📚 Combining Like Terms | 合并同类项
In IGCSE algebra, simplifying expressions is a core skill. It allows you to write long expressions in a shorter, clearer form. One of the most important tools for simplification is collecting like terms, often called combining like terms. This topic appears again and again in equations, formulas, sequences and graphs, so it is worth mastering early.
在 IGCSE 代数中，化简表达式是一项核心技能。它能让你把较长的表达式写成更短、更清晰的形式。化简最重要工具之一就是合并同类项。这个主题在方程、公式、数列和图像中反复出现，因此值得尽早掌握。
1. What Are Like Terms? | 什么是同类项
In algebra, a term is a single number, a variable, or a product of numbers and variables. Examples include 5, x, −3y, 2ab and 7x². Like terms are terms whose variable parts are exactly the same, including the same powers. The coefficients, or the numbers in front, can be different.
在代数中，项是一个单独的数、变量或数与变量的乘积。例如 5、x、−3y、2ab 和 7x²。同类项是指变量部分完全相同（包括相同幂）的项。系数，也就是前面的数字，可以不同。
For example, 4x and −7x are like terms because both contain x to the power 1. However, 4x and 4x² are not like terms because the powers of x are different. Likewise, 3xy and 5xy are like, but 3xy and 3x are not.
例如，4x 和 −7x 是同类项，因为两者都含有 x 的一次幂。然而，4x 和 4x² 不是同类项，因为 x 的幂不同。同样，3xy 和 5xy 是同类项，但 3xy 和 3x 不是。
2. Coefficient and Variable Part | 系数与变量部分
Every term can be split into its coefficient and its variable part. In the term 7a²b, the coefficient is 7 and the variable part is a²b. In the term −4x, the coefficient is −4 and the variable part is x. This split helps you see which terms can be combined.
每个项都可以拆分为系数和变量部分。在项 7a²b 中，系数是 7，变量部分是 a²b。在项 −4x 中，系数是 −4，变量部分是 x。这种拆分能帮助你判断哪些项可以合并。
When deciding whether two terms are like terms, compare only the variable part. The coefficients do not need to match. For instance, 6p²q and −2p²q both have variable part p²q, so they are like terms.
判断两项是否为同类项时，只比较变量部分。系数不需要相同。例如，6p²q 和 −2p²q 的变量部分都是 p²q，所以它们是同类项。
3. The Golden Rule: Only Add or Subtract the Coefficients | 黄金法则：只加减系数
icients and keep the variable part absolutely unchanged. For example, 4x + 3x means four x's plus three x's, giving seven x's. This is written as 4x + 3x = (4 + 3)x = 7x.
合并同类项时，只加减系数，变量部分保持完全不变。例如，4x + 3x 表示四个 x 加三个 x，得到七个 x。这可以写成 4x + 3x = (4 + 3)x = 7x。
riable y does not become y², and it does not disappear. Combining like terms is just repeated addition or subtraction of the same object.
同样，9y − 2y = (9 − 2)y = 7y。变量 y 不会变成 y²，也不会消失。合并同类项只是对同一个对象进行重复加法或减法。
4. Combining Positive and Negative Coefficients | 合并正负系数
nd −9. Add them: 5 + (−9) = −4. Therefore, 5x − 9x = −4x. The answer keeps the variable x.
负系数经常引起错误。对于 5x − 9x，把系数看作 5 和 −9。相加：5 + (−9) = −4。因此，5x − 9x = −4x。答案保留变量 x。
− 7y = −9y. A number line can help: starting at 5, move left 9 steps, and you land on −4.
如果两个系数都为负，例如 −2y − 7y，将 −2 与 −7 相加得到 −9，所以 −2y − 7y = −9y。数轴可能有帮助：从 5 出发，向左移动 9 步，到达 −4。
5. Constants Are Like Terms Too | 常数项也是同类项
le part is empty. This means 3 + 5 − 2 can be combined as ordinary numbers: 3 + 5 − 2 = 6.
常数项是不含任何变量的数。它们互为同类项，因为它们的变量部分为空。这意味着 3 + 5 − 2 可以像普通数字一样合并：3 + 5 − 2 = 6。
In a longer expression, group constants separately from variable terms. For example, simplify 2x + 5 + 3x − 2. Combine 2x and 3x to get 5x, and combine 5 and −2 to get 3. The simplified form is 5x + 3.
在较长的表达式中，将常数项与变量项分开合并。例如，化简 2x + 5 + 3x − 2。将 2x 与 3x 合并得到 5x，将 5 与 −2 合并得到 3。化简结果为 5x + 3。
6. Powers Matter: x, x², x³ | 幂很重要：x、x²、x³
Terms with different powers are not like terms, even if they contain the same letter. For example, x, x² and x³ represent different quantities. You cannot combine them into a single term with a new power.
幂不同的项不是同类项，即使它们含有相同的字母。例如，x、x² 和 x³ 代表不同的量。你不能把它们合并成一个具有新幂的项。
is 9x² − 3x. Writing this as 6x would be incorrect because x² and x are different objects.
化简 6x² + 2x + 3x² − 5x 时，先对同类项分组。首先合并 x² 项：6x² + 3x² = 9x²。然后合并 x 项：2x − 5x = −3x。结果是 9x² − 3x。把它写成 6x 是错误的，因为 x² 和 x 是不同的对象。
7. Multi-Variable Expressions | 多变量表达式
r example, 2xy and 5xy are like terms because both contain xy. Therefore, 2xy + 5xy = 7xy.
对于含有多个变量的项，每个变量及其幂都必须完全一致。例如，2xy 和 5xy 是同类项，因为两者都含有 xy。因此，2xy + 5xy = 7xy。
However, 3ab and 2a are not like terms because the variable parts ab and a differ. Similarly, 4x²y and x²y are like terms, but 4x²y and 4xy² are not, because the powers of x and y do not match.
然而，3ab 和 2a 不是同类项，因为变量部分 ab 和 a 不同。类似地，4x²y 和 x²y 是同类项，但 4x²y 和 4xy² 不是，因为 x 和 y 的幂不匹配。
8. Removing Brackets Before Combining | 先去括号再合并
ront of a bracket leaves all signs inside unchanged: 3x + (2x − 5) = 3x + 2x − 5 = 5x − 5.
如果表达式含有括号，先去掉括号再合并同类项。括号前是加号时，括号内所有符号保持不变：3x + (2x − 5) = 3x + 2x − 5 = 5x − 5。
− 3 = 3a − 3. This is the most common place for sign errors, so expand brackets carefully.
括号前是减号时，括号内每一项都要变号。例如，5a − (2a + 3) = 5a − 2a − 3 = 3a − 3。这是符号错误最常见的地方，所以去括号时要格外仔细。
9. Fractional Coefficients | 分数系数
e common denominator is 6, so 1/2 = 3/6 and 2/3 = 4/6. Adding gives (3/6 + 4/6)x = (7/6)x.
带分数系数的同类项遵循相同规则，但你必须正确进行分数加减。例如，合并 (1/2)x + (2/3)x。公分母为 6，所以 1/2 = 3/6，2/3 = 4/6。相加得到 (3/6 + 4/6)x = (7/6)x。
− 2/4)y = (3/4)y. Keep the variable part unchanged and simplify the fraction if possible.
对于减法，计算 (5/4)y − (1/2)y。把 1/2 写成 2/4，所以差为 (5/4 − 2/4)y = (3/4)y。变量部分保持不变，并尽可能化简分数。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
ng. The correct answer is 2x, because the exponent does not change when adding like terms.
最常见的错误是在合并项时改变幂。例如，把 x + x = x² 写出来是错误的。正确答案是 2x，因为合并同类项时指数不变。
ab. These terms have different variable parts, so they cannot be merged into one term. Always check that the variable part matches exactly before combining.
另一个错误是合
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图