📚 Collecting Like Terms | 合并同类项
In algebra, many questions begin by asking you to simplify an expression. The most important early skill is collecting like terms. This topic appears throughout IGCSE Mathematics, from basic simplification to solving equations and rearranging formulas. You need to know exactly which terms can be combined and which must stay separate.
在代数中，许多题目一开始就要求你化简表达式。最重要的早期技能就是合并同类项。这个主题贯穿 IGCSE 数学，从基础化简到解方程和公式变形都会出现。你必须准确知道哪些项可以合并，哪些项必须保持独立。
1. What Are Like Terms? | 什么是同类项
In algebra, an expression is built from terms. A term is a number, a letter, or a product of numbers and letters. Terms are separated by + or − signs. Like terms are terms whose variable parts are exactly the same. For example, 3x and −7x both have the variable part x, so they are like terms. Constants such as 5 and −2 are also like terms with each other.
在代数中，表达式由项组成。项是一个数、一个字母或数与字母的乘积。项由 + 或 − 号分隔。同类项是指变量部分完全相同的项。例如，3x 和 −7x 的变量部分都是 x，所以它们是同类项。常数项如 5 和 −2 彼此也是同类项。
Understanding this definition is the first step. If the letters or exponents are different, the terms cannot be combined by ordinary addition or subtraction.
理解这个定义是第一步。如果字母或指数不同，这些项就不能通过普通的加减法合并。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and focus on the variable part. For example, 4xy and −2xy are like terms because both contain xy. 5a²b and 3a²b are like terms because both contain a²b.
要识别同类项，忽略系数，只关注变量部分。例如，4xy 和 −2xy 是同类项，因为两者都含有 xy。5a²b 和 3a²b 是同类项，因为两者都含有 a²b。
However, 5a²b and 5ab² are not like terms. In 5a²b the factor a is squared, but in 5ab² the factor b is squared. The powers must match exactly. This is one of the most common places to lose marks.
但是，5a²b 和 5ab² 不是同类项。在 5a²b 中，a 是平方的，而在 5ab² 中，b 是平方的。幂必须完全一致。这是最容易失分的地方之一。
Like terms: 3x and −7x; 4xy and −2xy; 5a²b and 3a²b
Not like terms: 3x and 3x²; 5a²b and 5ab²
3. The Rule for Collecting Like Terms | 合并同类项的法则
The rule for collecting like terms is simple: add or subtract the coefficients, and keep the variable part exactly the same. This works because 3x + 5x means 3 lots of x plus 5 lots of x, which gives 8 lots of x.
合并同类项的法则很简单：将系数相加或相减，变量部分保持完全不变。这是因为 3x + 5x 表示 3 个 x 加上 5 个 x，得到 8 个 x。
3x + 5x = (3 + 5)x = 8x
ce, 2ab + 6ab = (2 + 6)ab = 8ab. The coefficient changes, but the ab part stays unchanged.
这个法则适用于任何同类项，无论变量部分有多复杂。例如，2ab + 6ab = (2 + 6)ab = 8ab。系数改变了，但 ab 部分保持不变。
4. Positive and Negative Terms | 正项与负项
fficient. Example: 7y − 3y + 2y. Add the coefficients: 7 − 3 + 2 = 6, so the answer is 6y.
当表达式中同时含有正的和负的同类项时，要把符号看作系数的一部分。例如：7y − 3y + 2y。将系数相加：7 − 3 + 2 = 6，所以答案是 6y。
7y − 3y + 2y = (7 − 3 + 2)y = 6y
ite 5a and forget the negative sign. Always include the sign of each term when you add the coefficients.
减法时要格外小心。4a − 9a = (4 − 9)a = −5a。一个常见错误是写成 5a 而忘记负号。合并系数时一定要带上每一项的符号。
4a − 9a = (4 − 9)a = −5a
5. Simplifying Expressions with One Variable | 化简含一个变量的表达式
To simplify an expression, group like terms before combining. Example: 5x + 3 − 2x + 8. First group x terms: 5x − 2x. Then group constants: 3 + 8.
要化简一个表达式，先把同类项分组，再合并。例：5x + 3 − 2x + 8。先把 x 项放在一起：5x − 2x。再把常数项放在一起：3 + 8。
5x + 3 − 2x + 8 = (5x − 2x) + (3 + 8) = 3x + 11
The final answer should usually be written with variable terms first and the constant last. You can write 11 + 3x, but 3x + 11 is the more standard form in IGCSE answers.
最终答案通常先写变量项，最后写常数项。你可以写成 11 + 3x，但在 IGCSE 答案中，3x + 11 是更标准的写法。
6. Expressions with Multiple Variables | 多变量表达式
re 3xy and −2xy, which give 1xy = xy. The 5x and 7y cannot be combined with anything else.
当表达式中有多个变量时，只合并完全匹配的项。例：3xy + 5x − 2xy + 7y。xy 项是 3xy 和 −2xy，得到 1xy = xy。5x 和 7y 不能与其他项合并。
3xy + 5x − 2xy + 7y = (3xy − 2xy) + 5x + 7y = xy + 5x + 7y
Do not combine terms such as xy and y, or x and xy. They may look
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