📚 Solving Linear Equations with Brackets | 解含括号的线性方程
Welcome to this KS3 Cambridge Mathematics revision guide on solving linear equations with brackets. In this article, you will learn how to expand brackets, collect like terms, and use the balancing method to find the value of an unknown. These skills are essential for algebra at KS3 and will prepare you for more advanced equation solving in IGCSE.
欢迎阅读这篇 KS3 剑桥数学复习指南，主题是解含括号的线性方程。在本文中，你将学习如何展开括号、合并同类项，并运用平衡法求出未知数的值。这些技能是 KS3 代数的核心，也会为你后续在 IGCSE 阶段学习更复杂的方程打下基础。
1. What Is a Linear Equation? | 什么是线性方程？
ns such as x² - 4 = 0 are quadratic, not linear, so they are studied later at IGCSE level.
线性方程是指变量最高次数为 1 的方程。例如，3x + 5 = 20 是线性方程，因为 x 的指数为 1。像 x² - 4 = 0 这样的方程是二次方程，不是线性方程，因此会在 IGCSE 阶段再学习。
Linear equations can contain brackets, fractions, decimals, and variables on both sides. The goal is always the same: find the value of the unknown that makes the equation true.
线性方程可以包含括号、分数、小数和两边都有的未知数。目标始终相同：求出使方程成立的未知数的值。
2. The Balancing Method: The Golden Rule | 平衡法：黄金法则
The balancing method is based on the idea that an equation is like a set of scales. Whatever you do to one side of the equation, you must also do to the other side. This keeps the equation balanced and allows you to isolate the variable.
平衡法基于这样一个想法：方程就像一架天平。对方程的一边做了什么，另一边也必须做同样的操作。这样可以保持方程平衡，并使你能够分离出变量。
For example, to solve x + 7 = 12, subtract 7 from both sides:
例如，解 x + 7 = 12 时，两边同时减去 7：
x + 7 − 7 = 12 − 7 → x = 5
Always apply inverse operations in the reverse order of BIDMAS: add or subtract first, then multiply or divide.
始终按照 BIDMAS 的逆序应用逆运算：先加减，后乘除。
3. Expanding Single Brackets | 展开单项括号
Before solving an equation with brackets, you usually need to expand them. To expand a single bracket, multiply each term inside the bracket by the term outside.
在解含括号的方程之前，通常需要先展开括号。展开单项括号时，将括号内的每一项都乘以括号外的项。
For example:
例如：
3(x + 2) = 3 × x + 3 × 2 = 3x + 6
utside the bracket, remember to multiply every term by −1. For example, −(x − 4) = −x + 4.
如果括号外是负号，记得将每一项都乘以 −1。例如，−(x − 4) = −x + 4。
4. Equations with One Bracket | 含一个括号的方程
ckets are expanded, solve the equation using the balancing method. Consider 3(x + 2) = 21.
展开括号后，用平衡法解方程。考虑 3(x + 2) = 21。
Step 1: Expand the bracket: 3x + 6 = 21.
步骤1：展开括号：3x + 6 = 21。
Step 2: Subtract 6 from both sides: 3x = 15.
步骤2：两边同时减去6：3x = 15。
Step 3: Divide both sides by 3: x = 5.
步骤3：两边同时除以3：x = 5。
This three-step process works for most single-bracket equations. Always expand first, then undo addition or subtraction, and finally undo multiplication or division.
这个三步过程适用于大多数含一个括号的方程。始终先展开括号，然后通过加减还原，最后通过乘除还原。
5. Equations with Two Brackets | 含两个括号的方程
When an equation has two brackets, expand both sides first. Example: 2(x + 1) = 3(x − 2).
当方程有两个括号时，先展开两边。例如：2(x + 1) = 3(x − 2)。
Expanding gives: 2x + 2 = 3x − 6.
展开后得到：2x + 2 = 3x − 6。
en collect like terms. Subtract 2x from both sides: 2 = x − 6. Add 6 to both sides: x = 8.
然后合并同类项。两边同时减去 2x：2 = x − 6。两边同时加 6：x = 8。
When expanding two brackets, be careful to multiply every term correctly. A common mistake is only multiplying the first term inside the bracket.
展开两个括号时，要小心正确乘以每一项。一个常见错误是只乘括号内的第一项。
6. Unknowns on Both Sides | 两边有未知数
When variables appear on both sides of the equation, gather them on one side. Usually, move the smaller variable term to the larger one to keep the coefficient positive.
当变量出现在方程两边时，将它们移到同一边。通常把较小的变量项移到较大的那边，使系数保持为正。
. Subtract 2x from both sides: 3x + 3 = 12. Then subtract 3 and divide by 3, giving x = 3.
例如：5x + 3 = 2x + 12。两边同时减去 2x：3x + 3 = 12。然后减去3并除以3，得到 x = 3。
If the larger variable term is on the right side, move the smaller one to the right instead. This keeps the variable coefficient positive and reduces sign errors.
如果较大的变量项在右边，就把较小的移到右边。这样可以使变量系数保持为正，减少符号错误。
7. Negative Coefficients and Careful Signs | 负系数与符号注意
care. Consider 14 − 2x = 4. You can add 2x to both sides to make the coefficient positive.
负系数需要特别小心。考虑 14 − 2x = 4。可以在两边同时加上 2x，使系数变为正。
14 − 2x + 2x = 4 + 2x → 14 = 4 + 2x
Then subtract 4: 10 = 2x. Divide by 2: x = 5.
然后减去4：10 = 2x。除以2：x = 5。
−(x + 3) = −x − 3. A missing negative sign is one of the most frequent errors in algebra.
展开括号外为负号的式子时要检查符号，例如 −(x + 3) = −x − 3。漏掉负号是代数中最常见的错误之一。
8. Fractional Coefficients in Linear Equations | 线性方程中的分数系数
ared by multiplying every term by the lowest common denominator. For example, x/2 + 3 = 7.
方程中的分数可以通过将每一项乘以最小公分母来消除。例如，x/2 + 3 = 7。
Multiply all terms by 2: x + 6 = 14. Then x = 8.
将所有项乘以2：x + 6 = 14。于是 x = 8。
For equations like (x + 1)/3 = 5, multiply both sides by 3 first: x + 1 = 15, so x = 14.
对于像 (x + 1)/3 = 5 的方程，先将两边乘以3：x + 1 = 15，所以 x = 14。
If there are two fractions, multiply by the least common multiple of the denominators. This removes all fractions in one step and makes the equation easier to solve.
如果有两个分数，就乘以所有分母的最小公倍数。这样一步就能去掉所有分数，使方程更容易求解。
9. Checking Your Answer by Substitution | 通过代入检验答案
or 3(x + 2) = 21 with x = 5, the left side is 3(5 + 2) = 21, which matches the right side.
一定要将值代回原方程来检验你的解。对于 3(x + 2) = 21，当 x = 5 时，左边是 3(5 + 2) = 21，与右边相等。
If the two sides are not equal, go back and look for an error in expanding, collecting like terms, or applying inverse operations. Checking takes only a few seconds and can save marks in tests.
如果两边不相等，就回去检查展开、合并同类项或应用逆运算时是否有错误。检验只需要几秒钟，却能在考试中帮你保住分数。
10. Common Misconceptions | 常见误区
Common errors include forgetting to multiply every term inside the bracket, mishandling negative signs, and doing operations in the wrong order.
常见错误包括忘记乘以括号内的每一项、处理负号时出错，以及运算顺序错误。
Write each step clearly and use a new line for every operation. This makes it easier to spot mistakes and helps you follow the balancing method correctly.
每一步都要写清楚，每一个运算换一行。这样更容易发现错误，也能帮助你正确使用平衡法。
olving equations such as 2x + 6 = 14. Always undo addition or subtraction before division.
另一个常见错误是在解像 2x + 6 = 14 这样的方程时先除后减。一定要先通过加减还原，再进行除法。
11. Practice Questions | 练习题
Try these practice questions. Check your answers using the table below.
试试这些练习题。你可以用下面的表格检查答案。
- 4(x − 3) = 20
- 2(x + 5) = 3(x − 1)
- 5x − 7 = 2x + 8
- (2x + 1)/3 = 5
| Equation | Solution |
| 4(x − 3) = 20 | x = 8 |
| 2(x + 5) = 3(x − 1) | x = 13 |
| 5x − 7 = 2x + 8 | x = 5 |
| (2x + 1)/3 = 5 | x = 7 |
Work through each question step by step. Show your expansion, balancing, and final substitution to build confidence and accuracy.
逐步完成每道题。写出展开、平衡和最终代入的过程，以建立信心和准确性。
12. Summary and Key Takeaways | 总结与要点
In summary, solving linear equations with brackets involves expanding, collecting like terms, and using inverse operations. Practise these steps consistently, and always check your final answer.
总之，解含括号的线性方程包括展开括号、合并同类项和使用逆运算。坚持练习这些步骤，并始终检查最终答案。
Remember the main steps: expand brackets, move variable terms
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