📚 Cambridge KS3 Maths: Solving Linear Equations (p89_2 Exercise) | 剑桥 KS3 数学：解一元一次方程（p89_2 练习）
ons with brackets, equations with the unknown on both sides, and simple fractional equations. It is designed to match the skills tested in exercises such as p89_2 and to build confidence step by step.
线性方程是剑桥 KS3 数学课程中最重要的主题之一。它们几乎出现在每一次评估中，从课堂测验到阶段末考试。本文讲解解一元一次方程的核心方法，包括带括号的方程、未知数在两边的情况以及简单的分数方程。本文内容对应 p89_2 等练习所考查的技能，旨在帮助学生逐步建立信心。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation is an algebraic statement that contains one unknown variable, usually written as x, and an equals sign. The word 'linear' means that the highest power of the variable is 1. The standard form can be written as:
线性方程是一个含有未知数（通常写作 x）和等号的代数等式。‘线性’一词表示未知数的最高次数为 1。标准形式可以写作：
ax + b = c
ause 7 + 5 = 12. The solution must keep the equation balanced, just like a pair of scales.
在这个形式中，a、b 和 c 代表已知数字，而 x 是需要求出的未知值。解一元一次方程就是找出使等号两边相等的唯一 x 值。例如，在方程 x + 5 = 12 中，解是 x = 7，因为 7 + 5 = 12。方程的解必须保持等号两边平衡，就像一架天平一样。
2. Inverse Operations | 逆运算
The most powerful tool for solving linear equations is the idea of inverse operations. Addition and subtraction are inverse operations, just as multiplication and division are inverse operations. To isolate the unknown, we perform the opposite operation on both sides of the equation at the same time. This keeps the equation balanced and moves us closer to the value of x.
解一元一次方程最有力的工具是逆运算的概念。加法和减法互为逆运算，乘法和除法也互为逆运算。为了把未知数单独留在等号一边，我们同时对等号两边进行相反的运算。这样可以保持方程平衡，并使我们更接近 x 的值。
| Operation | Inverse operation | 中文对照 |
|---|---|---|
| + 4 | − 4 | 加 4 的逆运算是减 4 |
| − 7 | + 7 | 减 7 的逆运算是加 7 |
| × 3 | ÷ 3 | 乘 3 的逆运算是除以 3 |
| ÷ 5 | × 5 | 除以 5 的逆运算是乘 5 |
Remember that whatever operation you do to one side of an equation, you must also do to the other side. If you only change one side, the equation is no longer balanced. This rule is often called the golden rule of algebra and it is the reason why every correct solution follows the same logical pattern.
请记住，无论你对等式的一边进行什么运算，都必须对另一边进行同样的运算。如果你只改变一边，方程就不再平衡。这条规则通常被称为代数的黄金法则，它也是每一个正确解法都遵循相同逻辑模式的原因。
3. Solving Basic Equations | 解基本方程
move the number 7 that is added to x. The inverse of adding 7 is subtracting 7, so we subtract 7 from both sides:
让我们从一个简单的加法方程开始。要求解 x + 7 = 15，我们需要去掉加在 x 上的数字 7。加 7 的逆运算是减 7，因此我们在等号两边同时减去 7：
x + 7 − 7 = 15 − 7
x = 8
is multiplied by x. The inverse operation is division by 3, so we divide both sides by 3:
对于乘法方程，例如 3x = 21，系数 3 与 x 相乘。它的逆运算是除以 3，因此我们在等号两边同时除以 3：
3x ÷ 3 = 21 ÷ 3
x = 7
When an equation involves both addition and multiplication, we undo them in the reverse order of the order of operations. BIDMAS tells us that multiplication should be done before addition, so when solving equations we reverse BIDMAS: first deal with addition or subtraction, then deal with multiplication or division.
当一个方程同时包含加法和乘法时，我们要按照运算顺序的相反顺序来逆运算。BIDMAS 法则告诉我们先乘除后加减，所以解方程时要反过来使用 BIDMAS：先处理加减，再处理乘除。
4. Equations with Brackets | 带括号的方程
+ 3) = 14. The first method is to expand the bracket by multiplying each term inside by 2:
括号在 KS3 线性方程中非常常见。解方程 2(x + 3) = 14 有两种主要方法。第一种方法是展开括号，将括号内的每一项都乘以 2：
2x + 6 = 14
Then subtract 6 from both sides and divide by 2:
然后两边同时减 6，再除以 2：
2x = 8
x = 4
The second method is to divide both sides by the number outside the bracket first. This is often faster when the number outside the bracket is a factor of the number on the right-hand side:
第二种方法是先将两边同时除以括号外的数字。当括号外的数字能整除右边的数时，这种方法通常更快：
2(x + 3) ÷ 2 = 14 ÷ 2
x + 3 = 7
x = 4
Both methods give the same answer. Choose the method that looks quicker for each question, but always write your working clearly so that marks can be awarded for method in exams.
两种方法得到的答案相同。每道题选择看起来更快的方法，但在考试中一定要清楚地写出计算过程，以便获得方法分。
5. Unknowns on Both Sides | 未知数在两边
o collect the x terms on one side and the number terms on the other side. To remove the smaller x term, subtract 2x from both sides:
许多 KS3 方程中，未知数出现在等号两边，例如 5x + 2 = 2x + 11。第一步是把含 x 的项移到一边，把数字项移到另一边。为了消去较小的 x 项，两边同时减去 2x：
5x − 2x + 2 = 2x − 2x + 11
3x + 2 = 11
Now the equation is in a simple form. Subtract 2 from both sides and then divide by 3:
现在方程变成了简单的形式。两边同时减 2，然后除以 3：
3x = 9
x = 3
A common mistake is to add x terms to the wrong side. You may move the variable to either side, as long as you perform the same operation on both sides. However, it is usually easier to move the smaller variable term so that the final coefficient of x is positive.
一个常见的错误是把 x 项加到了错误的一边。你可以把变量移到任意一边，只要你对两边进行相同的运算即可。但是，通常移走较小的变量项会更简单，这样最后 x 的系数为正。
6. Fractional Equations | 分数方程
1 = 3. The term x/4 means x divided by 4. To isolate x, first subtract 1 from both sides:
分数方程看起来可能很难，但它们遵循相同的原理。考虑方程 x/4 + 1 = 3。x/4 这一项表示 x 除以 4。为了把 x 单独留在一边，首先两边同时减 1：
x/4 = 2
Now x is being divided by 4. The inverse operation is multiplication by 4, so multiply both sides by 4:
现在 x 被 4 除。它的逆运算是乘以 4，因此两边同时乘以 4：
(x/4) × 4 = 2 × 4
x = 8
For equations with more than one fraction, you can multiply every term by the lowest common denominator to clear the fractions in one step. This often makes the equation much easier to solve. Always check that your final answer does not make any denominator zero.
对于含有多个分数的方程，你可以把每一项都乘以最小公分母，一步去掉所有分母。这通常会使方程更容易求解。最后一定要检查你的答案是否使任何分母为零。
7. Checking Your Solution | 检查答案
to the same number. For example, if you solve 2x + 3 = 13 and get x = 5, check as follows:
求出 x 的值后，检查这个值是否正确非常重要。你可以把这个值代回原方程进行验证。等号两边化简后应该得到相同的数字。例如，如果你解方程 2x + 3 = 13 得到 x = 5，可以这样检查：
2(5) + 3 = 10 + 3 = 13
The left-hand side equals the right-hand side, so the solution is correct. Checking your solution is useful in exams because it can help you catch small arithmetic errors. It also gives you confidence that your answer is accurate before moving on to the next question.
左边等于右边，所以这个解是正确的。在考试中检查答案非常有用，因为它可以帮助你发现小的计算错误。它还能让你在继续做下一题之前对自己的答案有信心。
8. Common Mistakes | 常见错误
Students often lose marks on linear equations because of a few repeated mistakes. The most common errors are listed below, together with the correct approach:
学生在解线性方程时经常因为一些重复出现的错误而失分。以下是最常见的错误以及正确的做法：
- Forgetting to do the same operation on both sides. Always apply every step to both the left-hand side and the right-hand side.
- 忘记对等号两边进行相同的运算。每一步都必须同时应用于左边和右边。
- Incorrect sign when moving a term. For example, changing 5 to −5 without a proper inverse step. Use inverse operations carefully.
- 移项时符号错误。例如，在没有进行正确逆运算的情况下把 5 变成 −5。请小心使用逆运算。
- s writing 2(x + 3) = 2x + 5. Multiply every term inside the bracket by the factor outside.
- 展开括号时出错，例如把 2(x + 3) 写成 2x + 5。必须用括号外的因数乘以括号内的每一项。
- Dividing at the wrong time. Remember to undo addition and subtraction before multiplication and division unless using the shortcut of dividing through a bracket.
- 除法的时机错误。除非使用先除以括号外因数的简便方法，否则要先逆算加减，再逆算乘除。
Reading the question carefully and writing every step on a new line will help you avoid these mistakes. A tidy layout also makes it easier for examiners to give you method marks.
仔细读题并把每一步写在新的一行可以帮助你避免这些错误。整洁的书写格式也便于考官给你方法分。
9. Word Problems | 文字题
Linear equations often appear inside word problems. The key skill is to translate the English sentence into an algebraic equation before solving it. Let the unknown number be x, then build the equation step by step. For example, the sentence 'three times a number plus five equals twenty' becomes:
线性方程经常出现在文字题中。关键技能是先把你看到的文字句子转化为代数方程，然后再求解。设未知数为 x，然后逐步建立方程。例如，句子‘一个数的三倍加五等于二十’可以写成：
3x + 5 = 20
Subtract 5 from both sides and divide by 3 to find the number:
两边同时减 5，再除以 3，即可求出这个数：
3x = 15
x = 5
Other common word problems involve ages, perimeter, cost, or consecutive numbers. In every case, identify the unknown quantity, write an equation that describes the relationship, then solve it using the same inverse-operation methods. Always answer the question with a sentence, not just a number.
其他常见的文字题涉及年龄、周长、费用或连续整数。在每种情况下，都要先确定未知量，写出描述数量关系的方程，然后用同样的逆运算方法求解。最后一定要用完整的句子回答问题，而不只是给出一个数字。
10. Practice and Exam Tips | 练习与考试技巧
To become confident with linear equations, regular practice is essential. Start with basic one-step equations, then move to two-step equations, equations with brackets, and equations with variables on both sides. As you improve, try mixed exercises under timed conditions so that you can work accurately and efficiently.
要想熟练掌握线性方程，定期练习是必不可少的。从简单的一步方程开始，然后练习两步方程、带括号的方程以及未知数在等号两边的方程。随着能力的提高，尝试在限时条件下做混合练习，以便在考试中做得又准又快。
| Exam tip | 考试技巧 |
|---|---|
| Show every step of working, even for easy questions. | 即使题目简单，也要写出每一步计算过程。 |
| Keep the equals signs in a vertical line. | 让等号在每一行中垂直对齐。 |
| Check your answer by substitution. | 用代回原方程的方法检查答案。 |
| If time remains, go back to the hardest questions. | 如果还有时间，回头检查最难的题目。 |
Linear equations are a core part of the Cambridge KS3 mathematics syllabus. By mastering the inverse operations, understanding brackets, and practising word problems, you will build a strong foundation for more advanced algebra in later stages.
线性方程是剑桥 KS3 数学课程的核心内容之一。通过掌握逆运算、理解括号并练习文字题，你将为进一步学习更高阶段的代数打下坚实的基础。
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