📚 Solving Linear Equations: KS3 Cambridge Guide | 解一元一次方程：KS3 剑桥指南
A linear equation is an equation in which the unknown variable, usually written as x or y, is not raised to any power other than 1. These equations are central to the KS3 Cambridge Mathematics curriculum and appear in a wide range of problems, from basic arithmetic puzzles to geometry and real-life contexts.
一元一次方程是指未知数（通常写作 x 或 y）的指数仅为 1 的方程。这类方程是剑桥 KS3 数学课程的核心内容，出现在从基础算术谜题到几何和现实生活情境的各类问题中。
In this guide we will develop a clear method for solving linear equations. You will learn how to use inverse operations, keep an equation balanced, remove brackets, clear fractions, and check your final answer.
在本指南中，我们将建立解一元一次方程的清晰方法。你将学会如何使用逆运算、保持方程平衡、去括号、清除分数以及检查最终答案。
1. What is a Linear Equation? | 什么是线性方程？
r than 1. For example, x + 5 = 12, 3n − 2 = 7, and 2(y + 4) = 18 are all linear equations.
线性方程表示两个表达式相等，并且未知数的指数不大于 1。例如 x + 5 = 12、3n − 2 = 7 和 2(y + 4) = 18 都是一元一次方程。
The goal is to find the value of the unknown that makes the statement true. In many KS3 questions, you are asked to solve an equation and show each step clearly.
目标是求出使等式成立的未知数的值。在许多 KS3 题目中，你都需要解方程并清楚地展示每一步。
2. The Balance Method | 平衡法
The balance method is a visual way to understand equations. Think of the equals sign as the pivot of a balance: whatever you do to one side, you must also do to the other side, otherwise the balance is destroyed.
平衡法是一种理解方程的直观方法。把等号想象成天平的中心支点：你对一边做的任何操作，也必须对另一边做，否则平衡就会被破坏。
to the right side as well. This rule is the foundation of all algebraic equation solving.
例如，如果你在 x + 5 = 12 的左边加 3，那么右边也必须加 3。这条规则是所有代数方程求解的基础。
3. Solving by Adding and Subtracting | 通过加减解方程
eed to remove the number that is added to x. The inverse of adding 7 is subtracting 7, so we subtract 7 from both sides.
要解 x + 7 = 15 这样的方程，我们需要去掉加在 x 上的数。加 7 的逆运算是减 7，所以我们在两边同时减去 7。
x + 7 − 7 = 15 − 7
ly, for x − 4 = 10, we add 4 to both sides because addition is the inverse of subtraction.
这样就得到 x = 8。我们已经将未知数分离出来，方程解出来了。同样地，对于 x − 4 = 10，我们在两边同时加 4，因为加法是减法的逆运算。
Remember: the operation must be done to both sides, not just one side, and you should write each new line clearly in your working.
记住：运算是两边都要做的，不是只做一边，并且你应该在解题过程中把每一行都写清楚。
4. Solving by Multiplying and Dividing | 通过乘除解方程
undo it. For example, in 4x = 28, the x is multiplied by 4, so we divide both sides by 4.
如果未知数乘以一个数，我们就用除法来还原。例如，在 4x = 28 中，x 乘以了 4，所以我们在两边同时除以 4。
4x ÷ 4 = 28 ÷ 4
s x/5 = 3, we multiply both sides by 5, because multiplication is the inverse of division.
这样得到 x = 7。如果未知数除以一个数，例如 x/5 = 3，我们就在两边同时乘以 5，因为乘法是除法的逆运算。
In Cambridge KS3 questions, you will often need to combine these inverse operations when the equation has more than one step.
在剑桥 KS3 题目中，当方程不止一步时，你通常需要结合这些逆运算。
5. Two-Step Equations | 两步方程
ied by 2 and then 3 is added. To solve it, we reverse the operations in the correct order.
两步方程中有两个运算作用于未知数。例如，在 2x + 3 = 11 中，x 先乘以 2，然后加 3。解这个方程时，我们要按正确的顺序逆运算。
rom both sides: 2x = 8. Then remove the multiplication by dividing both sides by 2: x = 4.
首先去掉加法或减法。在两边同时减去 3：2x = 8。然后在两边同时除以 2 去掉乘法：x = 4。
A common mistake is to divide by 2 before subtracting 3; that makes the working more difficult and can lead to errors. Always undo the addition or subtraction first.
一个常见的错误是在减 3 之前先除以 2；这会使解题更困难并可能导致错误。一定要先处理加法或减法。
6. Equations with Brackets | 含括号的方程
nd them first using the distributive law. For example, 3(x + 4) = 27 becomes 3x + 12 = 27.
当方程含有括号时，通常应先用分配律展开。例如，3(x + 4) = 27 变成 3x + 12 = 27。
ion: subtract 12 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.
现在解这个两步方程：两边同时减去 12 得到 3x = 15，然后两边同时除以 3 得到 x = 5。
t, giving x + 4 = 9, and then subtract 4. Both methods are correct when applied carefully.
或者，你也可以先在两边同时除以 3，得到 x + 4 = 9，然后再减去 4。只要认真运用，这两种方法都是正确的。
7. Equations with Fractions | 含分数的方程
ominator. For example, in x/3 + 2 = 6, we first subtract 2 from both sides to get x/3 = 4.
含有分数的方程可以通过消去分母来简化。例如，在 x/3 + 2 = 6 中，我们首先在两边同时减去 2，得到 x/3 = 4。
n is (2x)/5 = 4, multiply both sides by 5, giving 2x = 20, then divide by 2 to get x = 10.
然后两边同时乘以 3，得到 x = 12。如果方程是 (2x)/5 = 4，两边同时乘以 5，得到 2x = 20，再除以 2，得到 x = 10。
Always check whether the fraction bar applies to the whole expression. Writing clear brackets can help you avoid mistakes.
始终检查分数线是否适用于整个表达式。写清括号可以帮助你避免错误。
8. Forming Equations from Word Problems | 从文字题建立方程
Word problems require you to translate a situation into an equation. For example, "I think of a number, double it, add 6, and the result is 20." Let the unknown number be n, so the equation is 2n + 6 = 20.
文字题要求你将一个情境转化为方程。例如，“我想一个数，将它加倍，再加 6，结果是 20。” 设未知数为 n，那么方程就是 2n + 6 = 20。
tracting 6 and then dividing by 2: n = 7. This shows that the number you thought of was 7.
通过先减 6 再除以 2 来解这个方程：n = 7。这说明你最初想的数是 7。
For geometry problems, such as finding a missing side of a triangle when the perimeter is given, you can form an equation by adding all the side expressions and setting them equal to the known perimeter.
对于几何问题，比如在已知周长的情况下求三角形缺失的边，你可以把所有边的表达式相加，并令它们等于已知的周长来建立方程。
9. Checking Your Solution | 检查你的解
olved 5(x − 2) = 20 and got x = 6, substitute 6 into the left side: 5(6 − 2) = 5 × 4 = 20.
解完方程后，你应该总是把答案代回原方程进行检验。例如，如果你解 5(x − 2) = 20 得到 x = 6，就把 6 代入左边：5(6 − 2) = 5 × 4 = 20。
Since the left side equals the right side, the answer is correct. Checking is especially important in KS3 tests because it can catch careless arithmetic errors.
由于左边等于右边，答案是正确的。检验在 KS3 考试中尤其重要，因为它能发现粗心的计算错误。
Make checking part of your routine, and write "Check: 20 = 20" or a short note to show your reasoning.
把检验变成一种习惯，并写出 “检验：20 = 20” 或简短注释来说明你的推理。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One common mistake is forgetting to do the same operation on both sides of the equation. Another is subtracting a number before dealing with multiplication in a two-step equation. Also, when expanding brackets, remember to multiply every term inside the bracket.
常见的错误之一是忘记在方程两边做相同的运算。另一个错误是在两步方程中先处理乘法再处理减法。另外，去括号时，要记得乘以括号内的每一项。
For example, 4(x + 3) is not 4x + 3; it must be 4x + 12. In equations with fractions, avoid multiplying only part of the side by the denominator.
例如，4(x + 3) 不等于 4x + 3；它必须是 4x + 12。对于含分数的方程，避免只把一边的一部分乘以分母。
By practising these steps and checking your work, you will build accuracy and confidence for the Cambridge KS3 mathematics assessments.
通过练习这些步骤并检查你的解答，你将为剑桥 KS3 数学评估建立准确度和信心。
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