📚 Solving Linear Equations: From Basics to Unknowns on Both Sides | 解一元一次方程：从基础到两边含未知数
Linear equations are a core part of the Cambridge KS3 mathematics curriculum. They appear in number puzzles, geometry problems and real-life situations. Mastering how to balance and solve an equation builds a strong foundation for algebra at IGCSE and beyond.
一元一次方程是剑桥 KS3 数学课程的核心内容。它们出现在数字谜题、几何问题和现实生活情境中。掌握如何平衡并解方程，为 IGCSE 及以后的代数学习打下坚实基础。
1. What is a Linear Equation? | 什么是一元一次方程？
inear because x appears to the power 1. If the variable were squared or cubed, it would not be a linear equation.
一元一次方程是一个代数等式，两个表达式相等，且未知变量的指数为 1。例如 2x + 3 = 11 是线性方程，因为 x 的指数是 1。如果变量是平方或立方，它就不是一元一次方程。
The word 'linear' refers to a straight-line graph when the equation is plotted, but at KS3 we mainly solve equations algebraically.
'线性' 一词指方程绘图后为直线，但在 KS3 阶段我们主要用代数方法解方程。
2. Key Vocabulary and Notation | 关键术语与符号
Terms to know: variable (a letter standing for an unknown number), coefficient (the number multiplying the variable), constant term (a number on its own), and solution (the value that makes the equation true). An equation has an equals sign; an expression does not.
需要掌握的术语：变量（代表未知数的字母）、系数（乘以变量的数）、常数项（单独的数字）、解（使方程成立的值）。方程有等号；表达式没有等号。
ally write division as a fraction, for example x/4 or ¼x, rather than x ÷ 4 in algebraic equations.
3x 这样的记法表示 3 × x。在代数方程中，除法通常写成分数，如 x/4 或 ¼x，而不是 x ÷ 4。
3. Solving Simple One-Step Equations | 解简单一步方程
subtract 5 from both sides to get x = 7. If 4x = 20, divide both sides by 4 to get x = 5.
解一步方程时，两边同时进行逆运算。如果 x + 5 = 12，两边同时减去 5，得到 x = 7。如果 4x = 20，两边同时除以 4，得到 x = 5。
x + 5 = 12 → x = 12 − 5 = 7
4x = 20 → x = 20 ÷ 4 = 5
Always keep the equation balanced: whatever you do to one side, you must do to the other. This is the golden rule of algebra.
始终保持方程平衡：你对一边做什么，对另一边也必须做同样的事。这是代数的黄金法则。
4. Solving Two-Step Equations | 解两步方程
Reverse the order of operations: undo addition/subtraction before multiplication/division.
两步方程涉及两种运算。例如 3x + 2 = 14。先两边减去 2：3x = 12。然后两边除以 3：x = 4。要逆着运算顺序：先消去加减，再消去乘除。
3x + 2 = 14 → 3x = 12 → x = 4
ther example: x/5 − 3 = 2. Add 3 to both sides: x/5 = 5. Multiply both sides by 5: x = 25.
另一个例子：x/5 − 3 = 2。两边加 3：x/5 = 5。两边乘以 5：x = 25。
5. Expanding Brackets First | 先展开括号
or 2(x + 3) = 16, expand to 2x + 6 = 16. Then subtract 6: 2x = 10, and divide by 2: x = 5.
当方程含括号时，先用分配律展开。对于 2(x + 3) = 16，展开为 2x + 6 = 16。然后减 6：2x = 10，再除以 2：x = 5。
2(x + 3) = 16 → 2x + 6 = 16 → 2x = 10 → x = 5
s, be careful: 3(2x − 1) = 15 becomes 6x − 3 = 15. Add 3 to both sides: 6x = 18, so x = 3.
如果有负号，要小心：3(2x − 1) = 15 展开为 6x − 3 = 15。两边加 3：6x = 18，所以 x = 3。
6. Unknowns on Both Sides | 两边都含未知数
subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, and divide by 2: x = 4.
当变量出现在等号两边时，要合并同类项。对于 5x + 2 = 3x + 10，两边减 3x：2x + 2 = 10。再减 2：2x = 8，除以 2：x = 4。
5x + 2 = 3x + 10 → 2x + 2 = 10 → 2x = 8 → x = 4
e. For 2x + 9 = 7x − 1, subtract 2x from both sides: 9 = 5x − 1. Add 1: 10 = 5x, so x = 2.
通常先消去较小的 x 项，使剩余系数为正。对于 2x + 9 = 7x − 1，两边减 2x：9 = 5x − 1。加 1：10 = 5x，所以 x = 2。
7. Equations with Fractions | 含分数的方程
s by 3: x + 3 = 6, so x = 3. For (2x)/5 = 4, multiply both sides by 5: 2x = 20, so x = 10.
解含分数的方程时，每一项都乘以最小公分母。对于 x/3 + 1 = 2，所有项乘以 3：x + 3 = 6，所以 x = 3。对于 (2x)/5 = 4，两边乘以 5：2x = 20，所以 x = 10。
x/3 + 1 = 2 → x + 3 = 6 → x = 3
numerator, put brackets around it before multiplying: (x + 2)/4 = 3 → x + 2 = 12 → x = 10.
当分子是二项式时，先加括号再乘：(x + 2)/4 = 3 → x + 2 = 12 → x = 10。
8. Checking Your Solution | 检验你的解
3 = 12 − 3 = 9, which matches the right-hand side. This confirms the solution is correct.
始终将答案代回原方程检验。如果你解 4x − 3 = 9 得到 x = 3，检验：4(3) − 3 = 12 − 3 = 9，与右边相等。这确认解是正确的。
Checking also helps catch sign errors and arithmetic mistakes. In exams, show your substitution clearly for method marks.
检验也有助于发现符号错误和算术错误。考试中，清楚地写出代入过程以获得方法分。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
x = 12 is wrong because the 5 was not subtracted from both sides. Correct: x = 12 − 5 = 7.
不要忘记两边执行相同的运算。例如，写 x + 5 = 12 → x = 12 是错误的，因为没有从两边减去 5。正确：x = 12 − 5 = 7。
= 3, add x to both sides and subtract 3: 10 − 3 = x, so x = 7. Avoid guessing that x = 13.
另一个常见错误是处理负系数不当。对于 10 − x = 3，两边加 x 并减 3：10 − 3 = x，所以 x = 7。不要猜成 x = 13。
When expanding brackets, all terms inside must be multiplied by the factor outside: 2(x + 3) is 2x + 6, not 2x + 3.
展开括号时，括号内每一项都要乘以外面的因数：2(x + 3) 是 2x + 6，而不是 2x + 3。
10. Exam-Style Practice Ideas | 考试风格练习题思路
lve and check each one. For word problems, let the unknown be n and translate the words into an equation.
尝试混合练习：7x − 3 = 18，4(x + 2) = 32，6x + 1 = 2x + 17，以及 (x − 3)/5 = 4。解出并检验每一个。对于文字题，设未知数为 n，把文字翻译成方程。
Timed practice helps develop fluency. Aim to solve a two-step equation in under one minute, showing all working clearly.
限时练习有助于提高熟练度。目标是在一分钟内解出一个两步方程，并清楚写出所有过程。
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