Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, collecting like terms is one of the earliest and most powerful simplification tools. It allows you to rewrite a long expression in a shorter, clearer form without changing its value. Mastering this skill will help you solve equations, rearrange formulas, and handle more advanced algebra with confidence.

在 IGCSE 代数中，合并同类项是最早接触、也是最实用的化简工具之一。它可以让你把长表达式改写成更短、更清晰的形式，而不改变其值。掌握这一技能将帮助你解方程、变换公式，并自信地处理更高级的代数内容。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable part, including the same variables and the same powers. The coefficients, or the number parts, can be different. For example, 3x, −7x, and ½x are like terms because they all contain x to the power 1.

同类项是指变量部分完全相同的项，包括相同的变量和相同的指数。系数（数字部分）可以不同。例如，3x、−7x 和 ½x 是同类项，因为它们都含有 x 的一次方。

On the other hand, 3x and 3x² are not like terms, because the powers of x are different. Likewise, 2xy and 3x are not like terms because the variable combinations are not the same.

另一方面，3x 和 3x² 不是同类项，因为 x 的指数不同。同样，2xy 和 3x 也不是同类项，因为变量的组合不一样。

3x, −7x, ½x → like terms

2. Why Collecting Matters | 为什么要合并

Simplifying an expression by collecting like terms makes it shorter and easier to understand. It also reduces mistakes when substituting values into the expression. For example, 2a + 3b + 5a − b is much easier to evaluate once it is written as 7a + 2b.

通过合并同类项化简表达式，可以让它更短、更容易理解。它还可以减少代入数值时的错误。例如，2a + 3b + 5a − b 一旦写成 7a + 2b，计算起来就容易多了。

In exam questions, leaving a final answer simplified is often required for full marks. Expressions that are not simplified may be marked as incomplete or incorrect.

在考试题目中，最终答案通常要求化简后才能得满分。没有化简的表达式可能被视为不完整或错误。

3. The Golden Rule | 黄金法则

The golden rule of collecting like terms is simple: only change the coefficient, and keep the variable part exactly the same. For instance, 5a + 2a means 5 lots of a plus 2 lots of a, which gives 7 lots of a.

合并同类项的黄金法则很简单：只改变系数，变量部分保持完全不变。例如，5a + 2a 表示 5 个 a 加上 2 个 a，一共得到 7 个 a。

5a + 2a = (5 + 2)a = 7a

This rule works for subtraction too: 9m − 4m = (9 − 4)m = 5m. The variable m is not squared, multiplied, or removed.

这个法则同样适用于减法：9m − 4m = (9 − 4)m = 5m。变量 m 不会被平方、相乘或消失。

4. Single Variable Examples | 单变量示例

Let’s start with one variable. To simplify 4x + 9x − 3x, collect the coefficients: 4 + 9 − 3 = 10. The simplified expression is therefore 10x.

我们先从单变量开始。要化简 4x + 9x − 3x，先把系数合并：4 + 9 − 3 = 10。因此化简结果为 10x。

4x + 9x − 3x = 10x

Another example is 12y − 5y + y. Remember that y on its own has coefficient 1. So 12 − 5 + 1 = 8, giving 8y.

另一个例子是 12y − 5y + y。记住单独一个 y 的系数是 1。因此 12 − 5 + 1 = 8，得到 8y。

12y − 5y + y = 8y

5. Multiple Variables | 多变量

When an expression has more than one variable, collect each variable separately. For example, in 2x + 3y + 5x − y, the x terms are 2x and 5x, and the y terms are 3y and −y.

当表达式中含有多个变量时，要分别合并每一种变量。例如，在 2x + 3y + 5x − y 中，x 项是 2x 和 5x，y 项是 3y 和 −y。

Collect the x terms to get 7x, and collect the y terms to get 2y. The simplified answer is 7x + 2y.

合并 x 项得到 7x，合并 y 项得到 2y。化简结果是 7x + 2y。

2x + 3y + 5x − y = 7x + 2y

Be careful not to combine different variable types. For example, 4ab + 3a − 2ab + a simplifies to 2ab + 4a, not

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