Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that can be simplified. One of the most important basic skills is collecting like terms, also called combining like terms. This topic is frequently tested in IGCSE Mathematics because it is the foundation for solving equations, expanding brackets, and simplifying more complex formulas.

在代数中，表达式常常包含几个可以化简的项。最重要的一项基本技能就是合并同类项。这个主题在 IGCSE 数学中经常考查，因为它是解方程、展开括号以及化简更复杂公式的基础。

Mastering this skill allows you to rewrite expressions in their shortest and clearest form. It also reduces the chance of mistakes in later work, especially when signs and brackets are involved.

掌握这项技能可以让你把表达式写成最简短、最清晰的形式。它还能减少后续计算中的错误，尤其是在涉及符号和括号时。

1. What Is Collecting Like Terms? | 什么是合并同类项？

When an algebraic expression contains terms that have exactly the same variable part, we can add or subtract those terms to make the expression shorter. This process is called collecting like terms or combining like terms.

当一个代数表达式中含有变量部分完全相同的项时，我们可以把这些项相加或相减，使表达式更短。这个过程就叫做合并同类项。

For example, the expression 3x + 2x has two like terms because both terms contain the same variable x. They can be collected into a single term 5x.

例如，表达式 3x + 2x 含有两个同类项，因为这两项都含有相同的变量 x。它们可以合并成一项 5x。

3x + 2x = 5x

The expression becomes simpler, but its value has not changed for any value of x. Collecting like terms is a simplification, not a change to the original expression’s meaning.

表达式变得更简单，但对于任意 x 值，它的值并没有改变。合并同类项是一种化简，并不会改变原表达式的意义。

2. Identifying Like Terms | 识别同类项

Like terms must have exactly the same variable letters and the same exponents. The coefficients, or numbers in front, do not need to be the same.

同类项必须具有完全相同的变量字母和相同的指数。系数，也就是变量前面的数字，不需要相同。

Term 1	Term 2	Like?	Reason / 原因
3x	5x	Yes	Same variable x / 相同变量 x
4a	4b	No	Different variables / 不同变量
7x²	2x	No	Same variable but different exponent / 变量相同但指数不同
6xy	−3xy	Yes	Same product xy / 相同乘积 xy

A useful way to check is to ignore the number in front and compare the letter part only. If the letter part is identical, the terms are like terms.

一个有用的检查方法是忽略前面的数字，只比较字母部分。如果字母部分完全相同，那么它们就是同类项。

3. The Basic Rule: Add or Subtract Coefficients | 基本法则：系数相加或相减

When terms are like terms, keep the variable part unchanged and add or subtract the coefficients only.

当项是同类项时，保持变量部分不变，只对系数进行加法或减法。

This means 3x + 2x is not 5x². The x part stays as x, and only the numbers 3 and 2 are added.

这意味着 3x + 2x 不等于 5x²。x 部分仍然是 x，只有数字 3 和 2 相加。

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

7y − 4y = (7 − 4)y = 3y

In general, ax + bx = (a + b)x. The same rule applies to subtraction: ax − bx = (a − b)x.

一般来说，ax + bx = (a + b)x。同样的法则也适用于减法：ax − bx = (a − b)x。

4. Positive and Negative Coefficients | 正系数与负系数

Sign errors are very common when collecting terms. Treat the sign in front of each term as belonging to that term.

合并同类项时，符号错误非常常见。要把每一项前面的符号看作属于这一项。

For example, in the expression 7y − 3y + y, the first term is +7y, the second is −3y, and the third is +1y because a letter on its own has coefficient 1.

例如，在表达式 7y − 3y + y 中，第一项是 +7y，第二项是 −3y，第三项是 +1y，因为单独的字母系数为 1。

7y − 3y + y = (7 − 3 + 1)y = 5y

If the result has a negative coefficient, keep the minus sign clearly in front. For instance, 5a − 8a = −3a.

如果结果是负系数，要把负号清楚地写在前面。例如，5a − 8a = −3a。

5. Expressions with Several Variables | 含多个变量的表达式

An expression may contain different groups of like terms. Collect each group separately and then write the simplified expression.

一个表达式可能包含不同的同类项组。分别合并每一组，然后写出化简后的表达式。

For example, simplify 4a + 3b − 2a + b. The a terms are 4a and −2a, while the b terms are 3b and +b.

例如，化简 4a + 3b − 2a + b。a 项是 4a 和 −2a，而 b 项是 3b 和 +b。

4a + 3b − 2a + b = (4a − 2a) + (3b + b) = 2a + 4b

It is often helpful to underline or circle each group of like terms before collecting them. This makes it easier to avoid mixing different variables.

在合并之前，给每一组同类项画线或画圈通常会很有帮助。这样可以更容易避免混淆不同的变量。

6. Powers and Unlike Terms | 幂与不同类项

Terms such as x and x² look similar but are not like terms because the exponents are different. Do not combine them.

x 和 x² 这样的项看起来相似，但不是同类项，因为指数不同。不要把它们合并。

For example, 2x² + 3x − x² + 5x can be simplified by collecting x² terms together and x terms together separately.

例如，2x² + 3x − x² + 5x 可以化简，方法是分别合并 x² 项和 x 项。

2x² + 3x − x² + 5x = (2x² − x²) + (3x + 5x) = x² + 8x

Also remember that xy, x, and y are three different variable parts. They cannot be collected into a single term.

还要记住，xy、x 和 y 是三种不同的变量部分。它们不能合并成一项。

7. Removing Brackets Before Collecting | 先展开括号再合并

If an expression includes brackets, expand them first. Pay special attention when a negative sign or a negative coefficient is outside a bracket.

如果表达式含有括号，要先展开括号。当括号外面有负号或负系数时，要特别注意。

For example, simplify 2(x + 3) + 3(x − 1). Multiply out each bracket, then collect the x terms and the constant terms.

例如，化简 2(x + 3) + 3(x − 1)。先展开每个括号，然后合并 x 项和常数项。

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

When a bracket is multiplied by a negative number, every term inside changes sign. For example, simplify 5x − 2(x − 3).

当括号乘以一个负数时，括号内的每一项都要变号。例如，化简 5x − 2(x − 3)。

5x − 2(x − 3) = 5x − 2x + 6 = 3x + 6

Always expand before collecting like terms. Trying to collect first can lead to mistakes with the signs inside brackets.

一定要先展开括号，再合并同类项。如果先合并，可能会导致括号内符号出错。

8. Collecting Like Terms with Fractions | 分数形式的同类项合并

Fractions can be collected when they have the same denominator. Combine the numerators over that common denominator and simplify if possible.

当分数具有相同分母时，可以合并同类项。将分子在公分母上合并，并在可能时化简。

(2x)/5 + (3x)/5 = (2x + 3x)/5 = 5x/5 = x

(7y)/3 − (4y)/3 = (7y − 4y)/3 = 3y/3 = y

If the denominators are different, find the lowest common denominator first. For example, x/2 + x/3 becomes 3x/6 + 2x/6 = 5x/6.

如果分母不同，要先找到最小公分母。例如，x/2 + x/3 可以写成 3x/6 + 2x/6 = 5x/6。

In IGCSE questions, fractions often appear with x or other variables in the numerator. Treat the numerator as an algebraic expression and collect like terms there.

在 IGCSE 题目中，分数的分子中经常出现 x 或其他变量。把分子看作代数表达式，并在分子中合并同类项。

9. Common Mistakes to Avoid | 需要避免的常见错误

Watch out for the following errors when collecting like terms.

合并同类项时要注意以下错误。

Combining unlike terms, such as 2x + 3y = 5xy or 2x + 3 = 5x.

合并不同类项，例如 2x + 3y = 5xy 或 2x + 3 = 5x。
Forgetting that a letter on its own has coefficient 1, for example x + 2x = 3x, not 2x.

忘记单独的字母系数为 1，例如 x + 2x = 3x，而不是 2x。
Misapplying signs when expanding brackets, such as −2(x − 3) = −2x − 6 instead of −2x + 6.

展开括号时符号错误，例如 −2(x − 3) = −2x − 6 而不是 −2x + 6。
Incorrectly adding exponents when combining terms, for example x² + x² = x⁴ instead of 2x².

合并时错误地把指数相加，例如 x² + x² = x⁴ 而不是 2x²。

A good habit is to check each simplified term to make sure its variable part matches the original terms exactly.

一个好习惯是检查每一个化简后的项，确保它的变量部分与原项完全一致。

10. Worked Exam-Style Examples | 考试型例题解析

Let us apply the rules to several examples of the type commonly found in IGCSE exams.

让我们把这些法则应用到几个 IGCSE 考试中常见类型的例题上。

Example 1: Simplify 6m + 4n − 2m + 7n.

示例 1：化简 6m + 4n − 2m + 7n。

6m + 4n − 2m + 7n = (6m − 2m) + (4n + 7n) = 4m + 11n

Example 2: Simplify 3(2a − b) + 2(a + 3b).

示例 2：化简 3(2a − b) + 2(a + 3b)。

3(2a − b) + 2(a + 3b) = 6a − 3b + 2a + 6b = 8a + 3b

Example 3: Simplify 4x² + 3x − 2x² + 5x.

示例 3：化简 4x² + 3x − 2x² + 5x。

4x² + 3x − 2x² + 5x = (4x² − 2x²) + (3x + 5x) = 2x² + 8x

Example 4: Simplify (5x)/4 + (3x)/4.

示例 4：化简 (5x)/4 + (3x)/4。

(5x)/4 + (3x)/4 = (5x + 3x)/4 = 8x/4 = 2x

In the exam, always show the intermediate step of grouping like terms. This demonstrates understanding and helps you earn method marks even if a small arithmetic error occurs.

在考试中，一定要写出合并同类项的中间步骤。这样可以展示你的理解，即使出现小的算术错误，也能帮助你获得方法分。

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Collecting Like Terms | 合并同类项