📚 Combining Like Terms in IGCSE Mathematics | IGCSE 数学：合并同类项
In IGCSE Mathematics, simplifying algebraic expressions is a core skill. One of the first and most important steps is combining like terms. This allows you to make expressions shorter, clearer and easier to use in equations, graphs and problem solving.
在 IGCSE 数学中，化简代数式是一项核心技能。其中第一步也是最重要的步骤之一就是合并同类项。它能让表达式更短、更清晰，也更容易用于方程、图像和实际问题的求解。
1. What Are Like Terms? | 什么是同类项？
A term is a single number, a variable, or a product of numbers and variables. For example, 3x, −5y, 7, x² and 2xy are all terms. Terms are usually separated by + or − signs.
项是一个单独的数、一个变量，或数与变量的乘积。例如，3x、−5y、7、x² 和 2xy 都是项。项通常由 + 或 − 号分隔。
Like terms are terms that have exactly the same variable part, including the same variables and the same powers. The coefficients, or the numbers in front, can be different.
同类项是指变量部分完全相同的项，包括相同的变量和相同的次数。系数，也就是前面的数，可以不同。
For example, 3x and 5x are like terms because both contain x to the power 1. However, 3x and 3x² are not like terms because the powers of x are different.
例如，3x 和 5x 是同类项，因为它们都含有 x 的一次方。但 3x 和 3x² 不是同类项，因为 x 的次数不同。
Like terms: 4x and −9x | Unlike terms: 4x and 4x²
同类项：4x 和 −9x | 不同类项：4x 和 4x²
2. Recognising Like Terms | 识别同类项
To decide whether two terms are like terms, ignore the number in front and examine only the variable part. The coefficient does not decide whether terms are like or unlike.
判断两个项是否为同类项时，先忽略前面的数，只看变量部分。系数并不能决定两项是否同类。
Terms such as 2ab and −7ab are like terms. Terms such as 3a and 3b are not like terms, because the variables differ.
比如 2ab 和 −7ab 是同类项。而 3a 和 3b 不是同类项，因为变量不同。
A common trap is to think that 5x and 5y are like terms just because they have the same coefficient. They are not; the coefficient does not matter when identifying like terms.
一个常见误区是，因为 5x 和 5y 的系数相同就认为它们是同类项。事实并非如此；判断同类项时系数无关紧要。
| 6m and 2m | Yes, both have m | 是，都含有 m |
| 4x and 4y | No, variables differ | 否，变量不同 |
| 3x² and −x² | Yes, both have x² | 是，都含有 x² |
| 2ab and 2a²b | No, powers differ | 否，次数不同 |
3. Combining Single-Variable Terms | 合并单变量项
When you combine like terms, add or subtract the coefficients while keeping the variable part unchanged. The variable acts like a label that is carried through the calculation.
合并同类项时，把系数相加或相减，变量部分保持不变。变量就像一个标签，在整个计算过程中一直带着。
ax + bx = (a + b)x
For example, 3x + 5x = 8x. You add the coefficients 3 and 5, then keep the x unchanged.
例如，3x + 5x = 8x。先把系数 3 和 5 相加，再保持 x 不变。
One useful way to think about this is to treat x as an object: 3 apples plus 5 apples gives 8 apples. In the same way, 3x plus 5x gives 8x.
一种有用的理解方法是把 x 看作一种物品：3 个苹果加 5 个苹果等于 8 个苹果。同样，3x 加 5x 等于 8x。
For subtraction, 9y − 4y = 5y, because 9 − 4 = 5 and the y remains unchanged.
对于减法，9y − 4y = 5y，因为 9 − 4 = 5，y 保持不变。
4. Working with Negative Coefficients | 处理负系数
Negative coefficients follow the same rules as directed numbers. Always include the sign in front of each term when adding or subtracting coefficients.
负系数的运算规则与有向数相同。在加减系数时，一定要带上每一项前面的符号。
se 7 − 10 = −3. The result is negative because the negative coefficient is larger in size.
例如：7x − 10x = −3x，因为 7 − 10 = −3。结果是负的，因为负系数的绝对值更大。
d −2a − 6a = −8a. In both cases, you combine the coefficients using directed number rules.
又如：−4a + 9a = 5a；−2a − 6a = −8a。在这两种情况下，都要按照有向数规则合并系数。
5x − (−2x) = 5x + 2x = 7x. Subtracting a negative term is the same as adding its opposite.
当出现两个负号时要小心：5x − (−2x) = 5x + 2x = 7x。减去一个负项，等于加上它的相反项。
a − (−b) = a + b
5. Multi-Variable Terms | 含多个变量的项
Terms with more than one variable are like terms only if every variable appears with the same power. The order of variables can change, but the powers cannot.
含有多个变量的项，只有当每个变量及其次数都相同时才是同类项。变量的顺序可以改变，但次数不能改变。
For example, 3xy and −8xy are like terms, so 3xy − 8xy = −5xy.
例如，3xy 和 −8xy 是同类项，所以 3xy − 8xy = −5xy。
However, 3xy and 3x²y are not like terms, because the power of x is different. Likewise, 5ab and 5ab² are not like terms.
然而，3xy 和 3x²y 不是同类项，因为 x 的次数不同。同样，5ab 和 5ab² 也不是同类项。
ation is commutative, so ab = ba. You can reorder the variables without changing the term.
另外，4ab 和 4ba 是同类项，因为乘法满足交换律，所以 ab = ba。你可以重新排列变量，而不改变这个项。
But 4ab and 4a²b are not like terms, because the power of a is different in each term.
但 4ab 和 4a²b 不是同类项，因为两个项中 a 的次数不同。
6. Constants and Unlike Terms | 常数项与不同类项
Constants are numbers without variables. All constant terms are like terms with each other, so they can always be combined.
常数项是不含变量的数。所有常数项彼此都是同类项，因此它们总是可以合并。
and −3 + 8 = 5. This works even when constants appear in different parts of an expression.
例如：5 + 9 = 14；−3 + 8 = 5。即使常数项出现在表达式的不同位置，也可以这样合并。
Unlike terms cannot be combined into a single term. For example, 4x + 3y cannot be simplified further because x and y are different variables.
不同类项不能合并成一项。例如，4x + 3y 不能再化简，因为 x 和 y 是不同的变量。
The expression 2x + 5 can only be left as it is, since 2x and 5 are unlike terms. The plus sign simply shows that the expression has two separate terms.
表达式 2x + 5 只能保持原样，因为 2x 和 5 不是同类项。加号只是表示这个表达式有两个独立的项。
4x + 3y stays as 4x + 3y
4x + 3y 保持为 4x + 3y
7. Expanding Brackets Before Combining | 先去括号再合并
Sometimes expressions contain brackets. You must expand brackets before combining like terms, because multiplication must be completed before addition or subtraction.
有时表达式含有括号。你必须先去括号，再合并同类项，因为乘法必须在加减法之前完成。
Example: Simplify 2(x + 3) + 3x. First expand: 2x + 6 + 3x. Then combine the x terms: 5x + 6.
例如：化简 2(x + 3) + 3x。先展开：2x + 6 + 3x。再合并 x 项：5x + 6。
Example: Simplify 3(2a − b) + 2(a + b). Expand to get 6a − 3b + 2a + 2b. Then combine: 8a − b.
例如：化简 3(2a − b) + 2(a + b)。展开得 6a − 3b + 2a + 2b。再合并：8a − b。
If there is a negative sign in front of brackets, multiply every term inside by −1. This changes the sign of each term inside the brackets.
如果括号前是
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