📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying expressions is a core skill. One of the first and most important techniques is collecting like terms, also called combining like terms. This process makes expressions shorter and easier to use in equations, formulas and graphs.
在 IGCSE 代数中，化简表达式是一项核心技能。其中最初也最重要的技巧之一就是合并同类项，也叫合并同类项。这个过程可以让表达式更简短，也便于在方程、公式和图像中使用。
1. What Are Terms in Algebra? | 代数中的“项”是什么？
An algebraic expression is made up of terms separated by plus or minus signs. For example, in the expression 3x + 2y - 5, there are three terms: 3x, 2y and -5. A term can be a number, a variable, or a product of numbers and variables.
代数表达式由用加号或减号分隔的项组成。例如，在表达式 3x + 2y - 5 中，有三个项：3x、2y 和 -5。一个项可以是一个数、一个变量，或者是数与变量的乘积。
The coefficient is the numerical factor in a term. In 4xy, the coefficient is 4. A constant term has no variable part, such as -5 or 9.
系数是项中的数字因数。在 4xy 中，系数是 4。常数项没有变量部分，例如 -5 或 9。
3x + 2y - 5 has terms 3x, 2y, -5
2. Like Terms and Unlike Terms | 同类项与不同类项
Like terms have exactly the same variable part, including the same powers. For example, 3x and -7x are like terms because both contain x to the first power. However, 5x² and 2x are not like terms because the powers of x are different.
同类项具有完全相同的变量部分，包括相同的幂次。例如，3x 和 -7x 是同类项，因为两者都含有 x 的一次幂。然而，5x² 和 2x 不是同类项，因为 x 的幂次不同。
Terms such as 4ab and 3a²b are also unlike terms. The first has a and b each to the power 1, while the second has a² and b.
像 4ab 和 3a²b 这样的项也是不同类项。前者中 a 和 b 的幂次都是 1，而后者含有 a² 和 b。
- Like terms: 3x, -7x, 0.5x, x
- Unlike terms: 3x, 2x², 4y, 5
同类项示例：3x、-7x、0.5x、x；不同类项示例：3x、2x²、4y、5。
3. The Golden Rule of Collecting | 合并的核心规则
The golden rule is simple: you can only add or subtract the coefficients of like terms, while the variable part remains unchanged. This works because multiplication distributes over addition.
核心规则很简单：只能对同类项的系数进行加减，变量部分保持不变。这是因为乘法对加法有分配律。
ax + bx = (a + b)x
of x plus 3 lots of x, which gives 8 lots of x. So 5x + 3x = 8x. Similarly, 9y - 4y = 5y.
例如，5x + 3x 表示 5 个 x 加上 3 个 x，得到 8 个 x。所以 5x + 3x = 8x。同样，9y - 4y = 5y。
You must never change the variable part when collecting. For example, 2x + 3x is not 5x², it is 5x.
合并时绝不能改变变量部分。例如，2x + 3x 不是 5x²，而是 5x。
4. Collecting with One Variable | 单一变量的合并
To simplify an expression with one variable, first identify all like terms, then add or subtract their coefficients separately.
要化简含有一个变量的表达式，首先找出所有同类项，然后分别对它们的系数进行加减。
Simplify 7x - 2 + 3x + 8. The x terms are 7x and 3x, giving 10x. The constant terms are -2 and 8, giving 6. Therefore, the simplified expression is 10x + 6.
化简 7x - 2 + 3x + 8。x 项是 7x 和 3x，得到 10x。常数项是 -2 和 8，得到 6。因此，化简后的表达式是 10x + 6。
7x - 2 + 3x + 8 = 10x + 6
- 9a. Add the coefficients: 12 - 5 + 3 - 9 = 1, so the result is 1a, usually written as a.
另一个例子：化简 12a - 5a + 3a - 9a。将系数相加：12 - 5 + 3 - 9 = 1，所以结果是 1a，通常写成 a。
5. Collecting with Multiple Variables | 多变量合并
When an expression contains more than one variable, collect each type of like term independently. It is often helpful to write the answer in alphabetical order of variables.
当表达式中含有多个变量时，要分别合并每一类同类项。通常按变量的字母顺序书写答案会很有帮助。
terms are 4a - 2a = 2a, and the b terms are 3b + 5b = 8b. The simplified form is 2a + 8b.
化简 4a + 3b - 2a + 5b。a 项为 4a - 2a = 2a，b 项为 3b + 5b = 8b。化简后的形式是 2a + 8b。
4a + 3b - 2a + 5b = 2a + 8b
So the result is 2xy + 6x. Notice that xy and x are different, so they cannot be combined.
考虑 3xy + 2x - xy + 4x。xy 项为 3xy - xy = 2xy，x 项为 2x + 4x = 6x。所以结果是 2xy + 6x。注意 xy 与 x 不同，因此不能合并。
6. Powers and Like Terms | 幂次与同类项
Powers are part of the variable structure, so x and x² are never like terms. Only terms with exactly the same power of the same variable can be collected.
幂次是变量结构的一部分，因此 x 和 x² 绝不是同类项。只有相同变量的幂次完全相同的项才可以合并。
The x² terms are 5x² - 2x² = 3x². The x terms are 3x + 7x = 10x. The answer is 3x² + 10x.
化简 5x² + 3x - 2x² + 7x。x² 项为 5x² - 2x² = 3x²。x 项为 3x + 7x = 10x。答案是 3x² + 10x。
5x² + 3x - 2x² + 7x = 3x² + 10x
Be careful with expressions like 4a²b and 3ab². Although they contain the same letters, the powers are assigned differently, so they are not like terms.
注意像 4a²b 和 3ab² 这样的表达式。虽然它们含有相同的字母，但幂次分配不同，因此它们不是同类项。
7. Expanding Brackets Before Collecting | 先去括号再合并
If an expression contains brackets, expand them first before collecting like terms. Use the distributive law to multiply each term inside the bracket by the factor outside.
如果表达式中含有括号，要先展开括号，再合并同类项。使用分配律，将括号外的因数乘以括号内的每一项。
x + 6 + 2x - 2. Then collect like terms: 3x + 2x = 5x and 6 - 2 = 4. The result is 5x + 4.
化简 3(x + 2) + 2(x - 1)。先展开得到 3x + 6 + 2x - 2。然后合并同类项：3x + 2x = 5x，6 - 2 = 4。结果是 5x + 4。
3(x + 2) + 2(x - 1) = 5x + 4
When a negative sign is outside a bracket, it changes the sign of every term inside. For example, 5 - 2(x + 3) becomes 5 - 2x - 6, which simplifies to -2x - 1.
当括号外是负号时，它会改变括号内每一项的符号。例如，5 - 2(x + 3) 变为 5 - 2x - 6，化简为 -2x - 1。
8. Fractional and Decimal Coefficients | 分数与小数系数
Collecting like terms also works when coefficients are fractions or decimals. The same rule applies: add or subtract the coefficients and keep the variable part unchanged.
当系数是分数或小数时，合并同类项的方法同样适用。规则相同：对系数进行加减，变量部分保持不变。
For fractions, use a common denominator. Simplify 1/2 x + 1/3 x. The coefficients are 1/2 and 1/3, with sum 5/6, so the result is 5/6 x.
对于分数，要使用公分母。化简 1/2 x + 1/3 x。系数为 1/2 和 1/3，和为 5/6，因此结果是 5/6 x。
1/2 x + 1/3 x = 5/6 x
ractions by converting one form to the other, such as 0.5x + 1/4 x = 0.5x + 0.25x = 0.75x.
对于小数，直接把小数系数相加即可。例如，2.5x + 1.3x = 3.8x。也可以把小数和分数相互转换后再合并，例如 0.5x + 1/4 x = 0.5x + 0.25x = 0.75x。
9. Common Mistakes to Avoid | 常见错误提醒
such as writing 3x + 2x² = 5x². This is incorrect because x and x² have different powers.
一个常见错误是合并不同类项，例如把 3x + 2x² 写成 5x²。这是错误的，因为 x 和 x² 的幂次不同。
Another mistake is changing the variable part when adding coefficients. For example, 4x + 3x is not 7x², it is 7x. The variable remains x.
另一个错误是在系数相加时改变了变量部分。例如，4x + 3x 不是 7x²，而是 7x。变量仍然是 x。
- Mistake 1: 3x + 2x² = 5x² (wrong, unlike terms)
- Mistake 2: 4x + 3x = 7x² (wrong, variable changed)
- Mistake 3: 5x - 3 + 2x + 4 = 7x + 7 (wrong, signs ignored)
- Mistake 4: 2(x - 3) = 2x - 3 (wrong, distribution incomplete)
错误 1：3x + 2x² = 5x²（错误，不同类项）；错误 2：4x + 3x = 7x²（错误，变量被改变）；错误 3：5x - 3 + 2x + 4 = 7x + 7（错误，忽略了符号）；错误 4：2(x - 3) = 2x - 3（错误，分配不完整）。
x - 3 + 2x + 4 is 7x + 1, because -3 + 4 = 1. Always carry the sign in front of each term.
5x - 3 + 2x + 4 的正确结果是 7x + 1，因为 -3 + 4 = 1。始终要带着每一项前面的符号。
10. Applying to Perimeter and Word Problems | 周长与文字题应用
tangle has length 3x + 2 and width x - 1, its perimeter is given by P = 2(length + width).
合并同类项在几何中非常有用，特别是在求周长时。如果一个矩形的长是 3x + 2，宽是 x - 1，它的周长公式为 P = 2(长 + 宽)。
side the bracket: (3x + 2) + (x - 1) = 4x + 1. Then multiply by 2: P = 2(4x + 1) = 8x + 2.
先计算括号内的和：(3x + 2) + (x - 1) = 4x + 1。然后乘以 2：P = 2(4x + 1) = 8x + 2。
P = 2[(3x + 2) + (x - 1)] = 8x + 2
nd person C has n - 4 cards, the total number of cards is n + (2n + 3) + (n - 4) = 4n - 1.
文字题也可以通过合并同类项来解决。如果 A 有 n 张卡片，B 有 2n + 3 张卡片，C 有 n - 4 张卡片，那么卡片总数为 n + (2n + 3) + (n - 4) = 4n - 1。
11. Quick Practice with Answers | 快速练习与答案
Try these practice questions before looking at the answers. Simplify each expression fully.
在看答案之前，先试着做这些练习题。请完整化简每个表达式。
- Simplify 6x + 4 - 2x + 7 = 4x + 11
- Simplify 5a + 3b - 2a + 8b = 3a + 11b
- Simplify 4x² + 3x - x² + 2x = 3x² + 5x
- Simplify 2(3x - 1) - (x + 4) = 6x - 2 - x - 4 = 5x - 6
化简 6x + 4 - 2x + 7 = 4x + 11；化简 5a + 3b - 2a + 8b = 3a + 11b；化简 4x² + 3x - x² + 2x = 3x² + 5x；化简 2(3x - 1) - (x + 4) = 6x - 2 - x - 4 = 5x - 6。
If any answer does not match, re-check the signs and make sure every like term has been included exactly once.
如果任何答案不一致，请重新检查符号，并确保每个同类项都恰好包含了一次。
12. Summary and Exam Tips | 总结与考试技巧
To collect like terms successfully, remember these steps: identify terms with the same variable part, add or subtract only their coefficients, and keep the variable part unchanged. Expand brackets first when necessary.
要成功合并同类项，请记住以下步骤：找出具有相同变量部分的项，只对它们的系数进行加减，并保持变量部分不变。必要时先展开括号。
In the exam, use small marks or underlines to group like terms. Write answers in a neat order, usually alphabetical, and always double-check negative signs. Showing each step can gain method marks even if the final answer is wrong.
考试时，可以用小标记或下划线把同类项分组。按整齐的顺序书写答案，通常按字母顺序排列，并始终仔细检查负号。即使最终答案错误，写出每一步也能获得步骤分。
Identify → Expand → Group → Collect → Check
Identify the like terms, expand any brackets, group them, collect their coefficients, and check the signs. This simple routine will help you avoid common errors in IGCSE algebra.
识别同类项，展开所有括号，将它们分组，合并系数，并检查符号。这个简单的流程会帮助你避免 IGCSE 代数中的常见错误。
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