Combining Like Terms | 合并同类项

📚 Combining Like Terms | 合并同类项

In IGCSE Mathematics, combining like terms is one of the first algebra skills you must master. It means simplifying an algebraic expression by adding or subtracting terms that have exactly the same variable part. This topic appears in Foundation and Higher tier papers, often inside equation solving, expanding brackets, and factorising questions.

在IGCSE数学中，合并同类项是你必须掌握的第一批代数技能之一。它是指通过加减变量部分完全相同的项来化简代数式。这一主题常出现在基础卷和高级卷中，经常出现在解方程、展开括号和因式分解题目里。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms whose variable factors are identical. In other words, they must contain the same letters, and each letter must have the same power. The number in front, called the coefficient, can be different.

同类项是指变量因式完全相同的项。换句话说，它们必须包含相同的字母，而且每个字母的指数也必须相同。前面的数字，也就是系数，可以不同。

For example, 4x and -7x are like terms because both contain x to the power 1. The terms 3x² and 5x² are also like terms. However, 2x and 2x² are not like terms, because one has x to the power 1 and the other has x to the power 2.

例如，4x 和 -7x 是同类项，因为两者都含有 x 的一次方。3x² 和 5x² 也是同类项。但 2x 和 2x² 不是同类项，因为一个含有 x 的一次方，另一个含有 x 的二次方。

2. Why Combine Like Terms? | 为什么合并同类项？

Algebraic expressions often contain many repeat parts. Combining like terms makes an expression shorter, clearer, and easier to work with. It also helps you solve equations because the left-hand side or right-hand side can usually be simplified first.

代数式经常包含许多重复的部分。合并同类项可以让表达式更短、更清晰、更容易处理。它还能帮助你解方程，因为方程左边或右边通常可以先化简。

Without this skill, you may end up carrying too many terms through a calculation, which increases the chance of sign errors and arithmetic mistakes.

如果没有这个技能，你在计算中可能会携带过多项，这会增加符号错误和计算错误的风险。

3. Identifying Like Terms | 识别同类项

To identify like terms, first ignore the coefficient and look only at the variable part. Ask yourself: are the letters the same? Are the powers the same? If both answers are yes, the terms are like terms.

要识别同类项，先忽略系数，只看变量部分。问自己：字母相同吗？指数相同吗？如果两个答案都是肯定的，这些项就是同类项。

Constant terms such as 2, -5, and 8 are also like terms with each other, because they have no variable part at all.

常数项如 2、-5 和 8 也互为同类项，因为它们完全没有变量部分。

Be careful: terms like ab and a²b are not like terms, because the power of a is different. Similarly, xy and xz are not like terms because the letter sets differ.

注意：ab 和 a²b 不是同类项，因为 a 的指数不同。同样，xy 和 xz 不是同类项，因为字母组合不同。

4. Adding and Subtracting Like Terms | 加减同类项

When you add or subtract like terms, keep the variable part unchanged and simply add or subtract the coefficients.

当你加减同类项时，保持变量部分不变，只需加减系数。

For example, 4a + 9a means 4 lots of a plus 9 lots of a, which is 13 lots of a. Therefore, 4a + 9a = 13a.

例如，4a + 9a 表示 4 个 a 加上 9 个 a，也就是 13 个 a。因此，4a + 9a = 13a。

4a + 9