Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that look similar. Collecting like terms is the process of simplifying such expressions by adding or subtracting the coefficients of terms that have exactly the same variable part.

在代数中，表达式常常包含若干相似的项。合并同类项就是把变量部分完全相同的项，将其系数相加或相减，从而化简表达式的过程。

This is one of the most important foundations of algebra. Once you master it, you will find solving equations and working with formulas much easier.

这是代数最重要的基础之一。一旦掌握它，你会发现解方程和处理公式都变得轻松许多。

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 powers. For example, 4x and −2x are like terms because both contain the single variable x. Similarly, 5x² and 9x² are like terms because both contain x².

同类项是指变量部分完全相同（包括指数）的项。例如，4x 和 −2x 是同类项，因为都含有变量 x；5x² 和 9x² 也都是同类项，因为都含有 x²。

Constants are also like terms. The numbers 2, −5 and ½ are all constants and can be combined together.

常数项也是同类项。例如 2、−5、½ 都是常数，可以合并。

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

Collecting like terms turns a long, messy expression into a short and clear one. It makes substitution and equation-solving easier, and it is usually the first step in any simplification task.

合并同类项能把冗长杂乱的表达式化为简短清晰的形式，使代入求值和解方程更加方便，也通常是化简题的第一步。

Without collecting like terms, you may lose marks even if your final answer is numerically correct, because the expression is not fully simplified.

如果不合并同类项，即使最终数值正确，也可能因为未彻底化简而被扣分。

3. The Coefficient Rule | 系数的加减法则

To collect like terms, keep the variable part unchanged and add or subtract the coefficients. For instance, 2x + 3x = (2 + 3)x = 5x, and 8y − 5y = (8 − 5)y = 3y.

合并同类项时，变量部分保持不变，只对系数进行加减。例如 2x + 3x = (2 + 3)x = 5x，8y − 5y = (8 − 5)y = 3y。

ax + bx = (a + b)x

ax − bx = (a − b)x

This general rule works for any variable, including x, y, z, and also for expressions with powers such as x².

这个法则对任何变量都成立，包括 x、y、z，也适用于带指数的项，比如 x²。

4. Terms with Different Variables | 不同变量的处理

Terms with different variables are not like terms and cannot be combined. For example, 3x + 4y has no simpler form, and 5x² + 2x cannot be simplified further because the powers are different.

变量不同的项不是同类项，不能合并。例如 3x + 4y 已是最简形式；5x² + 2x 也不能再化简，因为指数不同。

Just like you cannot add apples and oranges directly, you cannot add different variables as if they were the same. In algebra, we write them side by side.

就像不能直接把苹果和橘子相加一样，不能把不同变量当成同一个量。在代数中，我们把它们分开书写。

5. Combining Multiple Terms in Practice | 多项合并实操

When an expression contains several groups of like terms, group them together and combine each group separately.

当表达式中包含多组同类项时，先把同类项分组，再分别合并。

Example: simplify 4a + 3b − 2a + 5b.

例如：化简 4a + 3b − 2a + 5b。

Group the a-terms: 4a − 2a = 2a. Group the b-terms: 3b + 5b = 8b. So the answer is 2a + 8b.

把 a 的项放在一起：4a − 2a = 2a；把 b 的项放在一起：3b + 5b = 8b。所以答案为 2a + 8b。

4a + 3b − 2a + 5b = 2a + 8b

6. Powers and Exponents: Be Careful | 幂与指数：务必小心

Like terms must have exactly the same power on each variable. x² is not the same as x, and x³ is not the same as x².

同类项的每个变量上的指数必须完全相同。x² 和 x 不是同一类，x³ 和 x² 也不是。

For example, 3x² + 2x² = 5x², but 3x² + 2x cannot be simplified into 5x³ or 5x. It stays as 3x² + 2x.

例如，3x² + 2x² = 5x²；但 3x² + 2x 不能化简成 5x³ 或 5x，只能保持 3x² + 2x。

Always ask: are these terms exactly the same apart from the coefficient? If not, they are not like terms.

请时刻自问：除了系数之外，这些项是否完全相同？如果不同，它们就不是同类项。

7. Negative Coefficients and Subtraction | 负系数与减法

Subtraction can be seen as adding a negative number. For example, 7y − 3y = 4y, and −5z + 2z = −3z.

减法可以看作加上一个负数。例如 7y − 3y = 4y，−5z + 2z = −3z。

When you have a term like −x, the coefficient is −1, not 0. So x − x = 0, and 2x − x = x.

当遇到像 −x 这样的项时，它的系数是 −1，而不是 0。所以 x − x = 0，2x − x = x。

Watch the sign in front of each term; it belongs to that term. A common error is ignoring the minus sign.

注意每一项前面的符号，符号属于这一项。常见的错误是忽略了负号。

8. Simplifying with Brackets | 含括号的化简

Before collecting like terms, you must expand any brackets using the distributive law. Only then can you combine terms.

在合并同类项之前，必须先利用分配律去掉括号，才能合并各项。

Example: simplify 2(x + 3) + 3(x + 1).

例如：化简 2(x + 3) + 3(x + 1)。

First expand: 2(x + 3) = 2x + 6, and 3(x + 1) = 3x + 3. The expression becomes 2x + 6 + 3x + 3. Now collect like terms: 2x + 3x = 5x, and 6 + 3 = 9. Final answer: 5x + 9.

先去括号：2(x + 3) = 2x + 6，3(x + 1) = 3x + 3。原式变为 2x + 6 + 3x + 3。再合并同类项：2x + 3x = 5x，6 + 3 = 9。最终答案是 5x + 9。

2(x + 3) + 3(x + 1) = 5x + 9

9. Common Mistakes and Pitfalls | 常见错误与陷阱

Here are some frequent errors that students make when collecting like terms.

以下是同学们在合并同类项时经常犯的错误。

Mistake 1: Combining 3x with 3x². They are not like terms because the powers differ.

错误一：把 3x 与 3x² 合并。它们的指数不同，不是同类项。
Mistake 2: Changing the variable part while combining. For example, writing 2a + 3b = 5ab is wrong.

错误二：合并时改变了变量部分。例如把 2a + 3b 写成 5ab 是错的。
Mistake 3: Forgetting the coefficient 1. For a term like x, the coefficient is 1, so x + x = 2x, not x.

错误三：忘记系数 1。对于 x 这样的项，系数是 1，所以 x + x = 2x，而不是 x。
Mistake 4: Mixing up signs when subtracting. For example, 3a − 2a = a, not 5a or −a.

错误四：做减法时符号混乱。例如 3a − 2a = a，不是 5a 也不是 −a。

10. Exam-Style Worked Example | 考试型例题精讲

Let us work through a full exam-style question step by step.

我们按步骤完整处理一道考试风格例题。

Question: Simplify 3a + 2a² − a + 4a² + 5.

题目：化简 3a + 2a² − a +

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Collecting Like Terms | 合并同类项