📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying expressions usually starts with collecting like terms. A like term has exactly the same variable part, so 3x and 5x can be combined, but 3x and 3x² cannot. Once you can collect like terms confidently, solving equations and rearranging formulae becomes much easier.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。同类项是指变量部分完全相同的项，因此 3x 和 5x 可以合并，但 3x 和 3x² 不能合并。熟练掌握合并同类项后，解方程和变换公式会变得容易得多。
1. What Is a Term? | 什么是项
In algebra, a term is a single number, a single letter, or a product of numbers and letters. Terms are separated by plus or minus signs. In the expression 4x + 7y − 3, there are three terms: 4x, 7y and −3.
在代数中，项是由加号或减号分隔的单个数字、单个字母或数字与字母的乘积。在表达式 4x + 7y − 3 中，有三个项：4x、7y 和 −3。
The number part of a term is called the coefficient. In 4x the coefficient is 4, and in 7y it is 7. A term with no variable, such as −3 or 5, is called a constant term.
项中的数字部分称为系数。在 4x 中系数是 4，在 7y 中系数是 7。不含变量的项，如 −3 或 5，称为常数项。
2. Recognising Like Terms | 识别同类项
Terms are like if they have exactly the same variable factors raised to exactly the same powers. Coefficients can be different. For example, 2x, −5x and x are like terms because they all contain x to the power 1.
如果各项所含的变量因式完全相同且次数也完全相同，这些项就是同类项。系数可以不同。例如，2x、−5x 和 x 是同类项，因为它们都含有一次变量 x。
However, 3x and 3x² are unlike because the powers differ. Also, 2xy and 3x are unlike because the variable combinations differ. Constants such as 4 and −9 are like terms with each other.
然而，3x 和 3x² 不是同类项，因为次数不同。另外，2xy 和 3x 也不是同类项，因为变量组合不同。常数项（如 4 和 −9）彼此之间是同类项。
3. The Golden Rule: Only Like Terms Combine | 黄金法则：只有同类项才能合并
es + 5 apples = 8 apples, but 3 apples + 5 bananas cannot be combined into one fruit term.
你只能对同类项进行加减。可以把同类项看作同一种物品：3 个苹果加 5 个苹果等于 8 个苹果，但 3 个苹果加 5 个香蕉不能合并成一种水果项。
3x + 5x = (3 + 5)x = 8x
When terms are unlike, leave them as separate terms in the simplified expression. The expression 2x + 3y is already in its simplest form because x and y are different variables.
当各项不是同类项时，在化简后的表达式中将它们保留为独立项。表达式 2x + 3y 已经是最简形式，因为 x 和 y 是不同的变量。
4. Adding and Subtracting Like Terms | 加减同类项
or example, 7m − 2m = (7 − 2)m = 5m. Be careful with negative coefficients: −4p + 9p = 5p.
合并同类项时，只对系数进行加减，变量部分保持不变。例如，7m − 2m = (7 − 2)m = 5m。注意负数系数：−4p + 9p = 5p。
left to right or rearrange sign by sign. For example, 6a − 3a + 10a = (6 − 3 + 10)a = 13a.
你可以通过重写表达式按任意顺序分组各项。可以从左到右计算，也可以带着符号重新排列。例如，6a − 3a + 10a = (6 − 3 + 10)a = 13a。
keep the minus sign in front of the variable part. For instance, 2x − 7x = (2 − 7)x = −5x.
如果合并后系数为负数，把减号保留在变量部分前面。例如，2x − 7x = (2 − 7)x = −5x。
5. Simplifying Mixed Expressions | 化简混合表达式
ng each term with its sign is helpful: 3x + 4y − 2x + y becomes 3x − 2x + 4y + y = x + 5y.
在较长表达式中，先找出并分组所有同类项，再分别合并各组。把每一项连同符号一起移动会很有帮助：3x + 4y − 2x + y 可变为 3x − 2x + 4y + y = x + 5y。
5a + 2b − 3a + 7b − 4 = 5a − 3a + 2b + 7b − 4 = 2a + 9b − 4
Notice that the constant −4 is kept on its own because there are no other constant terms to combine with it. The final expression usually lists unlike terms separately.
注意常数项 −4 被单独保留，因为没有其他常数项可以与之合并。最终表达式通常将不同类项分别列出。
6. Working with Brackets | 去括号并合并
When an expression contains brackets, use the distributive law first to remove them, then collect like terms. Multiply the term outside the bracket by every term inside the bracket.
当表达式包含括号时，先用分配律去括号，再合并同类项。将括号外的项分别乘以括号内的每一项。
3(2x + y) − 2(x − 2y) = 6x + 3y − 2x + 4y = 4x + 7y
Be especially careful when a negative sign sits before a bracket: it changes the sign of every term inside. This is one of the most important checking points in IGCSE algebra.
当括号前为负号时要格外小心：它会改变括号内每一项的符号。这是 IGCSE 代数中最重要的检查点之一。
7. Powers and Like Terms | 幂与同类项
A power is part of the variable factor, so x and x² are different variable parts. Therefore 3x + 4x² cannot be combined. Likewise, 2xy and 3x are unlike because 3x lacks the factor y.
幂是变量因式的一部分，因此 x 和 x² 属于不同的变量部分。所以 3x + 4x² 不能合并。同样，2xy 和 3x 也不是同类项，因为 3x 缺少因式 y。
tion order does not matter, so xy and yx are usually treated as like terms, but it is good practice to write variables in alphabetical order.
像 2xy 和 −3xy 这样的项是同类项，合并得 2xy − 3xy = −xy。乘法顺序不影响结果，因此 xy 和 yx 通常视为同类项，但最好按字母顺序书写变量。
5x² + 2x² = 7x², but 5x² + 2x cannot be combined.
8. Coefficients and Constants | 系数与常数项
Always combine all constant terms at the end of simplification. In 6 + 2x − 3 + 5x, the constants are 6 and −3, giving 3, while the x terms give 7x. The simplified form is 7x + 3.
化简时最后总是合并所有常数项。在 6 + 2x − 3 + 5x 中，常数项为 6 和 −3，合并得 3，而含 x 的项合并得 7x。化简后的形式为 7x + 3。
6 + 2x − 3 + 5x = 2x + 5x + 6 − 3 = 7x + 3
e coefficient, such as x or −y, the coefficient is 1 or −1. Thus x + x = 2x and y − y = 0.
如果一项没有显示系数，如 x 或 −y，则系数为 1 或 −1。因此 x + x = 2x，而 y − y = 0。
9. Common Mistakes | 常见错误
A very common error is to add the powers: x + x² is incorrectly written as x³. You cannot merge terms by changing exponents, so x + x² must stay as it is.
一个非常常见的错误是把指数相加：x + x² 被错误地写成 x³。你不能通过改变指数来合并项，因此 x + x² 必须保持原样。
y. This is wrong because unlike terms must stay separate. Only identical variable parts can be collected.
另一个错误是合并不同变量，例如把 2x + 3y 写成 5xy。这是错误的，因为不同类项必须保持分离。只有完全相同的变量部分才能合并。
Students also often lose signs when rearranging terms. Keep each sign attached
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