📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra at KS3 Cambridge level. They appear in nearly every checkpoint assessment and help you build the problem-solving skills needed for harder topics such as simultaneous equations, functions, and graphs. In this article, we will revise the key methods for solving one-step, two-step, bracket, and fraction equations, and we will check each solution carefully.
一元一次方程是 KS3 剑桥数学代数部分的基础。它们几乎出现在每一次 checkpoint 测试中，帮助你建立解决更复杂问题所需的技能，例如联立方程、函数和图像。本文将复习解一步方程、两步方程、含括号方程和含分数方程的核心方法，并且认真检验每一个解。
1. What Is a Linear Equation? | 什么是一元一次方程？
3 = 7, 2x − 5 = 9, and 4(x + 1) = 20 are all linear equations because the power of x is 1.
一元一次方程是指未知数只出现一次且次数为一的方程。它的一般形式是 ax + b = c，其中 a、b 和 c 是数字，x 是未知数。例如 x + 3 = 7、2x − 5 = 9 和 4(x + 1) = 20 都是一元一次方程，因为 x 的指数是 1。
In KS3 Cambridge Mathematics, you are expected to solve linear equations using inverse operations, the balance method, and clear algebraic working. The goal is always to isolate the unknown on one side of the equation.
在 KS3 剑桥数学中，你应能运用逆运算、平衡法和清晰的代数步骤来解一元一次方程。目标始终是将未知数单独移到方程的一边。
2. Balancing the Equation | 方程两边平衡
Think of an equation as a balance scale. The left side and the right side must stay equal. Whatever operation you perform on one side, you must also perform on the other side. This is called the balance method. For example, if you add 4 to the left side, you must add 4 to the right side as well.
把方程想象成一架天平。左边和右边必须保持相等。你对一边进行任何运算，另一边也必须进行同样的运算。这就是平衡法。例如，如果你在左边加 4，你也必须在右边加 4。
x + 4 = 10 → x + 4 − 4 = 10 − 4 → x = 6
Here we subtracted 4 from both sides to keep the equation balanced and to find the value of x. This method prevents mistakes and shows clear working for exam marks.
这里我们两边同时减去 4，保持方程平衡，并求出 x 的值。这种方法可以避免错误，并在考试中展示清晰的解题步骤以获得分数。
3. Solving One-Step Equations | 解一步方程
One-step equations need only one inverse operation to find the unknown. If the equation shows addition, use subtraction. If it shows subtraction, use addition. If it shows multiplication, use division. If it shows division, use multiplication.
一步方程只需要一次逆运算就能求出未知数。如果方程中是加法，就用减法；如果是减法，就用加法；如果是乘法，就用除法；如果是除法，就用乘法。
- Solve x + 5 = 12: subtract 5 from both sides, so x = 7. / 解 x + 5 = 12：两边减 5，所以 x = 7。
- Solve x − 3 = 9: add 3 to both sides, so x = 12. / 解 x − 3 = 9：两边加 3，所以 x = 12。
- Solve 3x = 18: divide both sides by 3, so x = 6. / 解 3x = 18：两边除以 3，所以 x = 6。
- Solve x ÷ 4 = 5: multiply both sides by 4, so x = 20. / 解 x ÷ 4 = 5：两边乘 4，所以 x = 20。
Always write the operation on both sides in your working. This shows the examiner that you understand the balance method and reduces careless errors.
解题时始终把两边进行的运算写出来。这向阅卷老师表明你理解平衡法，并能减少粗心错误。
4. Solving Two-Step Equations | 解两步方程
ion or subtraction first, and then undo multiplication or division. This is the reverse of the order of operations.
两步方程中未知数带有两种运算，例如 2x + 3 = 11。通常的规则是先处理加法或减法，再处理乘法或除法。这与运算顺序相反。
2x + 3 = 11 → 2x = 11 − 3 → 2x = 8 → x = 8 ÷ 2 = 4
Here we first subtracted 3 from both sides, and then divided both sides by 2. Reversing the order of operations helps you isolate x step by step without making sign errors.
这里我们首先两边减去 3，然后两边除以 2。反过来按照运算顺序处理，可以帮助你一步一步地求出 x，避免符号错误。
divide by 5 to get x = 5. Keep each step on a new line and keep the equals signs aligned.
对于方程 5x − 7 = 18，先加 7 得到 5x = 25，再除以 5 得到 x = 5。每一步换一行，等号保持对齐。
5. Equations with Brackets | 含括号的方程
and the brackets first or divide both sides by the coefficient in front of the bracket. Both methods give the same answer.
当方程含有括号时，例如 3(x + 2) = 15，你可以先展开括号，也可以两边同时除以括号前的系数。两种方法得到相同的答案。
Method 1: Expand the brackets. / 方法 1：展开括号。
3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3
Method 2: Divide by the bracket coefficient. / 方法 2：除以括号前的系数。
3(x + 2) = 15 → x + 2 = 5 → x = 3
In KS3 exams, expanding brackets is often the expected method because it links to simplification and expression work. However, the division method is quicker when the bracket coefficient divides the right side exactly.
在 KS3 考试中，展开括号通常是期望的方法，因为它与化简和代数表达式相关。不过，当括号前的系数能整除右边时，除法方法更快。
6. Equations with Unknowns on Both Sides | 未知数在方程两边
like terms so that x appears on only one side. A reliable method is to subtract the smaller x-term from both sides.
有时未知数同时出现在方程两边，例如 5x + 2 = 3x + 10。关键思想是合并同类项，使 x 只出现在一边。一种可靠的方法是两边同时减去较小的含 x 项。
5x + 2 = 3x + 10 → 5x − 3x + 2 = 10 → 2x + 2 = 10 → 2x = 8 → x = 4
By subtracting 3x from both sides, we keep the x-term positive. This reduces the chance of sign mistakes. You can also subtract the larger x-term, but then you would get negative coefficients, which are harder for many students.
通过两边减去 3x，我们保持 x 项为正。这减少了符号出错的机会。你也可以减去较大的含 x 项，但那样会得到负系数，对许多学生来说更难处理。
Always check your final answer by substituting it back into both sides of the original equation.
始终把最终答案代回原方程两边进行检验。
7. Equations Involving Fractions | 涉及分数的方程
olve these, first multiply every term by the lowest common denominator to clear the fractions. This makes the equation simpler and avoids fraction arithmetic.
一元一次方程可能包含分数，例如 x/3 + 1 = 5 或 (2x)/5 = 4。解这类方程时，先用最小公分母乘以每一项，消去分数。这会使方程更简单，并避免分数运算。
x/3 + 1 = 5 → x/3 = 4 → x = 4 × 3 = 12
tion like x/2 + 1/3 = 2, multiply every term by 6 to get 3x + 2 = 12, then solve normally.
这个例子中，先减去 1 消去常数项，然后乘以 3 即可解出方程。对于更复杂的方程，如 x/2 + 1/3 = 2，每一项都乘以 6 得到 3x + 2 = 12，然后按正常步骤求解。
Writing each multiplication step clearly helps you avoid missing a term, which is a common mistake in checkpoint exams.
清晰地写出每一步乘法运算可以帮助你避免漏乘某一项，这是 checkpoint 考试中的常见错误。
8. Checking Your Solution | 检验你的解
After solving a linear equation, always check your answer by substituting it back into the original equation. Both sides of the equation must give the same value. This is called verification and it is worth doing for every exam question.
解出一元一次方程后，一定要把答案代回原方程进行检验。方程的两边必须得到相同的值。这叫做验算，对每一道考试题都值得做。
e: 2(4) + 3 = 8 + 3 = 11. The left side equals the right side, so the solution is correct.
例如：对于 2x + 3 = 11，我们得到 x = 4。代入检验：2(4) + 3 = 8 + 3 = 11。左边等于右边，所以解是正确的。
If the two sides are not equal, go back through your working and look for sign errors, arithmetic slips, or a missed operation. Checking also helps you build confidence in multi-step questions.
如果两边不相等，回到解题过程中检查符号错误、计算错误或漏掉的运算。检验还能帮助你在多步骤题目中建立信心。
9. Word Problems Leading to Equations | 列方程解应用题
Many KS3 Cambridge questions ask you to form an equation from a written description and then solve it. Start by letting the unknown number be x, then translate words such as 'sum', 'difference', 'product', and 'is equal to' into algebraic operations.
许多 KS3 剑桥数学题要求你根据文字描述列出方程并求解。先设未知数为 x，然后把 '和'、'差'、'积' 和 '等于' 等词语转化为代数运算。
Example: A number multiplied by 5 and then reduced by 7 gives 18. Find the number. / 示例：一个数乘以 5 再减去 7 得到 18。求这个数。
5x − 7 = 18 → 5x = 25 → x = 5
Write the equation exactly as the sentence describes. If you define the variable clearly and work line by line, word problems become simple algebra questions.
按照句子的描述准确地写出方程。只要你清晰地定义变量并一步一步地计算，应用题就会变成简单的代数题。
10. Common Mistakes and Exam Tips | 常见错误与备考建议
Some common mistakes in linear equations include forgetting to do an operation on both sides, incorrectly combining unlike terms, losing a negative sign, and making arithmetic errors when dividing. In bracket equations, students often forget to multiply every term inside the bracket.
解一元一次方程的一些常见错误包括：忘了对两边进行同样的运算、错误地合并不同类项、丢失负号，以及在除法时出现算术错误。在含括号的方程中，学生经常忘记将括号内的每一项都相乘。
- Always use the balance method: never move a term without changing the operation. / 始终使用平衡法：不改变运算就不能随意移项。
- Write one step per line and keep equals signs aligned. / 每一步写一行，等号保持对齐。
- Expand brackets carefully before solving unless you use division. / 除非使用除法，否则先仔细展开括号。
- Check your answer by substitution every time. / 每次都用代入法检验答案。
- If the unknown is negative, multiply or divide by −1 as needed. / 如果未知数为负，根据需要乘以或除以 −1。
In Cambridge checkpoint exams, method marks are awarded for clear working even if the final answer is wrong. So always show your steps and avoid working only in your head.
在剑桥 checkpoint 考试中，即使最终答案错误，清晰的解题过程也能获得方法分。因此一定要展示步骤，不要只在脑海里心算。
11. Try These Practice Questions | 尝试这些练习题
Use the methods from this revision guide to solve the following questions. Try to write balanced working for each one, then check your solutions by substitution.
运用本复习指南中的方法解答以下题目。每道题都写出平衡的解题步骤，然后用代入法检验你的答案。
- Solve x + 9 = 15. / 解 x + 9 = 15。
- Solve 4x − 5 = 19. / 解 4x − 5 = 19。
- Solve 2(x + 3) = 16. / 解 2(x + 3) = 16。
- Solve 7x + 4 = 3x + 20. / 解 7x + 4 = 3x + 20。
- Solve x/5 + 2 = 6. / 解 x/5 + 2 = 6。
, x = 4, x = 20. Compare your working with the steps shown in the relevant sections above.
答案：x = 6、x = 6、x = 5、x = 4、x = 20。把你的解题过程与上面相关小节展示的步骤进行对照。
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