📚 Solving Linear Equations in Cambridge KS3 Maths | 剑桥KS3数学解一元一次方程
A linear equation is a key topic in Key Stage 3 mathematics and appears regularly in Cambridge Lower Secondary exercises. This article explains how to solve equations step by step, using the type of challenge you might meet in a textbook exercise such as page 275, question 2 from the Cambridge KS3 scheme.
一元一次方程是KS3数学中的核心主题，经常出现在剑桥初中练习中。本文逐步解释如何解方程，并使用你在剑桥KS3教材练习（例如第275页第2题）中可能遇到的题型作为示例。
1. Understanding What a Linear Equation Is | 理解什么是一元一次方程
mple, 3x + 5 = 20 and 2(x - 3) = 8 are linear equations because x is not squared or cubed.
一元一次方程是未知数只出现一次方（即指数为1）的方程。例如 3x + 5 = 20 和 2(x - 3) = 8 都是线性方程，因为 x 没有平方项或立方项。
ally asks you to find the value of the unknown that makes the equation true. This value is called the solution or root of the equation.
在剑桥KS3练习（例如 p275_2.pdf 中的题目）里，通常要求找出使等式成立的未知数的值。这个值叫作方程的解或根。
2. The Balance Method | 天平法
Think of an equation as a balanced set of scales. Whatever you do to one side of the equation, you must do exactly the same to the other side to keep the equation balanced.
把方程想象成一座平衡的天平。你对等式一边做的任何运算，必须对另一边做完全相同的运算，才能保持方程平衡。
This balance method helps you isolate the unknown without changing the solution. You can add, subtract, multiply or divide both sides by the same non-zero number.
天平法帮助你在不改变解的情况下把未知数单独分离出来。你可以在两边同时加、减、乘或除以同一个非零数。
3. Solving One-Step Equations | 解一步方程
tep equation such as x + 7 = 15, subtract 7 from both sides: x + 7 - 7 = 15 - 7, so x = 8.
对于一步方程如 x + 7 = 15，两边同时减去7：x + 7 - 7 = 15 - 7，因此 x = 8。
x + 7 = 15 → x = 8
If the equation is 5x = 35, divide both sides by 5: 5x ÷ 5 = 35 ÷ 5, so x = 7.
如果方程是 5x = 35，两边同时除以5：5x ÷ 5 = 35 ÷ 5，因此 x = 7。
5x = 35 → x = 7
Always use the inverse operation to undo what is being done to x. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.
始终使用逆运算来消除对 x 进行的运算。加法与减法互为逆运算，乘法与除法互为逆运算。
4. Solving Two-Step Equations | 解两步方程
First subtract 3 from both sides to get 4x = 20. Then divide both sides by 4 to get x = 5.
两步方程如 4x + 3 = 23 需要两步逆运算。首先从两边减去3得到 4x = 20。然后两边除以4得到 x = 5。
4x + 3 = 23
4x = 20
x = 5
The key idea is to undo the addition or subtraction first, and then undo the multiplication or division. This order makes the working clear and reduces mistakes.
关键思路是先消去加法或减法，然后再消去乘法或除法。这个顺序让过程更清晰，减少错误。
5. Removing Brackets First | 先去括号
ere is a good reason not to. For example, 2(x - 3) = 8 becomes 2x - 6 = 8 after expansion.
当方程含有括号时，通常先展开括号，除非有充分理由不这样做。例如 2(x - 3) = 8 展开后变成 2x - 6 = 8。
Then add 6 to both sides: 2x = 14. Finally divide by 2: x = 7.
然后两边加6：2x = 14。最后除以2：x = 7。
2(x - 3) = 8
2x - 6 = 8
2x = 14
x = 7
Remember that expanding a bracket must multiply every term inside the bracket by the number outside. So 3(x + 4) becomes 3x + 12, not 3x + 4.
记住展开括号必须用括号外的数乘以括号内的每一项。因此 3(x + 4) 变成 3x + 12，而不是 3x + 4。
6. Unknowns on Both Sides | 未知数在等式两边
s on one side first. For 5x + 2 = 3x + 10, subtract 3x from both sides to get 2x + 2 = 10.
如果未知数出现在等式两边，先把未知数项移到同一边。例如 5x + 2 = 3x + 10，两边减去 3x 得到 2x + 2 = 10。
Then subtract 2 from both sides: 2x = 8, so x = 4.
然后两边减去2：2x = 8，因此 x = 4。
5x + 2 = 3x + 10
2x + 2 = 10
2x = 8
x = 4
Always check that you have moved the correct term. Subtracting the smaller x-term from both sides keeps the coefficient positive, which is usually easier.
始终检查你是否移动了正确的项。从两边减去较小的 x 项可以让系数保持为正，通常更容易计算。
7. Equations Containing Fractions | 含分数的方程
example, x/3 + 1 = 5 can be solved by multiplying every term by 3: x + 3 = 15, so x = 12.
对于含分数的方程，将每一项乘以最小公分母来消去分母。例如 x/3 + 1 = 5，可以把每一项乘以3：x + 3 = 15，因此 x = 12。
x/3 + 1 = 5
x + 3 = 15
x = 12
Always check that the original denominators are not zero. In KS3 work, the denominators are usually positive whole numbers, so this is just a quick habit to develop.
始终检查原分母是否不为零。在KS3学习中，分母通常是正整数，因此这只是一个需要养成的好习惯。
8. Forming an Equation from a Word Problem | 从文字题列方程
Word problems often hide a linear equation. Read the problem carefully, define the unknown as a letter, and translate the words into algebra.
文字题通常隐藏着一元一次方程。仔细阅读题目，用字母定义未知数，并将文字转化为代数式。
Example: 'A number is tripled, then 5 is added, and the result is 26.' Let the number be x. Then 3x + 5 = 26.
示例：“一个数乘以3，再加上5，结果是26。”设这个数为 x。那么 3x + 5 = 26。
Solve: 3x = 21, so x = 7.
解方程：3x = 21，因此 x = 7。
3x + 5 = 26
3x = 21
x = 7
Writing the equation correctly is often more difficult than solving it. Underline the operations described in the problem and put them in the correct order.
正确列出方程通常比解方程更难。在题目中划出描述运算的词语，并按正确顺序转化为代数式。
9. Checking Your Answer by Substitution | 代入检验答案
After finding a solution, substitute it back into the original equation to check that the left-hand side equals the right-hand side.
求出解后，把解代入原方程，检验左边是否等于右边。
5 in 4x + 3 = 23, the left-hand side is 4 × 5 + 3 = 23, which matches the right-hand side.
在 4x + 3 = 23 中令 x = 5，左边为 4 × 5 + 3 = 23，与右边相等。
This checking step is especially important in exams because it catches careless arithmetic errors before you move on to the next question.
代入检验这一步在考试中特别重要，因为它能在你做下一题之前发现粗心的算术错误。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
changing x + 3 = 7 to x = 7 + 3 is wrong because the operation was not applied correctly.
一个常见错误是忘记对等式两边都执行运算。例如把 x + 3 = 7 变成 x = 7 - 3 是正确的，但把 x + 3 = 7 变成 x = 7 + 3 就错了，因为运算方向没有正确使用。
+ 4 = 10, you must write 2x = 6 after subtracting 4, not divide only one term prematurely.
另一个常见错误是只将某一项除以一个数，而应该将整个边除以该数。对于 2x + 4 = 10，你必须先减去4得到 2x = 6，而不是过早地只除某一项。
Also remember that expanding a bracket must multiply every term inside, so 2(x - 3) is 2x - 6, not 2x - 3.
还要记住展开括号必须乘以括号内的每一项，所以 2(x - 3) 是 2x - 6，而不是 2x - 3。
11. Practice Questions with Worked Solutions | 练习题与解答
Try these short practice questions, then compare your working with the solutions below. Each one follows the methods described in the sections above.
尝试以下短练习，然后与下面的解答对比。每道题都遵循上述各节中描述的方法。
- Question 1: 3x + 7 = 22 | 题目1：3x + 7 = 22
- Step: 3x = 15, so x = 5 | 步骤：3x = 15，因此 x = 5
- Question 2: 2(x + 4) = 18 | 题目2：2(x + 4) = 18
- Step: 2x + 8 = 18, 2x = 10, so x = 5 | 步骤：2x + 8 = 18，2x = 10，因此 x = 5
- Question 3: 5x - 3 = 2x + 9 | 题目3：5x - 3 = 2x + 9
- Step: 3x = 12, so x = 4 | 步骤：3x = 12，因此 x = 4
- Question 4: x/4 + 2 = 6 | 题目4：x/4 + 2 = 6
- Step: x/4 = 4, so x = 16 | 步骤：x/4 = 4，因此 x = 16
If you obtained the same solutions, you are ready to tackle similar questions in your Cambridge KS3 exercise file. If not, go back to the relevant section and retry the examples.
如果你得到了相同的答案，就说明你已经准备好应对剑桥KS3练习文件中的类似题目。如果没有，请回到相关小节重新尝试例题。
12. Key Summary | 关键总结
Remember to keep the equation balanced by doing the same operation to both sides. Use inverse operations to isolate the unknown. Always check your answer by substitution.
记住通过两边执行相同运算来保持方程平衡。使用逆运算分离未知数。始终代入检验答案。
Linear equations are a foundation for many later topics such as graphs, sequences and formulae. Building strong solving skills now will help you throughout your Cambridge KS3 course and beyond.
一元一次方程是后续许多主题的基础，例如图像、数列和公式。现在打下扎实的解题基础，将帮助你在整个剑桥KS3课程及以后的学习中受益。
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