Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, an expression such as 3x + 5x – 2 can be made much simpler. Collecting like terms means identifying terms that have identical variable parts and adding or subtracting their coefficients.

在 IGCSE 代数中，像 3x + 5x – 2 这样的式子可以化简很多。合并同类项就是找出变量部分完全相同的项，并把它们的系数相加或相减。

1. What Are Like Terms? | 什么是同类项

Like terms are terms whose variable parts are exactly the same. The coefficients can be different. For example, 3x and 5x are like terms because both contain the variable x to the power 1.

同类项是指变量部分完全相同的项。系数可以不同。例如，3x 和 5x 是同类项，因为它们都含有变量 x 的一次方。

Non-examples include 3x and 3x²: the powers of x are different, so they are not like terms. Also, 3x and 3y are not like terms because the variables differ.

非同类项的例子包括 3x 和 3x²：x 的指数不同，因此它们不是同类项。同样，3x 和 3y 也不是同类项，因为变量不同。

2. The Idea of a Term | 项的概念

A term is a single number, variable, or product of numbers and variables. In 4x – 7 + 2y, the terms are 4x, -7, and 2y. The sign before a term belongs to that term.

项是单个数字、变量或数字与变量的乘积。在 4x – 7 + 2y 中，各项分别是 4x、-7 和 2y。项前面的符号属于该项。

When collecting, always move the sign with the term: -7 is a constant term, not 7.

合并时，符号要跟着项一起移动：-7 是常数项，而不是 7。

3. Why Collect Like Terms? | 为什么要合并同类项

Collecting like terms reduces a long expression to its simplest equivalent form. This makes substitution, solving equations, and graphing much easier.

合并同类项可以把一个长表达式化为最简等价形式。这会让代入求值、解方程和画图变得容易得多。

For example, 2x + 3 + 5x – 1 simplifies to 7x + 2. Both expressions have the same value for any x, but the simplified form is shorter.

例如，2x + 3 + 5x – 1 可化简为 7x + 2。这两个式子对任意 x 的值都相同，但化简后的形式更短。

2x + 3 + 5x – 1 = 7x + 2

4. The Basic Rule: Add the Coefficients | 基本法则：系数相加

To combine like terms, keep the variable part unchanged and add only the coefficients. For 3x + 5x, add 3 and 5 to get 8, so the result is 8x.

合并同类项时，变量部分保持不变，只把系数相加。对于 3x + 5x，将 3 和 5 相加得到 8，所以结果是 8x。

3x + 5x = (3 + 5)x = 8x

For subtraction, treat the coefficient as negative: 7a – 2a = 5a.

对于减法，把系数看作负数：7a – 2a = 5a。

7a – 2a = (7 – 2)a = 5a

5. Collecting Terms with One Variable | 合并含一个变量的项

Start with expressions that contain only one variable. Scan the expression, group the x terms, then group the constant terms, and finally simplify each group.

从只含一个变量的式子开始。先扫描整个式子，把含 x 的项归为一组，再把常数项归为一组，最后分别化简。

Example: Simplify 6x + 3 – 2x + 8.

例子：化简 6x + 3 – 2x + 8。

Group the x terms: 6x – 2x = 4x. Group the constants: 3 + 8 = 11. The expression becomes 4x + 11.

将 x 项归组：6x – 2x = 4x。将常数项归组：3 + 8 = 11。原式变为 4x + 11。

6x + 3 – 2x + 8 = 4x + 11

6. Collecting Terms with Multiple Variables | 合并含多个变量的项

When an expression contains several different variables, collect each type separately. For example, 2x + 3y – x + 5y has x terms and y terms.

当式子中含有多个不同变量时，要按类型分别合并。例如，2x + 3y – x + 5y 中含有 x 项和 y 项。

The x terms give 2x – x = x. The y terms give 3y + 5y = 8y. So the simplified form is x + 8y.

x 项合并为 2x – x = x。y 项合并为 3y + 5y = 8y。所以化简结果是 x + 8y。

2x + 3y – x + 5y = x + 8y

Never add coefficients of different variables: 3x + 4y cannot become 7xy or 7(x + y).

绝不能把不同变量的系数相加：3x + 4y 不能写成 7xy 或 7(x + y)。

7. Collecting Terms with Powers | 合并含幂的项

Terms with different powers are not like terms. For instance, x and x² look similar, but x means x¹, so x¹ and x² have different exponents.

指数不同的项不是同类项。例如，x 和 x² 看起来相似，但 x 表示 x¹，因此 x¹ 和 x² 的指数不同。

Combine only the same powers: 3x² + 5x² = 8x², but 3x² + 5x cannot be combined into one term.

只能合并相同次幂的项：3x² + 5x² = 8x²，但 3x² + 5x 不能合并成一个项。

3x² + 5x² = 8x²

Example with mixed powers: 4x² + 2x – x² + 7x = 3x² + 9x.

混合次幂的例子：4x² + 2x – x² + 7x = 3x² + 9x。

8. Constants and Mixed Expressions | 常数项与混合表达式

Constants are numbers without variables. All constant terms are like terms with each other. In 5 + 3x – 2 + 7x, the constants 5 and -2 combine to 3, and the x terms combine to 10x.

常数是不含变量的数。所有常数项彼此都是同类项。在 5 + 3x – 2 + 7x 中，常数 5 和 -2 合并为 3，x 项合并为 10x。

5 + 3x – 2 + 7x = 3 + 10x = 10x + 3

In IGCSE answers, it is common to write the variable terms first and then the constant, though 3 + 10x is equivalent.

在 IGCSE 答案中，通常把变量项写在前面、常数项写在后面，不过 3 + 10x 也是等价的。

9. Common Mistakes | 常见错误

Students often combine unlike terms by mistake. For example, 2x + 3x² is sometimes written as 5x² or 5x, but both are wrong.

学生经常会错误地合并不同类项。例如，有人会把 2x + 3x² 写成 5x² 或 5x，但两种写法都是错的。

Another common error is losing the sign of a term. In 5 – 2x + 3x, the -2x must combine with +3x to give +x, not 5 + x incorrectly from changing signs.

另一个常见错误是丢掉项的符号。在 5 – 2x + 3x 中，-2x 必须与 +3x 合并得到 +x，不能因符号错误而得到 5 + x 的错误过程。

Also, 3x × 4x is multiplication, not collecting like terms: it gives 12x², while 3x + 4x gives 7x.

此外，3x × 4x 是乘法，不是合并同类项：结果是 12x²，而 3x + 4x 的结果是 7x。

10. Worked Examples | 例题解析

Work through examples carefully before practising. Each line shows one logical step.

练习前请先仔细阅读例题。每一行展示一个逻辑步骤。

Example 1: Simplify 8a + 3b – 5a + 2b – 4.

例 1：化简 8a + 3b – 5a + 2b – 4。

Collect a terms: 8a – 5a = 3a. Collect b terms: 3b + 2b = 5b. The constant is -4. So the answer is 3a + 5b – 4.

合并 a 项：8a – 5a = 3a。合并 b 项：3b + 2b = 5b。常数项为 -4。所以答案是 3a + 5b – 4。

8a + 3b – 5a + 2b – 4 = 3a + 5b – 4

Example 2: Simplify 6x² – 3x + 2x² + x – 5.

例 2：化简 6x² – 3x + 2x² + x – 5。

Collect x² terms: 6x² + 2x² = 8x². Collect x terms: -3x + x = -2x. Constant: -5. Answer: 8x² – 2x – 5.

合并 x² 项：6x² + 2x² = 8x²。合并 x 项：-3x + x = -2x。常数项：-5。答案：8x² – 2x – 5。

6x² – 3x + 2x² + x – 5 = 8x² – 2x – 5

11. Practice Checklist | 练习清单

Use this checklist when simplifying: first identify the terms, underline or circle like terms, move the sign with each term, combine coefficients, and write the simplified expression in order.

化简时请使用这份清单：先识别各项，给同类项画线或画圈，符号跟着项走，合并系数，最后按顺序写出化简后的式子。

Identify each term with its sign | 识别每一项及其符号
Group exactly matching variable parts | 将变量部分完全相同的项归组
Add or subtract coefficients only | 只对系数进行加减
Keep unlike terms separate | 不同类项保持分开
Check by substituting a simple value | 用一个简单数值代入检验

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Collecting Like Terms | 合并同类项