📚 Cambridge KS3 Algebra: Solving Linear Equations with Brackets | 剑桥 KS3 代数：解含括号的一元一次方程
x + 5. This article guides you through the key rules, worked examples and common pitfalls.
含括号的一元一次方程是剑桥初中数学（Cambridge Lower Secondary Mathematics）中的核心代数技能。你需要熟练掌握展开、化简并求解如 3(x + 2) = 21 或 2(3x − 4) = x + 5 这类方程。本文将带你梳理关键规则、典型例题和常见易错点。
1. What Is a Linear Equation? | 什么是线性方程
ctions or variables on both sides, but after simplifying it must take the form ax + b = c.
线性方程是指未知数的最高次数为 1 的代数等式。它可以带有括号、分数或等号两边都有未知数，但化简后必须能写成 ax + b = c 的形式。
ax + b = c, 其中 a ≠ 0
2. Balancing the Equation | 方程平衡原则
Equations are like a balance scale: whatever you do to one side, you must do to the other. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the opposite side to keep equality true.
方程就像一架天平：你对等号一边做的任何操作，都必须对另一边做同样的操作。无论是加、减、乘还是除以一个数，都必须在两边同时进行，等式才能保持成立。
Solve x + 5 = 12. Subtract 5 from both sides: x + 5 − 5 = 12 − 5, so x = 7.
例如解 x + 5 = 12。两边同时减去 5：x + 5 − 5 = 12 − 5，因此 x = 7。
3. Removing Brackets with the Distributive Law | 运用分配律去括号
stributive law: a(b + c) = ab + ac. Be careful to multiply the sign as well as the number.
去括号时，要用括号外的项分别乘以括号内的每一项。这就是分配律：a(b + c) = ab + ac。注意计算时既要乘数字，也要保留符号。
3(x + 2) = 3 × x + 3 × 2 = 3x + 6
write the negative sign with the multiplier: −2(x − 4) = −2x + 8, because −2 × (−4) = +8.
如果括号外的系数是负数，要带着负号一起去乘：−2(x − 4) = −2x + 8，因为 −2 × (−4) = +8。
4. Collecting Like Terms First | 先合并同类项
Before solving, simplify both sides as much as possible. Like terms are terms that have exactly the same variable part, such as 5x and −3x. Numbers without variables are also like terms with each other.
求解之前，要先把方程左右两边尽量化简。同类项是指含有相同变量部分的项，例如 5x 和 −3x；不含变量的常数项之间也是同类项。
Simplify 4x + 3 + 2x − 5. Combine 4x + 2x = 6x and 3 − 5 = −2, giving 6x − 2.
例如化简 4x + 3 + 2x − 5。先合并 4x + 2x = 6x，再合并 3 − 5 = −2，得到 6x − 2。
5. Unknowns on Both Sides | 未知数在等号两边
If the variable appears on both sides of the equation, collect all variable terms on one side and constants on the other. Choose the side where the variable term will have a positive coefficient to reduce sign mistakes.
如果未知数同时出现在等号两边，需要把含未知数的项移到一边，把常数项移到另一边。尽量选择让未知数系数为正的那一边，这样可以减少符号错误。
2x + 11. Subtract 2x from both sides: 3x + 2 = 11. Subtract 2: 3x = 9. Divide by 3: x = 3.
例如解 5x + 2 = 2x + 11。两边同时减去 2x：3x + 2 = 11。两边再减去 2：3x = 9。两边除以 3：x = 3。
6. Using Inverse Operations | 使用逆运算
To isolate x, undo each operation in reverse order. The inverse of addition is subtraction, the inverse of subtraction is addition, the inverse of multiplication is division, and the inverse of division is multiplication.
要单独解出 x，需要按逆序逐步还原运算。加法的逆运算是减法，减法的逆运算是加法，乘法的逆运算是除法，除法的逆运算是乘法。
For 3x + 6 = 21, first subtract 6, then divide by 3: 3x = 15, x = 5.
对于 3x + 6 = 21，先减 6，再除以 3：3x = 15，x = 5。
7. Common Sign Errors | 常见符号错误
A very common mistake is forgetting to distribute the negative sign across every term inside the bracket. Another is changing the sign of only one term when moving it to the other side. Always rewrite each line carefully and check signs before continuing.
一个非常常见的错误是去括号时忘记把负号分配给括号内的每一项。另一个错误是把项移到等号另一边时只改变一个项的符号。每一步都要仔细重写，并在继续之前检查符号。
−2(3x − 5) = −6x + 10, not −6x − 10. Both the 3x and the −5 must be multiplied by −2.
例如 −2(3x − 5) = −6x + 10，而不是 −6x − 10。3x 和 −5 都要乘以 −2。
8. Checking Your Solution | 检验答案
After finding x, substitute it back into the original equation. If the left side equals the right side, your solution is correct. This check is quick and can help you catch sign or arithmetic errors.
求出 x 后，要把它代回原方程进行检验。如果左边等于右边，说明答案正确。这种检验很快，还能帮助发现符号或计算错误。
For 3(x + 2) = 21 with x = 5: left = 3(5 + 2) = 21, right = 21, so correct.
例如 3(x + 2) = 21，当 x = 5 时：左边 = 3(5 + 2) = 21，右边 = 21，说明答案正确。
9. Worked Example: Two Brackets | 例题：含两个括号
= 3x − 3. Then collect variables on the left: 2x − 3x = −3 − 6, giving −x = −9, so x = 9.
解方程 2(x + 3) = 3(x − 1)。首先展开两边括号：2x + 6 = 3x − 3。然后把含未知数的项移到左边：2x − 3x = −3 − 6，得到 −x = −9，所以 x = 9。
Check: left = 2(9 + 3) = 24, right = 3(9 − 1) = 24, so the solution is valid.
检验：左边 = 2(9 + 3) = 24，右边 = 3(9 − 1) = 24，所以答案成立。
10. Practice Tips for Cambridge KS3 | 剑桥 KS3 备考提示
For Cambridge Lower Secondary Mathematics, practise writing every step clearly and avoid doing too many operations in your head. Show the expansion, the rearrangement and the inverse operations. In exams, marks are often awarded for method even if the final answer slips.
在剑桥初中数学中，要练习把每一个步骤写清楚，不要过多依赖心算。要展示去括号、移项和逆运算的过程。考试中即使最后答案出错，只要方法正确，也常常能拿到步骤分。
| Step | 步骤 | Action | 操作 |
|---|---|
| 1 | Expand brackets | 去括号 |
| 2 | Collect like terms | 合并同类项 |
| 3 | Move variable terms to one side | 将含未知数项移到一边 |
| 4 | Use inverse operations to solve | 使用逆运算求解 |
| 5 | Substitute to check | 代入检验 |
Remember to keep the equation balanced at all times. If you write each step on a separate line, revision and checking become much easier.
记住要始终保持方程两边平衡。如果把每一步单独写在下一行，复习和检查都会容易得多。
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