📚 Cambridge KS3 Mathematics: Linear Equations from p261 Exercise 2 | 剑桥 KS3 数学：从 p261 练习 2 看线性方程
Linear equations are one of the most important topics in the Cambridge KS3 mathematics syllabus. They appear in almost every test, checkpoint paper and later IGCSE module. This article is based on the types of questions often found on page 261, exercise 2 of the Cambridge KS3 practice book. You will learn how to solve one-step, two-step and bracket equations, and how to check your answers confidently.
线性方程是剑桥 KS3 数学大纲中最重要的主题之一。它们几乎出现在每一次测试、Checkpoint 考试以及后续 IGCSE 模块中。本文基于剑桥 KS3 练习册第 261 页练习 2 中常见的题型编写。你将学会如何解一步方程、两步方程和含括号的方程，并能自信地检验答案。
1. What is a Linear Equation? | 什么是线性方程？
ne-variable linear equation is ax + b = c, where a, b and c are numbers and a is not zero.
线性方程是指未知数（通常写作 x）的指数仅为 1 的方程。这意味着方程中没有 x² 项、没有 x³ 项，分母中也没有变量。一元一次线性方程的一般形式是 ax + b = c，其中 a、b 和 c 是数字，且 a 不等于零。
For example, the equation below is a linear equation because the highest power of x is 1.
例如，下面的方程是线性方程，因为 x 的最高次数是 1。
2x + 3 = 11
t makes the left side equal to the right side. In this case, x = 4 because 2 × 4 + 3 = 11.
解线性方程意味着求出使左边等于右边的 x 的值。在这个例子中，x = 4，因为 2 × 4 + 3 = 11。
2. The Balance Method | 天平法
Think of an equation as a balance scale. The left pan and the right pan must always have the same mass. If you add, subtract, multiply or divide one side, you must do exactly the same to the other side to keep the equation balanced.
把方程想象成一个天平。左盘和右盘必须始终保持相同的重量。如果你对一边进行加、减、乘、除，你必须对另一边做完全相同的操作，才能保持方程平衡。
inear equation at KS3 level. For example, to solve x + 4 = 10, subtract 4 from both sides.
这就是天平法。它是 KS3 阶段解每一个线性方程的基础。例如，要解 x + 4 = 10，需要在两边同时减去 4。
x + 4 − 4 = 10 − 4
x = 6
Using the balance method helps you avoid the common mistake of only changing one side of the equation. It also makes each step clear and easy to follow.
使用天平法可以帮助你避免只改变方程一边这一常见错误。它还能让每一步都清晰易循。
3. Solving Equations with Addition and Subtraction | 含加减的方程
dded to the variable, subtract that number from both sides. For example, solve x + 7 = 15.
如果方程中变量加上了一个数，就从两边同时减去这个数。例如，解方程 x + 7 = 15。
x + 7 − 7 = 15 − 7
x = 8
subtracted from the variable, add that number to both sides. For example, solve x − 5 = 9.
如果方程中变量减去了一个数，就在两边同时加上这个数。例如，解方程 x − 5 = 9。
x − 5 + 5 = 9 + 5
x = 14
Always write the operation on both sides before simplifying. This shows the examiner that you understand the balance method and not just guessing the answer.
一定要先写出两边的运算，然后再化简。这可以向阅卷人展示你理解天平法，而不是仅仅猜出答案。
4. Solving Equations with Multiplication and Division | 含乘除的方程
plied by a number, divide both sides by that number to find x. For example, solve 3x = 18.
当变量乘以一个数时，两边同时除以这个数即可求出 x。例如，解方程 3x = 18。
3x ÷ 3 = 18 ÷ 3
x = 6
is divided by a number, multiply both sides by that number. For example, solve x ÷ 4 = 5.
当变量除以一个数时，两边同时乘以这个数。例如，解方程 x ÷ 4 = 5。
x ÷ 4 × 4 = 5 × 4
x = 20
These are called one-step equations because only one operation is needed to isolate x. They are often the first step in longer multi-step questions.
这些被称为一步方程，因为只需要一步运算就可以把 x 单独分离出来。它们通常是更长的多步题中的第一步。
5. Two-Step Equations | 两步方程
15. First remove the addition or subtraction, then remove the multiplication or division.
两步方程包含两种运算，例如乘法和加法。例如，解方程 2x + 3 = 15。先消去加法或减法，再消去乘法或除法。
2x + 3 − 3 = 15 − 3
2x = 12
2x ÷ 2 = 12 ÷ 2
x = 6
The order is important: do the opposite of the order of operations. Since BIDMAS says multiplication comes before addition, the inverse order is to undo addition first, then multiplication.
顺序很重要：要逆着运算顺序来做。因为 BIDMAS 规则说先乘后加，所以逆运算顺序是先消去加法，再消去乘法。
Always write the new equation after each step. This reduces errors and helps you check your solution later.
每一步之后都要写出新的方程。这样可以减少错误，也便于你之后检验解。
6. Equations with Brackets | 含括号的方程
brackets, expand them first using the distributive law. For example, solve 3(x + 2) = 21.
当方程中含有括号时，首先使用分配律展开括号。例如，解方程 3(x + 2) = 21。
3 × x + 3 × 2 = 21
3x + 6 = 21
3x + 6 − 6 = 21 − 6
3x = 15
x = 5
Alternatively, you can divide both sides by 3 first, as long as the whole left side is divisible by 3. Both methods give the same answer, but expanding first is usually safer.
或者，你也可以先两边同时除以 3，只要左边整体都能被 3 整除。两种方法得到的答案相同，但先展开括号通常更稳妥。
with negative signs outside brackets: 2(x − 4) = 10 becomes 2x − 8 = 10, not 2x − 4 = 10.
注意括号外面的负号：2(x − 4) = 10 应变为 2x − 8 = 10，而不是 2x − 4 = 10。
7. Equations with Fraction Coefficients | 分数系数方程
hem, multiply every term on both sides by the denominator. For example, solve x/3 + 2 = 7.
有些线性方程含有分数。要解这类方程，需要把两边每一项都乘以分母。例如，解方程 x/3 + 2 = 7。
3 × (x/3 + 2) = 3 × 7
x + 6 = 21
x = 15
If there are two different denominators, find the lowest common denominator and multiply every term by it. This removes all fractions in one step.
如果有两个不同的分母，先找出最小公分母，然后把每一项都乘以它。这样就可以一步消去所有分数。
Keep the fraction bar as a division sign if you find that clearer. For example, x/3 means x ÷ 3, so multiplying by 3 undoes the division.
如果你觉得更清楚，可以把分数线看作除号。例如，x/3 表示 x ÷ 3，因此乘以 3 就抵消了除法。
8. Word Problems and Applications | 应用题与实际应用
Linear equations are often hidden inside word problems. The first job is to translate the words into an equation. Let x represent the unknown number, then build the equation step by step.
线性方程常常隐藏在文字题中。第一项任务是把文字转换成方程。设 x 表示未知数，然后逐步建立方程。
Example: A number is doubled, then 5 is added, and the result is 17. Find the number.
例子：一个数先加倍，再加上 5，结果是 17。求这个数。
2x + 5 = 17
2x = 12
x = 6
Common word clues: 'added to' means +, 'subtracted from' means −, 'multiplied by' means ×, 'divided by' means ÷, and 'is' or 'equals' means =.
常见的文字线索：“加上”表示 +，“减去”表示 −，“乘以”表示 ×，“除以”表示 ÷，“是”或“等于”表示 =。
Always reread the question after finding x to make sure your answer makes sense in the real situation.
求出 x 之后，一定要重新读一遍题目，确认你的答案在实际情境中是合理的。
9. Common Errors and How to Avoid Them | 常见错误与避免方法
and then x = 6 without showing −4 on both sides can lead to mistakes in harder equations.
一个常见错误是只对方程的一边进行运算。例如，写 x + 4 = 10，然后 x = 10 − 4 是正确的，但在更复杂的方程中，如果不在两边同时写出 −4，就容易出错。
ckets or fractions. For example, 2(x + 3) = 10 should become 2x + 6 = 10, not 2x + 3 = 10.
另一个常见错误是在去括号或去分母时忘记乘以每一项。例如，2(x + 3) = 10 应变为 2x + 6 = 10，而不是 2x + 3 = 10。
Sign errors are also frequent. When subtracting a negative term, change it to addition. Write each step clearly and double-check your signs.
符号错误也很常见。减去一个负项时，要把它变为加法。每一步写清楚，并仔细检查符号。
10. Check Your Answers | 检验你的答案
After solving an equation, always substitute your value of x back into the original equation. The left side and right side must be equal.
解完方程后，一定要把 x 的值代回原方程。左边和右边必须相等。
5x − 4 = 26. We find x = 6. Check: 5 × 6 − 4 = 30 − 4 = 26, which matches the right side.
例子：解方程 5x − 4 = 26。我们得到 x = 6。检验：5 × 6 − 4 = 30 − 4 = 26，与右边相等。
Checking is especially important in Cambridge Checkpoint papers because it helps you catch small arithmetic errors and gain full marks.
在剑桥 Checkpoint 考试中，检验尤为重要，因为它能帮助你发现小的计算错误，从而获得满分。
Make checking a habit after every practice question, even if the question does not ask for it.
即使题目没有要求检验，也要养成每道练习题都检验的习惯。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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