以 2。两种方法得出相同答案；选择你觉得更容易的那一种。
7. Equations with Variables on Both Sides | 两边都有变量的方程
Equations like 5x + 2 = 3x + 10 have the variable x on both sides of the equals sign. The strategy is to collect all the variable terms on one side and all the constant terms on the other. A good first step is to subtract the smaller variable term from both sides. Here, subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 10, leaving 2x + 2 = 10. Then subtract 2 and divide by 2 to find x = 4.
像 5x + 2 = 3x + 10 这样的方程，等号两边都有变量 x。解题策略是把所有变量项集中到一边，把所有常数项集中到另一边。一个好的第一步是两边同时减去较小的变量项。这里，两边减去 3x：5x − 3x + 2 = 3x − 3x + 10，剩下 2x + 2 = 10。然后减去 2 再除以 2，得出 x = 4。
Always aim to move the smaller variable term so that you end up with a positive coefficient for the variable. This avoids working with negative numbers unnecessarily. If you accidentally end up with −2x = −8, you can still divide both sides by −2 to get the correct answer, x = 4.
始终要移走较小的变量项，让变量的系数保持正数。这样可以避免不必要的负数运算。如果你不小心得到 −2x = −8，依然可以两边除以 −2 得到正确答案 x = 4。
8. Checking Your Solution | 检验你的解
Once you have found a value for the variable, substitute it back into the original equation to check. For example, if you solved 4(x − 3) = 12 and got x = 6, replace x with 6: left side becomes 4(6 − 3) = 4 × 3 = 12, which matches the right side. If both sides give the same number, your solution is correct.
找到变量的值后，将其代回原方程进行检验。例如，你解出 4(x − 3) = 12 并得到 x = 6，用 6 替换 x：左边变为 4(6 − 3) = 4 × 3 = 12，与右边一致。如果两边得出相同数字，你的解就是正确的。
Checking only takes a few seconds but can catch simple arithmetic mistakes. It is especially useful in exams when you have a little time left. Get into the habit of writing ‘Check: LHS = … RHS = …’ at the end of your working.
检验只需要几秒钟，却能发现简单的计算错误。考试时如果还有一点时间，检验尤其有用。养成在解题末尾写上“检验：左边 = …… 右边 = ……”的习惯。
9. Real-Life Applications | 实际应用
Linear equations are not just abstract puzzles – they model real situations. For example, a plumber charges a £30 call-out fee plus £25 per hour. The cost C for h hours is given by C = 25h + 30. If the total bill is £130, you can set up the equation 25h + 30 = 130. Solving gives h = 4, so the plumber worked for 4 hours.
线性方程不仅仅是抽象的谜题——它们可以模拟真实情境。例如，一位水管工收取 £30 的上门费，外加每小时 £25。工作 h 小时的费用 C 表示为 C = 25h + 30。如果总账单为 £130，你可以建立方程 25h + 30 = 130。解方程得 h = 4，因此水管工工作了 4 小时。
Other KS3 examples include calculating the number of items you can buy with a given budget, finding unknown angles in geometric problems where angles add up to 180° or 360°, and solving distance–speed–time questions. Recognising when to set up an equation is a valuable skill.
KS3 阶段的其他例子包括计算给定预算下可以购买的商品数量、在求和为 180° 或 360° 的几何问题中求未知角度，以及解决距离–速度–时间问题。懂得何时建立方程是一项很有价值的技能。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common mistake is forgetting to perform an operation on both sides. For instance, when solving x/2 = 10, some students write x = 10/2, which is wrong. The correct step is x = 10 × 2, so x = 20. Always ask yourself: ‘Have I done the same to both sides?’
一个常见错误是忘记对两边进行相同运算。例如，解 x/2 = 10 时，有些学生会写成 x = 10/2，这是错的。正确步骤是 x = 10 × 2，所以 x = 20。始终反问自己：“我是否对两边做了相同的操作？”
Another frequent error is mishandling negative signs, especially when expanding brackets like −2(3x − 4). The correct expansion is −6x + 8, not −6x − 8. Take extra care when multiplying negatives. It also helps to write the original equation down again after each step to avoid copying errors.
另一个常见错误是处理负号不当，尤其是在展开括号时，比如 −2(3x − 4)。正确的展开是 −6x + 8，而不是 −6x − 8。做负数乘法时要格外小心。每进行一步都重新写下方程也有助于避免抄写错误。
11. Practice Problems | 练习题
Try these equations to test your understanding. Write out full solutions using the balance method and check your answers. 1) 4y + 7 = 31. 2) 2(a − 5) = 12. 3) 6x − 3 = 2x + 13. 4) (m ÷ 3) + 4 = 10. 5) 5(2p + 1) = 45.
试着解这些方程来检验你的理解。用平衡法写出完整解题过程并检验答案。1) 4y + 7 = 31。2) 2(a − 5) = 12。3) 6x − 3 = 2x + 13。4) (m ÷ 3) + 4 = 10。5) 5(2p + 1) = 45。
Answers: 1) y = 6 (subtract 7, divide by 4). 2) a = 11 (divide by 2, add 5, or expand and solve). 3) x = 4 (subtract 2x, add 3, divide by 4). 4) m = 18 (subtract 4, multiply by 3). 5) p = 4 (divide by 5, subtract 1, divide by 2). If you didn’t get these, go back through the relevant section.
答案：1) y = 6（减 7，除以 4）。2) a = 11（除以 2，加 5；或展开后求解）。3) x = 4（减 2x，加 3，除以 4）。4) m = 18（减 4，乘 3）。5) p = 4（除以 5，减 1，除以 2）。如果你没有得到这些答案，请回到相应小节重新学习。
12. Key Takeaways and Next Steps | 要点总结与后续学习
Solving linear equations is all about balance: whatever you do to one side, do to the other. Start with one-step and two-step equations, then progress to brackets and variables on both sides. Always check your solution. As you gain confidence, you will be able to tackle fractional equations and more complex algebraic problems in higher years.
解线性方程的关键在于平衡：对一边做什么，对另一边就要做什么。从一步和两步方程开始，再进阶到带括号和两边有变量的方程。要经常检验你的解。随着信心的增强，你将能够在更高年级应对分数方程和更复杂的代数问题。
These skills are essential not only for algebra but for topics such as simultaneous equations, linear graphs, and even scientific formulas. Keep practising, write your steps clearly, and remember that every mistake is an opportunity to learn.
这些技能不仅对代数至关重要，对诸如联立方程、直线图像乃至科学公式等主题也同样重要。坚持练习，清晰地写出步骤，并记住：每一个错误都是一次学习的机会。
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