Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain terms that share the same variable part. Collecting like terms is the process of simplifying such expressions by adding or subtracting the coefficients of those identical terms. This skill is fundamental to solving equations, factorising, and working with formulas.

在代数中，表达式常常包含具有相同变量部分的项。合并同类项就是通过加减这些相同项的系数来化简表达式的过程。这一技能是解方程、因式分解以及处理公式的基础。

1. What Are ‘Like Terms’? | 什么是“同类项”？

Like terms are terms that have exactly the same variables raised to exactly the same powers. For example, 3x and 5x are like terms because both contain the variable x to the first power. Similarly, 2y² and 7y² are like terms because both contain y².

同类项是指变量完全相同、且变量对应的指数也完全相同的项。例如，3x 和 5x 是同类项，因为它们都包含一次变量 x。同样，2y² 和 7y² 是同类项，因为它们都包含 y²。

Only the coefficient (the number in front) may be different. The variable part must match exactly, including its exponent.

只有系数（前面的数字）可以不同。变量部分必须完全一致，包括其指数。

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

Collecting like terms makes an expression shorter and easier to understand. It also prepares the expression for substitution, graphing, or solving equations. For example, 4a + 3a is awkward, but 7a is simple and clear.

合并同类项可以使表达式更简短、更易于理解。它也为代入数值、绘制图像或解方程做好准备。例如，4a + 3a 很繁琐，而 7a 简洁明了。

In real-world problems, like terms often represent the same quantity measured in the same units. Combining them avoids unnecessary repetition and reduces the chance of calculation errors.

在实际问题中，同类项通常代表以相同单位度量的同一类数量。合并它们可以避免不必要的重复，并减少计算错误的机会。

3. The Basic Rule: Add or Subtract Coefficients | 基本规则：系数相加减

To collect like terms, keep the variable part unchanged and add or subtract only the coefficients. For instance:

合并同类项时，保持变量部分不变，只对系数进行加减。例如：

5x + 3x = (5 + 3)x = 8x

Here, the variable x is written once, and the coefficients 5 and 3 are added.

这里，变量 x 只写一次，系数 5 和 3 相加。

Similarly, 9m − 4m = 5m because 9 − 4 = 5 and the variable m stays the same.

类似地，9m − 4m = 5m，因为 9 − 4 = 5，变量 m 保持不变。

4. Different Variables Are Never Like Terms | 不同变量绝不是同类项

Terms with different variables cannot be combined. For example, 2x + 3y cannot be simplified to a single term because x and y represent different quantities.

具有不同变量的项不能合并。例如，2x + 3y 不能化简为一项，因为 x 和 y 代表不同的数量。

Even if the coefficients are equal, such as 5p + 5q, the expression remains 5p + 5q. They are not like terms because the variable letters differ.

即使系数相同，例如 5p + 5q，表达式仍然保持为 5p + 5q。由于变量字母不同，它们不是同类项。

However, we can still factorise such expressions if needed: 5p + 5q = 5(p + q). But that is a different operation, not collecting like terms.

然而，如果需要，我们仍然可以因式分解这类表达式：5p + 5q = 5(p + q)。但这是不同的运算，不是合并同类项。

5. Constant Terms: The ‘Number-Only’ Like Terms | 常数项：纯数字的同类项

Constant terms have no variable part. All constants are like terms with each other. For example, in the expression 7 + 3x − 2 + 5x, the numbers 7 and −2 are like terms.

常数项没有变量部分。所有常数彼此之间都是同类项。例如，在表达式 7 + 3x − 2 + 5x 中，数字 7 和 −2 是同类项。

We collect the constant terms separately from the variable terms:

我们应分别合并常数项和变量项：

7 − 2 = 5, and 3x + 5x = 8x, so 7 + 3x − 2 + 5x = 5 + 8x

Notice that the simplified expression usually places the variable term first: 8x + 5.

注意，化简后的表达式通常将变量项写在前面：8x + 5。

6. Coefficients Larger Than 1, Fractions, and Decimals | 大于1的系数、分数与小数

Coefficients can be fractions or decimals. For example, ½x + ⅓x cannot be added by simply combining the fractional numbers unless we find a common denominator:

系数可以是分数或小数。例如，½x + ⅓x 不能直接合并分数部分，除非找到公分母：

½x + ⅓x = (3⁄6 + 2⁄6)x = 5⁄6x

Decimal coefficients work the same way: 0.4a + 0.35a = 0.75a.

小数系数同样处理：0.4a + 0.35a = 0.75a。

Always keep your final answer in the simplest acceptable form. If the question asks for an exact answer, use fractions rather than rounded decimals.

始终将最终答案化为最简可接受的形式。如果题目要求精确答案，应使用分数而不是四舍五入的小数。

7. Subtractions and Negative Coefficients | 减法与负系数

When subtracting, remember that the sign belongs to the term that follows it. In the expression 6x − 2x + 4x, we treat it as 6x + (−2x) + 4x. The coefficients combine as 6 − 2 + 4 = 8, so the result is 8x.

做减法时，请记住符号属于它后面的项。在表达式 6x − 2x + 4x 中，我们将其视为 6x + (−2x) + 4x。系数合成为 6 − 2 + 4 = 8，因此结果是 8x。

Notice the negative sign is attached to the 2. This is why we write 6x − 2x rather than 6x + − 2x.

注意负号属于 2。这就是为什么我们写 6x − 2x 而不是 6x + − 2x。

For an expression such as 3a − 5a, the coefficient becomes 3 − 5 = −2, giving −2a.

对于 3a − 5a 这样的表达式，系数变为 3 − 5 = −2，结果为 −2a。

8. Terms with More Than One Variable | 含多个变量的项

Terms like 4xy and −2xy are like terms because the variable product xy is identical in both. We combine them by adding or subtracting coefficients only:

像 4xy 和 −2xy 这样的项是同类项，因为变量乘积 xy 在两者中完全相同。我们只对系数进行加减来合并它们：

4xy − 2xy = 2xy

However, 4xy and 4x²y are not like terms because the powers of x differ (one is x¹, the other is x²). Similarly, xy and yx are actually like terms because multiplication is commutative: xy = yx.

然而，4xy 和 4x²y 不是同类项，因为 x 的幂不同（一个是 x¹，另一个是 x²）。类似地，xy 和 yx 实际上是同类项，因为乘法满足交换律：xy = yx。

Always compare the full set of variables and their exponents, not just the first letter.

务必比较完整的变量集合及其指数，而不是只看第一个字母。

9. Collecting Like Terms Inside Brackets | 括号内的合并

Sometimes like terms appear inside brackets. Before collecting them, we may need to expand the brackets. For example, simplify 3(x + 2) + 2(x + 1):

有时同类项出现在括号内。在合并之前，我们可能需要先展开括号。例如，化简 3(x + 2) + 2(x + 1)：

3(x + 2) + 2(x + 1) = 3x + 6 + 2x + 2 = 5x + 8

First expand each bracket, then collect the x-terms and the constant terms.

先展开每个括号，然后合并含 x 的项和常数项。

If there is a minus sign before a bracket, multiply every term inside by −1. For example, 4a − (2a + 1) = 4a − 2a − 1 = 2a − 1.

如果括号前有负号，则用 −1 乘以括号内的每一项。例如，4a − (2a + 1) = 4a − 2a − 1 = 2a − 1。

10. Applying Collecting Like Terms in Geometry | 在几何中应用合并同类项

Perimeter problems often generate like terms. If a rectangle has sides (2x + 3) and (x + 5), its perimeter is:

周长问题常常会产生同类项。如果一个矩形的边长为 (2x + 3) 和 (x + 5)，则其周长为：

2(2x + 3) + 2(x + 5) = 4x + 6 + 2x + 10 = 6x + 16

Here we collect the x-terms (4x + 2x = 6x) and the constants (6 + 10 = 16).

这里我们合并含 x 的项（4x + 2x = 6x）和常数项（6 + 10 = 16）。

In area expressions, like terms may appear after expanding. For example, the combined area of two rectangles with areas 3x² and 5x² is simply 8x².

在面积表达式中，展开后可能会出现同类项。例如，两个矩形面积分别为 3x² 和 5x²，它们的总面积就是 8x²。

11. Common Mistakes to Avoid | 避免常见错误

Mistake: Combining 2x and 3x² into 5x³. Correct: They are not like terms because the exponents differ.

错误：将 2x 和 3x² 合并为 5x³。正确：它们不是同类项，因为指数不同。
Mistake: Forgetting to keep the variable part unchanged, e.g., writing 4x + 3x = 7x². Correct: 4x + 3x = 7x.

错误：忘记保持变量部分不变，例如写成 4x + 3x = 7x²。正确：4x + 3x = 7x。
Mistake: Mixing up signs. For example, 5a − 2a + 3a should be calculated as 5 − 2 + 3 = 6, not 0.

错误：弄混符号。例如，5a − 2a + 3a 应计算为 5 − 2 + 3 = 6，而不是 0。
Mistake: Treating 2x and 2y as like terms because they have the same coefficient. Correct: The variable letters must match.

错误：认为 2x 和 2y 是同类项，因为它们的系数相同。正确：变量字母必须相同。

Always double-check each term, especially when there are several variables and negative signs.

务必检查每一个项，尤其是当包含多个变量和负号时。

12. Practice and Summary | 练习与总结

Let us summarise the key steps:

我们来总结关键步骤：

Step 1	Identify all like terms in the expression.	识别表达式中所有同类项。
Step 2	Group them together, keeping their signs.	将它们分组到一起，保留其符号。
Step 3	Add or subtract the coefficients.	对系数进行加减。
Step 4	Write the simplified expression.	写出化简后的表达式。

Try these examples yourself:

请自己尝试这些例子：

a) 6x + 2y − 4x + 3y = ?

a) 6x + 2y − 4x + 3y = ?
b) 5a − 2a + 7 − 3 = ?

b) 5a − 2a + 7 − 3 = ?
c) 3p² + 2p − p² + 4p = ?

c) 3p² + 2p − p² + 4p = ?

Answers: a) 2x + 5y, b) 3a + 4, c) 2p² + 6p.

答案：a) 2x + 5y，b) 3a + 4，c) 2p² + 6p。

Collecting like terms is a quick, mechanical skill — but it must be done carefully. Once you master it, algebra becomes much more manageable.

合并同类项是一种快速而机械的技能——但必须仔细完成。一旦掌握它，代数就会变得容易得多。

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Collecting Like Terms | 合并同类项