Collecting Like Terms | IGCSE 数学：合并同类项

📚 Collecting Like Terms | IGCSE 数学：合并同类项

In IGCSE Algebra, simplifying expressions is one of the first skills you need to master. Collecting like terms allows you to reduce a long algebraic expression into a shorter, equivalent form, making later work with equations, graphs and formulae much easier.

在 IGCSE 代数中，化简表达式是你需要掌握的第一批技能之一。合并同类项可以把一个很长的代数式化简为更短的等价形式，让你之后学习方程、图像和公式时轻松得多。

1. What Are Like Terms? | 什么是同类项？

Like terms are algebraic terms that have exactly the same variable part and the same exponent on each variable. For example, 3x and -5x are like terms because both contain the variable x to the power of 1. The coefficients, 3 and -5, can be different.

同类项是指变量部分完全相同、每个变量的指数也相同的代数项。例如 3x 和 -5x 是同类项，因为它们都含有变量 x 的一次方。系数 3 和 -5 可以不同。

In contrast, 3x and 3x² are not like terms, because the powers of x are different. Likewise, 4xy and 4x are not like terms, because the variable parts xy and x do not match exactly.

相反，3x 和 3x² 不是同类项，因为 x 的指数不同。同样，4xy 和 4x 也不是同类项，因为变量部分 xy 和 x 并不完全一致。

2. Key Vocabulary: Coefficient, Variable, Exponent | 核心词汇：系数、变量、指数

Before simplifying expressions, you should be confident with three words. The coefficient is the number multiplying a variable, such as 7 in 7y. The variable is the letter symbol, such as y. The exponent or power tells you how many times the variable is multiplied by itself, such as the 2 in y².

在化简表达式之前，你应该对三个词很有把握。系数是乘以变量的数，例如 7y 中的 7。变量是字母符号，例如 y。指数或幂表示变量与自身相乘的次数，例如 y² 中的 2。

When collecting like terms, you only add or subtract the coefficients. The variable part stays exactly the same. This is why 5a + 3a = 8a, not 8a².

合并同类项时，你只需要把系数相加或相减。变量部分保持完全不变。因此 5a + 3a = 8a，而不是 8a²。

3. The Golden Rule: Same Variable, Same Power | 黄金法则：同变量同次幂

A term is only ‘like’ another if the variable letters and their powers match exactly. Use a simple checklist: same letter, same exponent, same number of variables. For example, 2m²n and -6m²n are like terms, but 2mn² is not like 2m²n because the exponent has moved from m to n.

一个项只有与另一个项在变量字母及其指数完全一致时才是“同类”。可以使用一个简单清单：同样的字母、同样的指数、同样数量的变量。例如，2m²n 和 -6m²n 是同类项，但 2mn² 不是 2m²n 的同类项，因为指数从 m 移到了 n 上。

In IGCSE exams, examiners often test this distinction by mixing terms such as x, x², xy and x²y. You must resist the temptation to combine them unless their variable parts are identical.

在 IGCSE 考试中，出题人经常通过混合 x、x²、xy 和 x²y 等项来考查这一区别。你必须抵制把它们合并在一起的冲动，除非它们的变量部分完全相同。

4. Adding and Subtracting Like Terms | 合并同类项的加法和减法

To collect like terms, add or subtract the coefficients and keep the variable part unchanged. For the expression 6p + 2p – 3p, add 6, 2 and subtract 3 to get 5, so the simplified form is 5p.

要合并同类项，就把系数相加或相减，变量部分保持不变。对于表达式 6p + 2p – 3p，把 6、2 相加再减去 3 得到 5，所以化简结果是 5p。

6p + 2p − 3p = (6 + 2 − 3)p = 5p

For terms with negative coefficients, use a number line or think of temperature. Simplify -4k + 9k – 2k: -4 + 9 gives 5, then 5 – 2 gives 3, so the result is 3k.

对于带有负系数的项，可以使用数轴或想象温度变化。化简 -4k + 9k – 2k：-4 + 9 得到 5，然后 5 – 2 得到 3，所以结果为 3k。

Remember that the sign in front of a term belongs to its coefficient. The expression 8t – 3t means 8t + (-3t), so the result is 5t.

请记住，项前面的符号属于它的系数。表达式 8t – 3t 表示 8t + (-3t)，所以结果为 5t。

5. Combining More Than Two Types | 合并多种类型的项

A polynomial may contain several different groups of like terms. Simplify each group separately, then write the simplified groups together. For example, 2x + 3y – x + 5y = (2x – x) + (3y + 5y) = x + 8y.

一个多项式可能包含几组不同的同类项。分别化简每一组，然后把化简后的各组写在一起。例如，2x + 3y – x + 5y = (2x – x) + (3y + 5y) = x + 8y。

2x + 3y − x + 5y = x + 8y

It can help to underline or circle matching terms in different colours or shapes before simplifying. Grouping mentally reduces errors when the expression is long.

化简前

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