Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, collecting like terms is one of the most fundamental skills. It helps you simplify expressions, solve equations, and work with formulas more efficiently. This topic is essential for IGCSE Mathematics and appears in almost every exam paper.

合并同类项是代数中最基本的技能之一。它帮助你简化表达式、解方程以及更高效地处理公式。该主题是 IGCSE 数学的核心内容，几乎出现在每份试卷中。

1. What Are Like Terms? | 什么是同类项

Like terms are terms that have exactly the same variable parts. The variable part includes the variable and its power. For example, 3x and 5x are like terms because both have the variable x raised to the power 1.

同类项是指变量部分完全相同的项。变量部分包括变量及其指数。例如，3x 和 5x 是同类项，因为它们都含有指数为 1 的变量 x。

Similarly, 2x² and 7x² are like terms because both have x squared. However, 3x and 3x² are not like terms because the powers of x are different.

同样，2x² 和 7x² 是同类项，因为它们都含有 x 的平方。然而，3x 和 3x² 不是同类项，因为 x 的指数不同。

Numbers without variables are called constant terms. All constant terms are like terms. For instance, 5 and -3 are like terms because neither contains a variable.

不含变量的数字称为常数项。所有常数项都是同类项。例如，5 和 -3 是同类项，因为它们都不含变量。

2. Why Do We Collect Like Terms? | 为什么合并同类项

Collecting like terms makes an expression shorter and easier to understand. A simplified expression is less likely to cause calculation mistakes when you substitute numbers into it.

合并同类项使表达式更简短、更容易理解。简化后的表达式在进行数值代入计算时，更不容易引发计算错误。

For example, the expression 4x + 3 + 2x + 5 can be simplified to 6x + 8. The simplified version is much clearer and easier to work with.

例如，表达式 4x + 3 + 2x + 5 可以简化为 6x + 8。简化后的形式更加清晰，也更便于后续处理。

In solving equations, collecting like terms is a key step before isolating the variable. Without this step, the equation may remain messy and difficult to solve.

在解方程时，合并同类项是在分离变量之前的关键步骤。没有这一步，方程会显得杂乱，难以求解。

3. The Basic Rules | 基本规则

To collect like terms, you add or subtract their coefficients. The coefficient is the number in front of the variable. The variable part stays exactly the same.

合并同类项时，你对它们的系数进行加或减。系数是变量前面的数字。变量部分保持完全不变。

ax + bx = (a + b)x

ax − bx = (a − b)x

For example, 2x + 3x = 5x. Here, we add the coefficients 2 and 3 to get 5, and keep x unchanged.

例如，2x + 3x = 5x。这里，我们将系数 2 和 3 相加得到 5，并保持 x 不变。

If the coefficient is 1, it is usually not written. So x means 1x. When collecting terms such as x + 2x, remember that x has a hidden coefficient of 1.

如果系数是 1，通常省略不写。因此 x 就表示 1x。当处理像 x + 2x 这样的项时，请记住 x 有一个隐藏的系数 1。

4. Adding and Subtracting Like Terms | 同类项的加减

When adding like terms, simply add the coefficients and keep the variable part the same. For instance, 3y + 4y = 7y.

当加法合并同类项时，只需将系数相加，并保持变量部分不变。例如，3y + 4y = 7y。

When subtracting like terms, subtract the coefficients. For example, 9p − 4p = 5p. Notice that the order of subtraction must be respected.

当减法合并同类项时，将系数相减。例如，9p − 4p = 5p。注意减法的顺序必须遵循。

If there is a negative term, treat it as adding a negative number. For example, 5x + (−3x) = 2x, which is the same as 5x − 3x.

如果存在负项，可以将其视为加上一个负数。例如，5x + (−3x) = 2x，这等同于 5x − 3x。

5. Combining Terms with Different Variables | 合并不同变量的项

Terms with different variables are not like terms. They cannot be combined by adding or subtracting their coefficients. For example, 2x + 3y must remain as 2x + 3y.

具有不同变量的项不是同类项。它们不能通过系数相加或相减来合并。例如，2x + 3y 必须保持为 2x + 3y。

Terms with the same variable but different powers are also not like terms. For example, 4x + 5x² cannot be simplified further.

变量相同但指数不同的项也不是同类项。例如，4x + 5x² 不能再进一步化简。

Constant terms can only be combined with other constant terms. So 6 + 2x + 4 simplifies to 2x + 10, not 8x.

常数项只能与其他常数项合并。所以 6 + 2x + 4 简化为 2x + 10，而不是 8x。

6. Dealing with Coefficients and Constants | 处理系数和常数项

In an expression, constants are the numbers without variables. They can be added or subtracted freely. For example, 3a + 7 + 2a − 4 simplifies to 5a + 3.

在表达式中，常数是不含变量的数字。它们可以自由相加或相减。例如，3a + 7 + 2a − 4 简化为 5a + 3。

When a term has no visible number before the variable, the coefficient is 1. For example, b means 1b, and −c means −1c. Be careful when combining such terms.

当一项的变量前面没有可见数字时，系数就是 1。例如，b 表示 1b，−c 表示 −1c。合并这类项时要小心。

If a coefficient is a fraction or decimal, the same rule applies. For example, ½x + ½x = x, and 0.5y + 0.3y = 0.8y.

如果系数是分数或小数，同样适用此规则。例如，½x + ½x = x，以及 0.5y + 0.3y = 0.8y。

7. Simplifying Longer Expressions | 简化长表达式

Longer expressions may contain several groups of like terms. To simplify them, first identify the different variable groups and constant group. Then combine each group separately.

较长的表达式可能包含几组同类项。为了简化它们，首先要识别不同的变量组和常数组，然后分别合并每一组。

For example, simplify 4x + 3y − 2x + 5y. Group the x terms: 4x − 2x = 2x. Group the y terms: 3y + 5y = 8y. So the result is 2x + 8y.

例如，化简 4x + 3y − 2x + 5y。将 x 项分组：4x − 2x = 2x。将 y 项分组：3y + 5y = 8y。因此结果是 2x + 8y。

It is often helpful to rearrange the terms so that like terms are next to each other. For instance, 7a − 2b + 3a + b can be rearranged to 7a + 3a − 2b + b, then simplified to 10a − b.

将项重新排列，使同类项相邻，通常会很有帮助。例如，7a − 2b + 3a + b 可以重新排列为 7a + 3a − 2b + b，然后简化为 10a − b。

8. Common Mistakes to Avoid | 避免常见错误

One common mistake is combining terms that are not like terms, such as adding x and x² to get 2x². This is incorrect. x and x² must remain separate.

一个常见错误是合并不是同类项的项，例如将 x 和 x² 相加得到 2x²。这是错误的。x 和 x² 必须保持分开。

Another mistake is forgetting the sign in front of a term. For example, in 3x − 2x + 5, the minus sign belongs to 2x, so we calculate 3x − 2x = x, not 3x + 2x.

另一个错误是忘记项前面的符号。例如，在 3x − 2x + 5 中，负号属于 2x，所以我们计算 3x − 2x = x，而不是 3x + 2x。

Students sometimes drop the coefficient 1. For example, x + 2x should be 3x, not 2x² or 2x. Always remember that x is 1x.

学生有时会遗漏系数 1。例如，x + 2x 应该是 3x，而不是 2x² 或 2x。永远记住 x 就是 1x。

Finally, do not forget to combine constant terms. In 4x + 6 − 2x + 3, the constants 6 and 3 also need to be combined to give an answer of 2x + 9.

最后，不要忘记合并常数项。在 4x + 6 − 2x + 3 中，常数 6 和 3 也需要合并，最终答案是 2x + 9。

9. Worked Examples | 例题解析

Example 1: Simplify 5x + 3y − 2x + 4y.

例题 1：化简 5x + 3y − 2x + 4y。

Solution: Combine x terms: 5x − 2x = 3x. Combine y terms: 3y + 4y = 7y. The simplified expression is 3x + 7y.

解答：合并 x 项：5x − 2x = 3x。合并 y 项：3y + 4y = 7y。化简后的表达式是 3x + 7y。

Example 2: Simplify 2a² + 3a + 4a² − a + 7.

例题 2：化简 2a² + 3a + 4a² − a + 7。

Solution: Combine a² terms: 2a² + 4a² = 6a². Combine a terms: 3a − a = 2a. The constant is 7. So the result is 6a² + 2a + 7.

解答：合并 a² 项：2a² + 4a² = 6a²。合并 a 项：3a − a = 2a。常数项是 7。所以结果是 6a² + 2a + 7。

Example 3: Simplify 4x − 2y + 6 − 3x + 5y − 2.

例题 3：化简 4x − 2y + 6 − 3x + 5y − 2。

Solution: x terms: 4x − 3x = x. y terms: −2y + 5y = 3y. Constants: 6 − 2 = 4. Therefore the simplified form is x + 3y + 4.

解答：x 项：4x − 3x = x。y 项：−2y + 5y = 3y。常数项：6 − 2 = 4。因此简化形式为 x + 3y + 4。

10. Practice Questions | 练习

Try these questions yourself. Check your answers in the table below.

请自行尝试以下练习。在下面的表格中核对你的答案。

Question	Answer
1. Simplify 7x + 2x	9x
2. Simplify 4a − a + 3a	6a
3. Simplify 2x + 3y + 5x − y	7x + 2y
4. Simplify 3x² + 2x + 4x²	7x² + 2x
5. Simplify 6 + 2m − 4 + m	3m + 2
6. Simplify 5p − 2q + 3p + 4q	8p + 2q

If you got any wrong, identify which group of like terms caused the difficulty and review the rules again.

如果你做错了任何题目，请找出哪一组同类项导致困难，并重新复习相关规则。

11. Applications | 应用

Collecting like terms is used in solving linear equations. For example, solve 3x + 5 = x + 9. First, subtract x from both sides: 2x + 5 = 9. Then subtract 5: 2x = 4, so x = 2.

合并同类项用于解线性方程。例如，解 3x + 5 = x + 9。首先，两边同时减去 x：2x + 5 = 9。然后两边同时减去 5：2x = 4，所以 x = 2。

In geometry, perimeter expressions often involve collecting like terms. For a rectangle with length 3x + 2 and width x + 1, the perimeter is 2(3x + 2) + 2(x + 1) = 8x + 6.

在几何中，周长表达式经常涉及合并同类项。对于一个长为 3x + 2、宽为 x + 1 的长方形，周长是 2(3x + 2) + 2(x + 1) = 8x + 6。

In probability and statistics, expressions like n + 2n + 3n simplify to 6n, which helps in calculating weighted totals quickly.

在概率和统计中，像 n + 2n + 3n 这样的表达式简化为 6n，这有助于快速计算加权总数。

12. Summary | 总结

Collecting like terms is the process of simplifying an expression by combining terms with identical variable parts. Always add or subtract coefficients while keeping the variables unchanged.

合并同类项是通过合并具有相同变量部分的项来简化表达式的过程。始终对系数进行加或减，同时保持变量不变。

Remember these key points: only like terms can be combined; constant terms combine together; different variables or different powers cannot be combined; and keep the signs of terms correct.

记住这些要点：只有同类项才能合并；常数项相互合并；不同变量或不同指数不能合并；并保持各项的符号正确。

With regular practice, collecting like terms will become quick and automatic. It is a skill that you will use throughout your IGCSE course and beyond.

通过经常练习，合并同类项会变得快速而自然。这项技能在 IGCSE 课程及今后的学习中都会反复用到。

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Collecting Like Terms | 合并同类项