Combining Like Terms & Algebraic Simplification | 合并同类项与代数式化简

📚 Combining Like Terms & Algebraic Simplification | 合并同类项与代数式化简

In algebra, almost every expression can be rewritten in a simpler form. The key skill is to recognise and combine like terms — terms that share the exact same variable part. This article explains the rules, shows worked examples, and highlights the traps that appear in IGCSE examinations.

在代数中，几乎每一个代数式都可以改写为更简洁的形式。核心技能是识别并合并同类项，也就是合并那些”字母部分完全相同”的项。本文将解释规则、展示例题，并指出 IGCSE 考试中常见的陷阱。

1. What Are Like Terms? | 什么是同类项？

A term is a single number, a single variable, or a product of numbers and variables, for example 3x, -5xy², or 7. Two terms are like terms if they contain exactly the same variables raised to exactly the same powers. Only the numerical coefficients may differ.

项（term）是一个单独的数字、一个单独的字母，或数字与字母的乘积，例如 3x、-5xy²、7。如果两个项含有完全相同的字母，并且每个字母的指数也完全相同，那么它们就是同类项。只有数字系数可以不同。

For example, 3x and 5x are like terms because both have the variable part x. However, 4x² and 4x are not like terms, because the powers of x are different.

例如，3x 和 5x 是同类项，因为二者的字母部分都是 x。而 4x² 和 4x 不是同类项，因为 x 的幂不同。

3x and 5x → like terms, variable part x; 2xy and -7xy → like terms, variable part xy

3x 与 5x → 同类项，字母部分为 x；2xy 与 -7xy → 同类项，字母部分为 xy

2. Why Do We Combine Like Terms? | 为什么要合并同类项？

Combining like terms is the algebraic version of tidying up. It turns long, messy expressions into shorter equivalent ones, which makes substitution, factorisation and equation solving much easier.

合并同类项就是给代数式”做整理”。它能把冗长杂乱的代数式化成更短的等价形式，从而使代入求值、因式分解和解方程都变得更加容易。

For instance, x + 2x + 3x is exactly 6x. Writing the short form saves time and reduces errors when you evaluate the expression, for example with x = 7.

例如，x + 2x + 3x 正好等于 6x。写出短形式不仅节省时间，还能在代入 x = 7 等数值时减少计算错误。

3. Combining Like Terms with Addition and Subtraction | 用加法与减法合并同类项

To combine like terms, add or subtract their coefficients and keep the variable part unchanged. The variable does not gain a new exponent or a new letter.

合并同类项的方法是：把同类项的系数相加或相减，字母部分保持不变。字母不会因此获得新的指数或新的字母。

Example 1: Simplify 4x + 6x. Both terms have variable part x, so add the coefficients: 4 + 6 = 10. Hence 4x + 6x = 10x.

例 1：化简 4x + 6x。两项的字母部分都是 x，因此把系数相加：4 + 6 = 10。所以 4x + 6x = 10x。

Example 2: Simplify 8a – 3a. The coefficient difference is 8 – 3 = 5, so 8a – 3a = 5a.

例 2：化简 8a – 3a。系数之差为 8 – 3 = 5，所以 8a – 3a = 5a。

When several terms are involved, group all like terms first, write them side by side, then work only on the coefficients.

当项数较多时，应先把同类项归组、写到相邻位置，再只对系数进行运算。

3x + 7x – 2x = (3 + 7 – 2)x = 8x

3x + 7x – 2x = (3 + 7 – 2)x = 8x

4. Simplifying Products of Terms | 化简项与项的乘积

When multiplying two terms, multiply the coefficients together and multiply the variable parts together using the index law xᵐ × xⁿ = x⁽ᵐ⁺ⁿ⁾. This is often called simplifying a product, which is also a form of merging

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