📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression often begins with collecting like terms. An expression such as 3x + 5y + 2x can be made shorter and clearer by combining the x terms into 5x + 5y. This topic focuses on recognising which terms may be combined and which must remain separate.
在 IGCSE 代数中，化简代数式通常从合并同类项开始。例如，3x + 5y + 2x 可以通过合并 x 项简化为 5x + 5y。本主题的重点是识别哪些项可以合并，哪些项必须保持独立。
1. What Are Like Terms? | 什么是同类项
A term is a single number, variable, or product of numbers and variables. Examples of terms include 7, x, 3y, −2a² and 5xy. Two or more terms are called like terms when their variable parts are exactly the same. The coefficients, which are the number parts, may be different.
项是指一个单独的数、变量或数与变量的乘积。项的例子包括 7、x、3y、−2a² 和 5xy。当两个或多个项的变量部分完全相同时，它们就称为同类项。系数，也就是数字部分，可以不同。
For example, 3x and 5x are like terms because both contain the variable part x. Similarly, 2y² and −4y² are like terms because both contain y². However, 3x and 3y are not like terms, because one variable part is x and the other is y.
例如，3x 和 5x 是同类项，因为它们都含有变量部分 x。同样，2y² 和 −4y² 也是同类项，因为它们都含有 y²。但是，3x 和 3y 不是同类项，因为一个的变量部分是 x，另一个是 y。
- Like terms: 3x and 5x — 同类项
- Like terms: 2y² and −4y² — 同类项
- Not like terms: 3x and 3y — 不是同类项
- Not like terms: x and x² — 不是同类项
2. Identifying Like Terms in Expressions | 识别代数式中的同类项
To identify like terms in a longer expression, focus on the variable part of each term. Ignore the coefficient at first, because only the variable part determines whether terms can be combined. This makes grouping much easier.
要在较长代数式中识别同类项，首先要关注每一项的变量部分。先忽略系数，因为只有变量部分决定这些项是否可以合并。这样分组会容易得多。
| Term | Variable part | Like with |
|---|---|---|
| 3x | x | 5x, −2x |
| 4y² | y² | 7y², −y² |
| 2ab | ab | 5ab, −3ab |
| 5 | constant | 3, −8 |
Constants such as 5, 3 and −8 are always like terms with each other. This is because they have no variable part, so their variable part can be thought of as empty or identical.
常数项如 5、3 和 −8 总是彼此为同类项。这是因为它们没有变量部分，因此可以认为它们的变量部分为空或完全相同。
3. The Core Rule of Collecting Like Terms | 合并同类项的核心规则
Once you have found like terms, the rule is simple: add or subtract the coefficients, and keep the variable part unchanged. This works because like terms represent the same type of quantity.
一旦找到同类项，规则就很简单：系数相加或相减，变量部分保持不变。这是因为同类项表示同一种类型的量。
ax + bx = (a + b)x
ax − bx = (a − b)x
For example, 4x + 5x means four x plus five x, giving nine x altogether, written as 9x. In the same way, 7y − 2y means seven y take away two y, giving five y, written as 5y.
例如，4x + 5x 表示四个 x 加上五个 x，总共九个 x，写作 9x。同样，7y − 2y 表示七个 y 减去两个 y，得到五个 y，写作 5y。
4. Single Variable Examples | 单变量例题
x + 3x − 2x. The variable part is x in every term, so add the coefficients: 6 + 3 − 2 = 7.
许多 IGCSE 题目从简单的单变量代数式开始。例如，化简 6x + 3x − 2x。每一项的变量部分都是 x，所以将系数相加：6 + 3 − 2 = 7。
6x + 3x − 2x = 7x
Another example is 5a + 2a. Since both terms have the variable part a, add the coefficients 5 and 2 to get 7a. Remember that if no coefficient is written, it is 1.
另一个例子是 5a + 2a。因为两项的变量部分都是 a，所以将系数 5 和 2 相加得到 7a。请记住，如果项前没有写出系数，系数就是 1。
5a + 2a = 7a
Also, do not change the variable part when collecting. The answer to 9m − 4m is 5m, not 5m² or 5.
此外，合并时不要改变变量部分。9m − 4m 的答案是 5m，而不是 5m² 或 5。
5. Multi-Variable Terms | 多变量同类项
Terms can contain more than one variable. For example, 2xy, 5xy and −xy are like terms because each has the variable part xy. The order of variables in a product does not matter, so xy and yx represent the same variable part.
项可以包含多个变量。例如，2xy、5xy 和 −xy 是同类项，因为每项的变量部分都是 xy。乘积中变量的顺序无关紧要，因此 xy 和 yx 表示相同的变量部分。
5 − 1 = 6. The coefficient of xy by itself is 1, even though the 1 is not usually written.
要化简 2xy + 5xy − xy，请将系数相加：2 + 5 − 1 = 6。单独的 xy 的系数是 1，尽管 1 通常不写出来。
2xy + 5xy − xy = 6xy
On the other hand, 3ab and 4a²b are not like terms, because the power of a is different. Always compare the full variable part, including all powers and variables.
另一方面，3ab 和 4a²b 不是同类项，因为 a 的指数不同。请始终比较完整的变量部分，包括所有的指数和变量。
6. Combining Signed Terms | 合并带符号的项
When terms have negative coefficients, the sign must travel with the term. A common method is to group terms with their signs before adding the coefficients.
当项带有负系数时，符号必须随项一起移动。一个常用的方法是先将项连同符号分组，然后再相加系数。
b. Group the a terms: 5a + 2a = 7a. Group the b terms: −3b + b = −2b, because −3 + 1 = −2.
化简 5a − 3b + 2a + b。将 a 项分组：5a + 2a = 7a。将 b 项分组：−3b + b = −2b，因为 −3 + 1 = −2。
5a − 3b + 2a + b = 7a − 2b
mplify 4x − 7 + 3x + 2. The x terms give 4x + 3x = 7x, and the constants give −7 + 2 = −5.
常数项可以按同样的方式合并。例如，化简 4x − 7 + 3x + 2。x 项得到 4x + 3x = 7x，常数项得到 −7 + 2 = −5。
4x − 7 + 3x + 2 = 7x − 5
7. Terms with Powers and Indices | 含幂与指数的项
Only terms with exactly the same power of each variable can be collected. This means x and x² are not like terms, because the powers are 1 and 2. They must remain as separate terms.
只有当每个变量的指数完全相同时，项才能合并。这意味着 x 和 x² 不是同类项，因为指数分别是 1 和 2。它们必须保持为单独的项。
x + x² ≠ 2x²
x², 2x² and −x² are like terms. The coefficient of −x² is −1, so the sum is 5 + 2 − 1 = 6.
但是，像 5x²、2x² 和 −x² 这样的项是同类项。−x² 的系数是 −1，所以系数之和为 5 + 2 − 1 = 6。
5x² + 2x² − x² = 6x²
- Like: 4x³ and −2x³ — 同类项
- Not like: 3x² and 3x — 不是同类项
- Not like: xy² and x²y — 不是同类项
- Like: 3a²b and −5a²b — 同类项
8. Brackets Before Collecting | 先去括号再合并
If an expression contains brackets, you must expand the brackets before collecting like terms. Expanding first helps reveal all terms clearly, which reduces sign errors.
如果代数式中含有括号，你必须先展开括号，然后再合并同类项。先展开有助于清楚地显示所有项，从而减少符号错误。
nd: 3x + 6 + 2x − 2. Then collect the x terms: 3x + 2x = 5x, and the constants: 6 − 2 = 4.
化简 3(x + 2) + 2(x − 1)。首先展开：3x + 6 + 2x − 2。然后合并 x 项：3x + 2x = 5x，常数项：6 − 2 = 4。
3(x + 2) + 2(x − 1) = 5x + 4
mplify 2(3x − 1) − (x + 4). Expand: 6x − 2 − x − 4. Collect: 6x − x = 5x, and −2 − 4 = −6.
当括号外有负号时，它必须作用于括号内的每一项。化简 2(3x − 1) − (x + 4)。展开：6x − 2 − x − 4。合并：6x − x = 5x，−2 − 4 = −6。
2(3x − 1) − (x + 4) = 5x − 6
9. Fractions and Decimals | 分数与小数系数
implify 1/2 x + 1/3 x. The common denominator of 2 and 3 is 6, so 1/2 = 3/6 and 1/3 = 2/6.
合并同类项同样适用于分数系数。要将分数相加，先找到公分母。例如，化简 1/2 x + 1/3 x。2 和 3 的公分母是 6，因此 1/2 = 3/6，1/3 = 2/6。
1/2 x + 1/3 x = 5/6 x
decimal coefficients: 0.2 + 0.8 = 1.0, so the answer is 1x, which is written simply as x.
对于小数，过程与整数相同。例如，0.2x + 0.8x 表示将小数系数相加：0.2 + 0.8 = 1.0，因此答案是 1x，通常直接写作 x。
0.2x + 0.8x = x
10. Common Mistakes to Avoid | 常见错误
because x and y are different variable parts, so they cannot be combined into one product.
最常见的错误之一是将不同类项相加，例如将 3x + 2y 写成 5xy。这是错误的，因为 x 和 y 是不同的变量部分，不能合并成一个乘积。
3x + 2y ≠ 5xy
should be 3x, because x is the same as 1x. Writing x + 2x = 2x loses the original x term.
另一个错误是忘记系数 1。例如，x + 2x 应该等于 3x，因为 x 等同于 1x。将 x + 2x 写成 2x 会丢失原来的 x 项。
x + 2x = 3x
Sign errors are also common when collecting negative terms. Always attach the sign to the coefficient and calculate carefully. Also, do not combine a constant with a variable term: 5 + 2x cannot equal 7x.
合并负项时，符号错误也很常见。请始终把符号和系数连在一起，并仔细计算。此外，不要把常数项与含变量的项合并：5 + 2x 不等于 7x。
5 + 2x ≠ 7x
11. Exam-Style Worked Example | 考试型例题
Simplify 6x + 4y − 2x + y − 3 + 8. First group the like terms: the x terms are 6x and −2x, the y terms are 4y and y, and the constants are −3 and 8.
化简 6x + 4y − 2x + y − 3 + 8。首先将同类项分组：x 项为 6x 和
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