📚 Solving Linear Equations with Brackets: Cambridge KS3 Exercise p214 Q2 | 剑桥 KS3 数学：含括号的一次方程求解（第214页第2题）
Linear equations with brackets are a core skill in the Cambridge KS3 mathematics curriculum. In this revision guide, we work through the type of question found on page 214, question 2, where you must expand brackets, collect like terms, and solve for the unknown variable.
含括号的一次方程是剑桥 KS3 数学课程中的核心技能。在本复习指南中，我们将讲解第214页第2题这类题目，你需要先去括号、合并同类项，然后求出未知数的值。
1. Understanding the Problem | 理解题意
x + 5. The aim is to find the value of x that makes the left side equal to the right side.
含括号的一次方程是指像 3(x - 2) = 2x + 5 这样的等式。我们的目标是求出使左边等于右边的 x 的值。
On page 214 question 2, you may be asked to solve, simplify, or check a solution. Always read the instruction carefully before starting your working.
在第214页第2题中，你可能会被要求求解、化简或检验答案。开始计算前，一定要仔细阅读题目要求。
2. Key Vocabulary | 关键术语
Linear equation: an equation where the unknown appears only to the power of 1, such as 4x + 7 = 19.
一次方程：未知数的最高次数为 1 的方程，例如 4x + 7 = 19。
Brackets: symbols like ( ) that show multiplication or grouping. Expanding brackets means multiplying out the terms inside.
括号：像 ( ) 这样的符号，表示乘法或分组。去括号就是把括号内的每一项乘出来。
3. Expanding Brackets | 去括号
When a number is written directly outside a bracket, multiply every term inside the bracket by that number.
当一个数字直接写在括号外时，要用这个数字乘以括号内的每一项。
a(b + c) = ab + ac
For example, 3(x - 2) expands to 3x - 6 because 3 × x = 3x and 3 × (-2) = -6.
例如，3(x - 2) 去括号后得到 3x - 6，因为 3 × x = 3x，而 3 × (-2) = -6。
4. Collecting Like Terms | 合并同类项
Like terms contain exactly the same variable part. You can only add or subtract like terms.
同类项含有完全相同的字母部分。只有同类项才能相加或相减。
ple, 5x - 2x = 3x but 5x - 2 cannot be simplified further because they are not like terms.
例如，5x - 2x = 3x，但 5x - 2 不能再化简，因为它们不是同类项。
5. Isolating the Unknown | 移项求解未知数
After expanding and collecting like terms, isolate the unknown on one side of the equation.
去括号并合并同类项后，要把未知数单独移到方程的一边。
- If a term is added, subtract it from both sides.
- 如果某一项是加上的，就在等号两边同时减去它。
- If a term is subtracted, add it to both sides.
- 如果某一项是减去的，就在等号两边同时加上它。
- If the unknown is multiplied by a number, divide both sides by that number.
- 如果未知数乘以一个数，就在等号两边同时除以这个数。
6. Worked Example 1: 3(x - 2) = 2x + 5 | 例题一：3(x - 2) = 2x + 5
Step 1: Expand the bracket on the left side.
第一步：展开左边的括号。
3x - 6 = 2x + 5
Step 2: Subtract 2x from both sides to collect the x terms on the left.
第二步：两边同时减去 2x，把含 x 的项移到左边。
3x - 2x - 6 = 5
Step 3: Simplify and then add 6 to both sides.
第三步：化简，然后两边同时加上 6。
x = 11
7. Worked Example 2: 5(2x - 1) = 3(x + 4) | 例题二：5(2x - 1) = 3(x + 4)
Expand both brackets first.
首先展开两边的括号。
10x - 5 = 3x + 12
Subtract 3x from both sides and add 5 to both sides.
两边同时减去 3x，并同时加上 5。
10x - 3x = 12 + 5
Simplify and solve.
化简并求解。
7x = 17
x = 17 ÷ 7 = 2 3/7
8. Checking Your Answer | 检验答案
Substitute your value of x back into the original equation to check it works.
把求出的 x 值代回原方程，检验是否成立。
he right side is 2 × 11 + 5 = 22 + 5 = 27. Both sides are equal, so the answer is correct.
例如，对于 3(x - 2) = 2x + 5，当 x = 11 时，左边是 3(11 - 2) = 3 × 9 = 27，右边是 2 × 11 + 5 = 22 + 5 = 27。两边相等，因此答案正确。
9. Common Mistakes | 常见错误
Mistake 1: Forgetting to multiply the second term inside the bracket by the outside number.
错误一：忘记用括号外的数去乘括号内的第二项。
Incorrect: 3(x - 2) = 3x - 2. Correct: 3(x - 2) = 3x - 6.
错误写法：3(x - 2) = 3x - 2。正确写法：3(x - 2) = 3x - 6。
Mistake 2: Moving a term to the other side without changing its sign.
错误二：把项移到另一边时没有变号。
Incorrect: x - 6 = 5 so x = 5 - 6. Correct: x - 6 = 5 so x = 5 + 6.
错误做法：x - 6 = 5，所以 x = 5 - 6。正确做法：x - 6 = 5，所以 x = 5 + 6。
10. Practice Questions | 练习题目
Try these questions to test your understanding.
尝试以下问题来检验你的理解。
- 4(x + 3) = 2x + 18
- 2(3x - 5) = 4(x + 1)
- 7(x - 2) = 5x + 6
For each one, expand the brackets, collect like terms, solve, and check your answer by substitution.
对每一题，先去括号，再合并同类项，求解，然后用代入法检验答案。
11. Exam Tips | 考试技巧
Show every step of your working. Even if your final answer is wrong, you can still earn method marks for correct expansion and balancing.
写出每一步计算过程。即使最终答案错误，只要去括号和移项过程正确，你仍然可以得到方法分。
Write your answer clearly, and if the answer is a fraction, simplify it fully unless the question says otherwise.
答案要写清楚；如果答案是分数，除非题目另有说明，否则要化简到最简形式。
12. Summary | 总结
To solve a linear equation with brackets, expand the brackets first, collect like terms, isolate the unknown, and check your solution by substitution.
解含括号的一次方程时，先去掉括号，合并同类项，移项求出未知数，最后用代入法检验答案。
This method is exactly what you need for Cambridge KS3 questions like page 214 question 2, and it prepares you for more advanced algebra in later stages.
这种方法正是剑桥 KS3 第214页第2题这类题目所需要的，也为后续更高阶段的代数学习打下基础。
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