Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions are built from terms that may contain variables, coefficients, and constants. Collecting like terms is the process of simplifying an expression by combining terms that have exactly the same variable part. This skill is fundamental to nearly every topic in IGCSE Mathematics, from solving equations to manipulating formulae.

在代数中，表达式由含有变量、系数和常数的项构成。合并同类项是将具有完全相同变量部分的项合并从而简化表达式的过程。这一技能是IGCSE数学中几乎所有主题的基础，从解方程到公式变形都离不开它。

1. Terms and Coefficients | 项与系数

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y − 2, there are three terms: 3x, 5y, and −2. The number multiplying the variable is called the coefficient. In the term 3x, the coefficient is 3; in −2y, the coefficient is −2.

项是单个数字、单个变量，或数字与变量的乘积。例如，在表达式 3x + 5y − 2 中，共有三个项：3x、5y 和 −2。与变量相乘的数字称为系数。在项 3x 中，系数是 3；在 −2y 中，系数是 −2。

A constant term has no variable part. Constants such as 4, −7, and ½ are always like terms with each other, because they all have the same variable part — none.

常数项没有变量部分。诸如 4、−7 和 ½ 这样的常数彼此之间永远是同类项，因为它们具有相同的变量部分——即没有变量。

2. What Are Like Terms | 什么是同类项

Like terms are terms that have exactly the same variable part, including the same powers. The coefficient does not need to be the same. For example, 2x and 5x are like terms; 4x² and −3x² are like terms; but 2x and 2x² are not like terms, because the powers of x are different.

同类项是指变量部分完全相同的项，包括相同的幂次。系数不需要相同。例如，2x 和 5x 是同类项；4x² 和 −3x² 是同类项；但 2x 和 2x² 不是同类项，因为 x 的幂次不同。

Similarly, 3xy and −7xy are like terms because both contain the product xy. However, 3xy and 3x²y are not like terms, since the power of x differs. The variable part must match exactly for two terms to be combinable.

同样地，3xy 和 −7xy 是同类项，因为两者都含有乘积 xy。然而，3xy 和 3x²y 不是同类项，因为 x 的幂次不同。两个项要能合并，变量部分必须完全一致。

3. The Basic Rule of Collecting | 合并的基本规则

To collect like terms, add or subtract their coefficients while keeping the variable part unchanged. For instance, 2x + 5x = (2 + 5)x = 7x. The variable x remains exactly the same; only the coefficients are added.

合并同类项时，只对其系数进行加减运算，变量部分保持不变。例如，2x + 5x = (2 + 5)x = 7x。变量 x 完全不变；只有系数相加。

ax + bx = (a + b)x

This rule applies to any variable and any power, as long as the variable parts match. If the variable parts do not match, the terms must be written separately and cannot be combined.

只要变量部分一致，该规则适用于任何变量和任何幂次。如果变量部分不一致，这些项必须分开书写，不能合并。

4. Working with Different Variables | 处理不同变量的项

When an expression contains more than one variable, collect terms separately for each variable. Consider the expression 3x + 4y − x + 2y. First identify x-terms: 3x and −x. Then identify y-terms: 4y and 2y. Combine each group: 3x − x = 2x, and 4y + 2y = 6y. The simplified result is 2x + 6y.

当表达式中含有多于一个变量时，需要对每个变量分别合并。考虑表达式 3x + 4y − x + 2y。首先找出 x 项：3x 和 −x。再找出 y 项：4y 和 2y。分别合并：3x − x = 2x，4y + 2y = 6y。化简结果为 2x + 6y。

Notice that x and y terms are never combined with each other. The final answer keeps each variable group separate. A common check is to count how many x-terms and y-terms appeared in the original expression and confirm they match the simplified form.

注意 x 项和 y 项永远不会相互合并。最终答案中各组变量保持分离。一个常见的检验方法是数一数原表达式中有多少个 x 项和 y 项，并确认与化简后的形式一致。

5. Dealing with Constant Terms | 常数项的处理

Constant terms are always like terms with each other, regardless of their values. In the expression 5x + 3 + 2x − 8, first combine the x-terms: 5x + 2x = 7x. Then combine the constants: 3 − 8 = −5. The simplified expression is 7x − 5.

常数项彼此之间永远是同类项，无论其数值大小。在表达式 5x + 3 + 2x − 8 中，先合并 x 项：5x + 2x = 7x。再合并常数项：3 − 8 = −5。化简后的表达式为 7x − 5。

When writing the final answer, it is conventional to place variable terms first, followed by the constant term. This ordering is not mathematically required but helps maintain clarity and consistency, especially when dealing with longer expressions.

书写最终答案时，习惯上把变量项写在前面，常数项写在后面。这种排序在数学上并非必须，但有助保持清晰与一致，尤其在处理较长表达式时。

6. Subtraction and Negative Coefficients | 减法与负系数

Subtraction must be handled carefully when collecting like terms. The minus sign belongs to the term that follows it. In the expression 4x − 3x + 2x, each term is positive or negative as written: +4x, −3x, +2x. Adding the coefficients: 4 − 3 + 2 = 3, so the result is 3x.

合并同类项时必须小心处理减法。负号属于其后的项。在表达式 4x − 3x + 2x 中，每项的符号分别为 +4x、−3x、+2x。将系数相加：4 − 3 + 2 = 3，结果为 3x。

Consider 6a − 2b − 4a + b. Group a-terms: 6a − 4a = 2a. Group b-terms: −2b + b = −b. The simplified expression is 2a − b. Notice that the negative sign must be kept in front of b in the final answer.

考虑 6a − 2b − 4a + b。合并 a 项：6a − 4a = 2a。合并 b 项：−2b + b = −b。化简结果为 2a − b。注意最终答案中 b 前面的负号必须保留。

7. Combining Squares and Higher Powers | 合并平方项与更高次幂

Terms with the same variable but different powers are not like terms. The expression x² + 3x + 2x² − x contains x² terms (x² and 2x²) and x terms (3x and −x). Combine them separately: x² + 2x² = 3x², and 3x − x = 2x. The result is 3x² + 2x.

变量相同但幂次不同的项不是同类项。表达式 x² + 3x + 2x² − x 中含有 x² 项（x² 和 2x²）以及 x 项（3x 和 −x）。分别合并：x² + 2x² = 3x²，3x − x = 2x。结果为 3x² + 2x。

The same logic applies to cubes and other powers. For example, 5x³ + 2x² − 3x³ + x² simplifies to 2x³ + 3x². Do not attempt to combine x³ with x² — their exponents are different.

同样的逻辑适用于立方项和其他幂次。例如，5x³ + 2x² − 3x³ + x² 化简为 2x³ + 3x²。不要试图将 x³ 与 x² 合并——它们的指数不同。

8. Products Like xy and Mixed Terms | xy 等乘积项与混合项

Terms such as xy, ab, and pq are treated as single variable parts. In the expression 3xy + 2x + 5xy − x, the xy-terms are 3xy and 5xy, and the x-terms are 2x and −x. Combining gives 8xy + x.

像 xy、ab、pq 这样的项被视为单一的变量部分。在表达式 3xy + 2x + 5xy − x 中，xy 项是 3xy 和 5xy，x 项是 2x 和 −x。合并后得到 8xy + x。

Be careful with expressions that contain both x and xy. Since x and xy have different variable parts, they cannot be combined. Similarly, 2xy and 2x²y are not like terms because the power of x differs.

对于同时含有 x 和 xy 的表达式要格外小心。由于 x 与 xy 的变量部分不同，它们不能合并。同样地，2xy 与 2x²y 不是同类项，因为 x 的幂次不同。

9. Simplifying Before Collecting: Removing Brackets | 合并前先化简：去括号

Sometimes brackets must be expanded before like terms can be collected. For example, 2(x + 3) + 4x expands to 2x + 6 + 4x. Then combine x-terms: 2x + 4x = 6x, giving the simplified result 6x + 6.

有时需要先去括号，然后才能合并同类项。例如，2(x + 3) + 4x 展开为 2x + 6 + 4x。然后合并 x 项：2x + 4x = 6x，得到化简结果 6x + 6。

When brackets are preceded by a negative sign, every term inside the bracket changes sign. For instance, 3y − 2(y − 1) = 3y − 2y + 2 = y + 2. Failing to change all signs is a common source of errors.

当括号前是负号时，括号内的每一项都要变号。例如，3y − 2(y − 1) = 3y − 2y + 2 = y + 2。未能将所有项正确变号是常见错误来源。

10. Common Mistakes and Checking Answers | 常见错误与答案检查

Mistake 1: Combining unlike terms. Writing 3x + 2 = 5x is incorrect, because x and 2 are not like terms. They must remain separate.

错误一：合并非同类项。 写成 3x + 2 = 5x 是不对的，因为 x 和 2 不是同类项，它们必须保持分离。

Mistake 2: Missing negative signs. When simplifying 5x − 3x, the result is 2x, not 8x. Always keep track of the sign in front of each term.

错误二：遗漏负号。 化简 5x − 3x 时，结果是 2x，而不是 8x。始终注意每一项前面的符号。

Mistake 3: Confusing powers. x + x² cannot be combined into one term. If the exponents differ, the terms are not like terms.

错误三：混淆幂次。 x + x² 不能合并成一项。如果指数不同，这些项就不是同类项。

To check your answer, substitute a small number such as x = 2 into both the original expression and your simplified expression. If the two values are equal, the simplification is most likely correct.

要检查答案，可将一个小数如 x = 2 分别代入原表达式和化简后的表达式。如果两个值相等，说明化简很可能是正确的。

11. Worked Examples | 典型例题

Example 1: Simplify 7x + 2 − 3x + 5.

例1： 化简 7x + 2 − 3x + 5。

x-terms: 7x − 3x = 4x. Constants: 2 + 5 = 7. Answer: 4x + 7.

x 项：7x − 3x = 4x。常数项：2 + 5 = 7。答案：4x + 7。

Example 2: Simplify 4a + 3b − 2a + b.

例2： 化简 4a + 3b − 2a + b。

a-terms: 4a − 2a = 2a. b-terms: 3b + b = 4b. Answer: 2a + 4b.

a 项：4a − 2a = 2a。b 项：3b + b = 4b。答案：2a + 4b。

Example 3: Simplify 5(x + 2) − 3x.

例3： 化简 5(x + 2) − 3x。

First expand: 5x + 10 − 3x. Then combine x-terms: 5x − 3x = 2x. Answer: 2x + 10.

先展开：5x + 10 − 3x。再合并 x 项：5x − 3x = 2x。答案：2x + 10。

12. Summary | 总结

Collecting like terms is a straightforward but essential algebraic skill. Identify like terms by matching variable parts exactly, combine coefficients with careful attention to signs, and never combine terms with different variables or powers. With consistent practice, this process becomes automatic and error-free.

合并同类项是一项基础而重要的代数技能。通过精确匹配变量部分来识别同类项，仔细注意符号来合并系数，绝不合并变量或幂次不同的项。通过持续练习，这一过程将变得熟练而准确。

Remember the key rule: only coefficients change during collection; the variable part remains unchanged. Apply this rule consistently, and you will build a solid foundation for all future algebra topics.

记住关键规则：合并过程中只有系数改变，变量部分保持不变。始终如一地应用这一规则，你将为今后所有代数主题打下坚实的基础。

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Collecting Like Terms | 合并同类项