📚 Collecting Like Terms: Simplify Algebraic Expressions | 合并同类项：化简代数式
In IGCSE Mathematics, simplifying an algebraic expression often begins with collecting like terms. This topic is one of the most important foundation skills in algebra, because it appears every time you solve equations, expand brackets, factorise or rearrange formulae.
在 IGCSE 数学中，化简代数式往往从合并同类项开始。这个主题是代数中最基础也最重要的技能之一，因为你在解方程、去括号、因式分解或变换公式时都会用到它。
1. What Are Terms in Algebra? | 代数中的项是什么？
An algebraic expression is made up of terms. A term is a number, a variable, or a product of numbers and variables, separated by plus or minus signs. For example, in 3x + 5y – 7, the terms are 3x, 5y and -7.
代数式由项组成。项是一个数、一个变量，或数与变量的乘积，项与项之间用加号或减号分隔。例如在 3x + 5y – 7 中，项分别是 3x、5y 和 -7。
Each term has a coefficient and a variable part. In 3x, the coefficient is 3 and the variable part is x. The constant term -7 has no variable part, because it is just a number.
每一项都有系数和变量部分。在 3x 中，系数是 3，变量部分是 x。常数项 -7 没有变量部分，因为它只是一个数。
When you see an expression such as 4xy – 2x + 9, you should be able to split it into three separate terms: 4xy, -2x and 9. Recognising terms correctly is the first step before any simplifying can happen.
当你看到 4xy – 2x + 9 这样的表达式时，应该能把它拆分成三个独立的项：4xy、-2x 和 9。正确识别各项是进行化简前的第一步。
2. Like Terms and Unlike Terms | 同类项与异类项
Like terms have exactly the same variable part, including the same powers. For instance, 4x and -2x are like terms, while 4x and 4x² are unlike terms because the powers of x are different.
同类项是指变量部分完全相同、包括相同幂次的项。例如 4x 和 -2x 是同类项，而 4x 和 4x² 不是同类项，因为 x 的幂次不同。
Similarly, 7xy and -3xy are like terms, but 7xy and 7x are unlike terms since the variable combinations differ. Constants are also like terms with each other, so 5 and -12 can be combined.
同样地，7xy 和 -3xy 是同类项，但 7xy 和 7x 不是同类项，因为变量的组合不同。常数项之间也互为同类项，因此 5 和 -12 可以合并。
Because multiplication is commutative, 4xy and 2yx are usually treated as like terms too, since xy and yx represent the same product. However, x and y alone are not the same as xy.
因为乘法满足交换律，4xy 和 2yx 通常也视为同类项，因为 xy 和 yx 表示相同的乘积。但单独的 x 和 y 与 xy 并不相同。
3. The Core Rule: Only Like Terms Combine | 核心规则：只有同类项才能合并
You can add or subtract like terms by combining their coefficients while keeping the variable part unchanged. For example, 3x + 5x = (3 + 5)x = 8x.
你可以通过合并系数来加减同类项，同时保持变量部分不变。例如 3x + 5x = (3 + 5)x = 8x。
Unlike terms cannot be combined into a single term. 3x + 4y must stay as 3x + 4y because x and y represent different quantities.
异类项不能合并成一项。3x + 4y 必须保留为 3x + 4y，因为 x 和 y 表示不同的量。
ax + bx = (a + b)x
This rule applies to any like terms, whether they involve x, y, x², ab or any other variable part. The variable part never changes when you collect like terms.
这个规则适用于所有同类项，无论涉及 x、y、x²、ab 还是其他任何变量部分。合并同类项时，变量部分永远不会改变。
4. Collecting Positive Like Terms | 合并正同类项
When all coefficients are positive, simply add them. For 2a + 5a + 3a, add 2 + 5 + 3 = 10, so the result is 10a.
当所有系数都是正数时，只需把它们相加。对于 2a + 5a + 3a，先算 2 + 5 + 3 = 10，所以结果是 10a。
Example: Simplify 6m + 3 + 2m + 8. The m terms give 6m + 2m = 8m, and the constants give 3 + 8 = 11, so the answer is 8m + 11.
例题：化简 6m + 3 + 2m + 8。m 项合并为 6m + 2m = 8m，常数项合并为 3 + 8 = 11，因此答案是 8m + 11。
Always write the final answer in a tidy order. Usually variables are written before constants, so 8m + 11 is preferred over 11 + 8m, although both are mathematically equivalent.
总是把最终答案写成整洁的顺序。通常变量写在常数前面，所以 8m + 11 比 11 + 8m 更常用，尽管两者在数学上是等价的。
5. Collecting Negative Like Terms | 合并负同类项
With negative coefficients, follow the rules of directed numbers. For -7x + 2x, combine -7 + 2 = -5, giving -5x.
处理负系数时，要遵循正负数运算规则。对于 -7x + 2x，合并 -7 + 2 = -5，得到 -5x。
Example: Simplify 9p – 4 – 3p – 6. The p terms: 9p – 3p = 6p; the constants: -4 – 6 = -10. The answer is 6p – 10, not 6p + 10.
例题：化简 9p – 4 – 3p – 6。p 项为 9p – 3p = 6p；常数项为 -4 – 6 = -10。答案是 6p – 10，而不是 6p + 10。
Always carry the sign in front of each term when rearranging. In 5 – 3x + 2, the term before 2 is +2, and the x term is -3x, so the constants combine to 7 and the x term remains -3x.
重新排列项时，一定要带上每项前面的符号。在 5 – 3x + 2 中，2 前面的符号是 +2，x 项是 -3x，所以常数项合并为 7，x 项保持为 -3x。
6. Mixed Terms: Keeping Different Powers Separate | 混合项：区分不同幂次
Terms with different powers are not like terms. For example, x² and x are different, so 4x² + 3x – 2x² + 5x simplifies to 2x² + 8x.
幂次不同的项不是同类项。例如 x² 和 x 不同，所以 4x² + 3x – 2x² + 5x 化简为 2x² + 8x。
A common exam expression is ax² + bx + c. You collect x² terms, then x terms, then constants separately. For 5x² + 7x + 4 – 3x² + 2x – 1, the result is 2x² + 9x + 3.
考试中常见的形式是 ax² + bx + c。你要分别合并 x² 项、x 项和常数项。对于 5x² + 7x + 4 – 3x² + 2x – 1，合并结果是 2x² + 9x + 3。
x² terms: 5x² – 3x² = 2x²; x terms: 7x + 2x = 9x; constants: 4 – 1 = 3
Never combine x² with x, because they represent different powers. Writing 2x² + 9x + 3 as 11x + 3 or 11x² + 3 would be mathematically incorrect.
千万不要把 x² 和 x 合并，因为它们代表不同的幂次。把 2x² + 9x + 3 写成 11x + 3 或 11x² + 3 在数学上都是错误的。
7. Simplifying Expressions Step by Step | 逐步化简表达式
Follow a consistent method: identify terms, group like terms, combine coefficients, write the simplified expression. You can underline or circle matching terms to avoid missing any.
采用固定的方法：找出各项、将同类项分组、合并系数、写出化简后的表达式。你可以用下划线或圆圈标出同类项，避免遗漏。
Example: Simplify 5x + 3y – 2x + 7y – 4. Step 1: group x terms: 5x – 2x = 3x. Step 2: group y terms: 3y + 7y = 10y. Step 3: constant: -4. Answer: 3x + 10y – 4.
例题：化简 5x + 3y – 2x + 7y – 4。第一步：合并 x 项 5x – 2x = 3x。第二步：合并 y 项 3y + 7y = 10y。第三步：常数项为 -4。答案：3x + 10y – 4。
Show the grouping step in your working, even if you can do it mentally. This helps you track negative signs and reduces careless errors in longer expressions.
即使你能心算，也要在解题过程中写出分组步骤。这有助于你跟踪负号，减少较长表达式中的粗心错误。
8. Dealing with Brackets Before Collecting | 先去括号再合并
If an expression includes brackets, expand them first. For 3(a + 2b) + 2a, multiply 3 into the bracket to get 3a + 6b + 2a, then collect like terms: 5a + 6b.
如果表达式含有括号，要先展开括号。对于 3(a + 2b) + 2a，先把 3 乘入括号得到 3a + 6b + 2a，再合并同类项：5a + 6b。
For subtraction, distribute the negative sign carefully: 4(x + 2) – (x – 1) = 4x + 8 – x + 1 = 3x + 9.
遇到减法时，要仔细分配负号：4(x + 2) – (x – 1) = 4x + 8 – x + 1 = 3x + 9。
A common error is to write -(x – 1) as -x – 1. The correct expansion is -x + 1 because the negative sign changes every term inside the bracket.
一个常见错误是把 -(x – 1) 写成 -x – 1。正确的展开是 -x + 1，因为负号会改变括号内每一项的符号。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Mistake 1: adding unlike terms such as 2x + 3x². This cannot be simplified because the powers differ.
错误一：把异类项相加，例如 2x + 3x²。这不能化简，因为幂次不同。
Mistake 2: losing the negative sign. In 6 – 2x + 5, the constant total is 11, not 1; and -2x stays as -2x.
错误二：丢掉负号。在 6 – 2x + 5 中，常数项合并为 11，而不是 1；-2x 仍保持为 -2x。
Mistake
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