Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In IGCSE Mathematics, algebraic expressions often contain several terms that can be simplified. Combining like terms is the first essential skill for simplifying expressions and solving equations accurately.

在 IGCSE 数学中，代数式经常含有多个可以化简的项。合并同类项是化简代数式、准确解方程的第一项核心技能。

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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 can be different.

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次数。系数可以不同。

For example, 3x and -5x are like terms because both contain x to the power 1. However, 3x and 3x² are not like terms because the powers of x are different.

例如，3x 和 -5x 是同类项，因为两者都含有 x 的一次方。然而，3x 和 3x² 不是同类项，因为 x 的次数不同。

Like terms	Not like terms
3x and -5x	3x and 3x²
2ab and 7ab	2ab and 2a
x² and -4x²	x² and x³

The table shows that coefficients do not matter when deciding whether terms are like terms. Only the variable part must match.

上表说明，判断是否为同类项时系数并不重要，只有变量部分必须一致。

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2. Why Combining Matters | 为什么要合并同类项

Combining like terms makes an expression shorter and easier to use. It is required before substituting values, drawing linear graphs, solving equations, and rearranging formulas.

合并同类项能让代数式更短、更易使用。在代入数值、画线性图像、解方程和变形公式之前，通常都需要先合并同类项。

9 would require more steps and can lead to errors. With combining, it becomes 5x + 4 = 19.

如果不合并，解 2x + 3x + 4 = 19 需要更多步骤且容易出错。合并后，它变为 5x + 4 = 19。

2x + 3x + 4 = 19 → 5x + 4 = 19

In examinations, simplified expressions often gain method marks even if the final numerical answer is wrong, so this skill is worth practising carefully.

在考试中，即使最终数值答案错误，化简后的表达式也常常能获得方法分，因此这项技能值得认真练习。

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3. Identifying Like Terms | 识别同类项

To identify like terms, first ignore the coefficient and look only at the variable part. The variable letters and their exponents must match exactly.

识别同类项时，首先忽略系数，只看变量部分。变量字母及其指数必须完全一致。

Terms such as 5x²y and -3x²y are like terms because both contain x²y. Terms such as 5xy and 5x²y are not like terms because the power of x differs.

5x²y 和 -3x²y 是同类项，因为两者都含有 x²y。5xy 和 5x²y 不是同类项，因为 x 的指数不同。

Also, xy and yx are like terms because multiplication is commutative, but it is often helpful to write variables in alphabetical order.

此外，xy 和 yx 是同类项，因为乘法满足交换律，但通常按字母顺序书写更清晰。

Always write the variable part in a consistent order, for example 3xy rather than 3yx, so that like terms are easier to spot.

始终用一致的顺序书写变量部分，例如写 3xy 而不是 3yx，这样更容易识别同类项。

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4. Single-Variable Examples | 单变量例子

For expressions with one variable, combine the coefficients and keep the variable unchanged.

对于只含一个变量的代数式，合并系数并保持变量不变。

Simplify 7n + 2n - 3n.

化简 7n + 2n - 3n。

Add the coefficients: 7 + 2 - 3 = 6, so the result is 6n.

将系数相加：7 + 2 - 3 = 6，所以结果是 6n。

7n + 2n − 3n = 6n

Simplify 9p - 4p + p.

化简 9p - 4p + p。

Remember that p means 1p, so 9 - 4 + 1 = 6. The answer is 6p.

记住 p 表示 1p，所以 9 - 4 + 1 = 6。答案是 6p。

9p − 4p + p = 6p

If the coefficient is 1, you do not need to write it. For example, 1x is simply written as x.

如果系数是 1，就不需要写出 1。例如，1x 写作 x。

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5. Multiple Variables and Powers | 多变量与幂

When an expression contains several different variable parts, combine only the terms whose variable parts match exactly.

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