Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, simplifying expressions is one of the most important skills. Collecting like terms means putting together terms that have exactly the same variables and the same exponents, so that the expression becomes shorter and easier to work with.

在代数中，化简表达式是最重要的技能之一。合并同类项就是把具有完全相同变量和指数的项放在一起，使表达式更简洁、更易于处理。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that contain exactly the same variable(s) raised to the same power(s). The coefficients (numbers in front) can be different.

同类项是指包含完全相同变量且变量指数也相同的项。它们前面的系数（数字）可以不同。

3x and 5x are like terms because both contain the variable x.
3x 与 5x 是同类项，因为它们都含有变量 x。
4y² and −2y² are like terms because both contain y².
4y² 与 −2y² 是同类项，因为它们都含有 y²。
2x and 2x² are NOT like terms because the exponents are different.
2x 与 2x² 不是同类项，因为指数不同。
3 and 7 are like terms because both are constants (no variable).
3 与 7 是同类项，因为它们都是常数（没有变量）。

Remember: the variable part must be identical. Only the number in front may change.

请记住：变量部分必须完全相同，只有前面的数字可以不同。

2. The Distributive Property | 分配律

Collecting like terms is based on the distributive property. For example, 2x + 3x = (2 + 3)x = 5x. This works because both terms share the same variable x.

合并同类项基于分配律。例如，2x + 3x = (2 + 3)x = 5x。这之所以成立，是因为两项都含有相同的变量 x。

a x + b x = (a + b) x

Similarly, a x − b x = (a − b) x.

类似地，a x − b x = (a − b) x。

This property allows us to add or subtract coefficients while keeping the variable part unchanged.

这一性质允许我们直接对系数做加法或减法，同时保持变量部分不变。

3. Adding and Subtracting Like Terms | 同类项的加减

To add like terms, simply add their coefficients. To subtract, subtract the coefficients. Keep the variable exactly the same.

合并同类项时，只需将它们的系数相加；若是相减，则系数相减。变量部分保持不变。

7x + 2x = 9x
7x + 2x = 9x
5x − 3x = 2x
5x − 3x = 2x
4a + a = 5a (because a = 1a)
4a + a = 5a（因为 a = 1a）
−2b + 7b = 5b
−2b + 7b = 5b

If a term has no number in front, the coefficient is 1 or −1. For example, x = 1x and −y = −1y.

如果某项前面没有数字，其系数就是 1 或 −1。例如，x = 1x，−y = −1y。

4. Simplifying Expressions with Multiple Terms | 多种项的化简

When an expression contains several different like groups, handle each group separately. For example:

当一个表达式包含多个不同的同类组时，应分别对每组进行合并。例如：

3x + 2y − x + 5y = (3x − x) + (2y + 5y) = 2x + 7y

Step: identify all x terms and all y terms, then combine their coefficients.

步骤：先找出所有含 x 的项与所有含 y 的项，然后分别合并系数。

Always leave the answer in alphabetical order of variables, with the constant term at the end.

答案通常按变量字母顺序排列，并把常数项写在最后。

5. Collecting Like Terms in Polynomials | 多项式中的合并同类项

In a polynomial, there can be terms with different powers of the same variable. You must collect terms with the same power, but never combine different powers.

在多项式中，同一变量可能以不同幂次出现。你只能合并相同幂次的项，绝不能把不同幂次合并。

For example, simplify 4x² + 3x − x² + 2x + 5.

例如，化简 4x² + 3x − x² + 2x + 5。

x² terms: 4x² − x² = 3x²
x² 项：4x² − x² = 3x²
x terms: 3x + 2x = 5x
x 项：3x + 2x = 5x
Constants: 5 stays as 5
常数项：5 保持不变为 5

4x² + 3x − x² + 2x + 5 = 3x² + 5x + 5

Always arrange the polynomial in descending powers of the variable.

通常按变量的降幂顺序排列多项式。

6. Dealing with Coefficients and Constants | 系数与常数的处理

Constants are like terms with each other. For example, 4 + 3 = 7. You can also combine negative constants: −2 − 5 = −7.

常数项之间互为同类项。例如，4 + 3 = 7；负常数也可以合并：−2 − 5 = −7。

When coefficients are fractions or decimals, follow the same rules. For example:

当系数是分数或小数时，规则完全相同。例如：

½x + ¼x = (½ + ¼)x = ¾x

0.3x − 0.1x = 0.2x

Do not change fractions into decimals unless necessary; keep the answer in the form given.

除非必要，不要将分数改成小数；应按题目给定形式保留答案。

7. Common Mistakes to Avoid | 常见错误避免

Many students make simple errors when collecting like terms. Here are the most frequent ones.

学生在合并同类项时常犯一些简单错误。下面是最常见的几类。

Mistake: 3x + 4y = 7xy. This is wrong because x and y are different variables.
错误：3x + 4y = 7xy。这是错误的，因为 x 与 y 是不同变量。
Mistake: 2x + 3x² = 5x³. Wrong because x and x² are not like terms.
错误：2x + 3x² = 5x³。错误，因为 x 与 x² 不是同类项。
Mistake: forgetting the invisible 1 in front of x.
错误：忘记 x 前面隐藏的系数 1。
Mistake: subtracting coefficients incorrectly with negative signs.
错误：处理负号时系数相减有误。

Always re-read the original expression after simplifying to make sure no term is lost.

化简后务必回头核对原式，确保没有遗漏任何项。

8. Worked Examples | 例题解析

Let’s work through a few typical examples step by step.

让我们逐步完成几个典型例题。

Example 1: Simplify 6x + 3 − 2x + 4.

例 1：化简 6x + 3 − 2x + 4。

6x − 2x + 3 + 4 = 4x + 7

Here we first collect x terms and constants separately.

这里我们先把 x 项与常数项分别合并。

Example 2: Simplify 5a² − 3a + 2a² + 7a − 1.

例 2：化简 5a² − 3a + 2a² + 7a − 1。

(5a² + 2a²) + (−3a + 7a) − 1 = 7a² + 4a − 1

Example 3: Simplify 4(x + 3) − 2(x − 1).

例 3：化简 4(x + 3) − 2(x − 1)。

First expand brackets: 4x + 12 − 2x + 2. Then collect like terms: 2x + 14.

先去括号：4x + 12 − 2x + 2。然后合并同类项：2x + 14。

9. Mixed Practice | 混合练习

Try these questions on your own, then check the answers below.

请独立尝试以下题目，然后对照下方答案。

1. 8x + 3x	2. 6y − 4y + 2y	3. 2x + 3 + 4x − 1
4. x² + 2x + 3x² − x	5. 5a + 2b − a + b	6. 3p − 2q − p + 5q

Answers: 1. 11x; 2. 4y; 3. 6x + 2; 4. 4x² + x; 5. 4a + 3b; 6. 2p + 3q.

答案：1. 11x；2. 4y；3. 6x + 2；4. 4x² + x；5. 4a + 3b；6. 2p + 3q。

10. Exam Tips | 考试技巧

In IGCSE exams, collecting like terms often appears in paper 1 as a short question, or as part of longer algebraic manipulation.

在 IGCSE 考试中，合并同类项常出现在试卷一的短题中，或作为较长代数运算的一部分。

Always write variables in alphabetical order and in descending powers.
始终按字母顺序排列变量，并按降幂排列。
Use a rough pencil to underline like terms before combining.
用铅笔先划出同类项，再进行合并。
Check the sign in front of each term carefully, especially after removing brackets.
仔细检查每项前的符号，尤其是在去括号之后。
If an expression contains decimal or fraction coefficients, keep them consistent throughout.
若表达式含有小数或分数系数，整个过程需保持一致。

With regular practice, collecting like terms will become automatic and you will avoid unnecessary errors.

通过日常练习，合并同类项会逐渐变得得心应手，从而避免不必要的错误。

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Collecting Like Terms | 合并同类项