Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the first real step into algebra, where you learn to find the value of an unknown number. The goal is simple: isolate the variable on one side of the equation using inverse operations. In KS3 Cambridge Mathematics, you will meet equations like 2x + 3 = 11, and by the end of this guide you will be able to solve them confidently, showing all working clearly. Let’s explore the key methods step by step.

一元一次方程是代数入门的第一关，你需要求出未知数的值。核心思路很简单：利用逆运算把变量单独移到等号的一边。在 KS3 剑桥数学课程中，你会遇到像 2x + 3 = 11 这样的方程。学完这一篇，你将能自信地解答这类题目，并清楚地写出每一步过程。下面我们就逐步讲解关键方法。

1. What is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation where the variable, often denoted by a letter such as x, is raised to the power of 1. There are no x² or x³ terms. The equation will have an equals sign, and both sides represent the same quantity. For instance, 3x − 4 = 2x + 5 is linear because the highest power of x is 1.

一元一次方程是指未知数（通常用字母 x 表示）的次数为 1 的方程，里面没有 x² 或 x³ 这样的项。方程含有等号，左右两边表示相等的量。例如 3x − 4 = 2x + 5 就是一个一元一次方程，因为 x 的最高次数是 1。

2. Understanding Inverse Operations | 理解逆运算

To solve an equation, you must undo the operations that have been applied to the variable. Addition and subtraction are inverse operations, as are multiplication and division. If the equation says 3x + 2 = 14, you first subtract 2 from both sides, then divide by 3. Always perform the same operation on both sides to keep the equation balanced — think of a pair of old‑fashioned scales.

要解方程，你必须把施加在未知数上的运算一步步撤销。加法和减法互为逆运算，乘法和除法也是。如果方程是 3x + 2 = 14，你会先两边同时减去 2，再同时除以 3。始终在等号两边做相同的运算，才能保持等式平衡——就像一架老式天平一样。

3. Solving with One Operation | 只有一种运算的方程

Start with the simplest type: x + 5 = 12. To get x on its own, subtract 5 from both sides. This gives x = 7. If the equation is x − 6 = 3, add 6 to both sides to obtain x = 9. For multiplication, such as 4x = 20, divide both sides by 4 to find x = 5. And for division, x ÷ 3 = 7, multiply both sides by 3, so x = 21. These one‑step equations build your confidence with inverse operations.

从最简单的类型开始：x + 5 = 12。为了让 x 单独在左边，两边同时减去 5，得到 x = 7。如果方程是 x − 6 = 3，两边加 6 就得出 x = 9。遇到乘法，比如 4x = 20，两边同时除以 4 得 x = 5。遇除法，如 x ÷ 3 = 7，两边同时乘以 3，于是 x = 21。这些一步方程能帮你熟练运用逆运算。

4. Solving Two‑Step Equations | 两步方程求解

Two‑step equations involve two operations. For example, 2x + 3 = 11. First, subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4. Notice the order: undo addition or subtraction first, then undo multiplication or division. This follows the order of operations in reverse — like ‘undoing’ BIDMAS. Always write your steps vertically, with a new line for each operation, so your working is clear and easy to check.

两步方程包含两种运算。例如 2x + 3 = 11。首先，两边同时减去 3：2x = 8。再同时除以 2：x = 4。请注意顺序：先撤销加减法，再撤销乘除法。这相当于把运算顺序（BIDMAS）倒过来使用。建议纵式书写每一步，每一步新起一行，这样解题过程清晰，检查起来也方便。

5. Solving Equations with Variables on Both Sides | 未知数在等号两侧的方程

Sometimes the variable appears on both sides, like 5x − 2 = 3x + 6. The strategy is to collect all variable terms on one side and all numbers on the other. Subtract 3x from both sides: 2x − 2 = 6. Then add 2: 2x = 8. Divide by 2 to find x = 4. Always aim to get the smaller variable term moved first — it reduces the chance of negative coefficients, though negatives are fine with practice.

有时未知数同时出现在等号两边，例如 5x − 2 = 3x + 6。策略是把所有带未知数的项移到一边，把纯数字移到另一边。两边同时减去 3x：2x − 2 = 6。然后加 2：2x = 8。除以 2 得 x = 4。通常先移较小的未知数项，这样能减少出现负系数的机会，当然熟练之后负数也完全没问题。

6. Dealing with Brackets | 含有括号的方程

When brackets appear, expand them first using the distributive law. For instance, 3(2x − 1) = 15 becomes 6x − 3 = 15. Then add 3 to both sides: 6x = 18. Divide by 6 to get x = 3. If the equation has more complex bracketing, such as 2(3x + 1) = 4(x − 2), expand both sides: 6x + 2 = 4x − 8. Then collect like terms: 2x = −10, so x = −5. Always be careful with negative signs when expanding — multiplying two negatives gives a positive.

遇到括号时，先用乘法分配律展开。例如 3(2x − 1) = 15 展开得 6x − 3 = 15。然后两边加 3：6x = 18，除以 6 得 x = 3。如果方程括号更复杂，如 2(3x + 1) = 4(x − 2)，两边分别展开：6x + 2 = 4x − 8。再移项合并同类项：2x = −10，解得 x = −5。展开时务必注意负号——负负得正。

7. Using Balancing to Check Your Answer | 用代回检验答案

After finding a solution, always substitute it back into the original equation to verify. For x = 4 in the equation 2x + 3 = 11, left side becomes 2(4) + 3 = 8 + 3 = 11, which matches the right side. This step is crucial — it proves your solution is correct and catches any arithmetic mistakes. In exams, making a habit of checking can save valuable marks.

求出解之后，一定把它代回原方程检验。对于方程 2x + 3 = 11 的解 x = 4，左边等于 2×4 + 3 = 8 + 3 = 11，与右边相等。这一步非常关键——它能证明你的解是正确的，并帮你发现计算错误。考试中养成检验的习惯，能避免丢分。

8. Equations Involving Fractions | 含分数的方程

When an equation contains fractions, it is often easiest to eliminate the denominators early on. Take (x/3) + 2 = 5. Multiply every term by 3: x + 6 = 15. Then x = 9. For an equation like (2x + 1)/4 = 3, multiply both sides by 4: 2x + 1 = 12, so 2x = 11 and x = 5.5 or 11/2. If there are multiple fractions, find the lowest common denominator and multiply each term. Always apply the multiplication to the entire side, not just selected parts.

当方程里有分数时，通常先消去分母最简单。例如 (x/3) + 2 = 5，每一项都乘以 3：x + 6 = 15，得 x = 9。对于 (2x + 1)/4 = 3，两边同乘 4：2x + 1 = 12，所以 2x = 11，x = 5.5 或 11/2。若有多个分母，找出最小公分母，再乘以各项。注意，乘法必须作用于整个一边，不能只挑部分来乘。

9. Forming Equations from Word Problems | 根据应用题建立方程

Real‑life problems often require you to write an equation before solving. Read the problem carefully, identify the unknown quantity, and let x represent it. Translate each phrase: “three more than a number” becomes x + 3, “twice a number” is 2x, “the sum of a number and 7 is 15” translates to x + 7 = 15. Then solve the equation you have formed. Always answer the question in context — for example, ‘the number is 8’, not just ‘x = 8’.

现实生活问题常常需要你先列出方程再求解。仔细读题，找出未知量，用 x 表示。转译每一句话：“比一个数多 3”就是 x + 3，“一个数的两倍”是 2x，“一个数与 7 的和是 15”可译为 x + 7 = 15。然后解出你建立的方程。最后要结合题意作答——比如“这个数是 8”，而不只是写“x = 8”。

10. Common Mistakes and How to Avoid Them | 常见错误及如何避免

Many students forget to keep the equation balanced, applying an operation to only one side. Others mishandle negative signs, especially when subtracting a negative. A classic error is writing 3x = 15 as x = 15 − 3 instead of dividing. To avoid these, write every step neatly, do the same thing to both sides, and double‑check signs. Practising with simple numbers first builds a strong foundation.

很多学生容易忘记保持等式平衡，只在一边做运算。还有的同学处理负号时容易出错，尤其是减去负数的时候。一个经典错误是把 3x = 15 写成 x = 15 − 3，而没有用除法。要避免这些错误，每一步都要工整书写，等号两边做同样的处理，并仔细检查符号。先从简单的数字练习，打好基础。

11. Summary of the Solving Strategy | 解题策略总结

The golden rule: whatever you do to one side, do to the other. Simplify each side first — expand brackets, combine like terms. Then move variable terms to one side and constants to the other by adding or subtracting. Finally, divide or multiply to find the unknown. This systematic approach works for any linear equation, no matter how complicated it looks at first glance.

黄金法则是：对等式一边做什么，对另一边也做同样的事。先化简每一侧——展开括号，合并同类项。然后通过加减法把含未知数的项移到一边，常数项移到另一边。最后用除法或乘法求出未知数。这种系统的解题思路适用于任何一元一次方程，无论它初看起来有多复杂。

12. Practice Makes Progress | 练习成就进步

Now that you have the tools, practise with a variety of equations — those with fractions, brackets, and variables on both sides. Start with small numbers, then move to more challenging ones. Use online resources, past paper questions, or create your own equations to solve with a friend. The more you practise, the more automatic the process becomes, and soon you’ll tackle equations with ease in your KS3 assessments and beyond.

现在你已经掌握了这些工具，接下来就用多种方程来练习吧——包括带分数、带括号、未知数在等号两边的题型。可以先从小数字入手，再挑战更难的题目。利用线上资源、历年真题，或者自己编题和朋友一起解。练得越多，解题过程就越自动化，很快你就能在 KS3 考试乃至以后的学习中轻松应对各类方程。

Published by TutorHao | Cambridge KS3 Mathematics Revision Series | aleveler.com

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