Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is one of the most fundamental skills in algebra. It is the process of simplifying an algebraic expression by adding or subtracting terms that have the same variable parts. Mastering this skill is essential for solving equations, factorising expressions, and working with formulas throughout your IGCSE mathematics course.

合并同类项是代数中最基础的技能之一。它通过加减具有相同变量部分的项来简化代数表达式。掌握这一技能对于在IGCSE数学课程中解方程、因式分解表达式以及处理公式都至关重要。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable(s) raised to the same power(s). The coefficients (the numbers in front of the variables) can be different, but the variable parts must match exactly.

同类项是指具有完全相同变量且变量指数也完全相同的项。变量前面的系数（数字）可以不同，但变量部分必须完全一致。

For example, 3x and 5x are like terms because both have the variable x raised to the power 1. Similarly, 2x² and -7x² are like terms. However, 3x and 3x² are NOT like terms, because the powers of x are different.

例如，3x和5x是同类项，因为两者都含有一次方的变量x。同样，2x²和-7x²也是同类项。但是，3x和3x²不是同类项，因为x的指数不同。

Like terms: 4x and -2x ✓ | 4x² and 8x² ✓ | 4x and 4x² ✗

2. Identifying Like Terms | 识别同类项

To identify like terms, look at the variable part of each term. Ignore the coefficient and focus only on the variables and their exponents. The terms 5xy and -3xy are like terms because both contain xy. The terms 5xy and 5x are NOT like terms because one contains xy and the other contains only x.

要识别同类项，请观察每一项的变量部分。忽略系数，只关注变量及其指数。5xy和-3xy是同类项，因为两者都包含xy。而5xy和5x不是同类项，因为一个包含xy，另一个只包含x。

It is also important to note that the order of variables does not matter for identifying like terms. Since multiplication is commutative, xy and yx represent the same product. Therefore, 3xy and 5yx are like terms.

还需注意，变量的顺序不影响同类项的识别。由于乘法满足交换律，xy和yx表示相同的乘积。因此，3xy和5yx是同类项。

3xy and 5yx → same variable part xy → like terms

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

To collect like terms, simply add or subtract their coefficients and keep the variable part unchanged. For example, 4x + 3x = (4 + 3)x = 7x. The variable x remains exactly the same; only the coefficients are combined.

合并同类项时，只需对系数进行加减运算，变量部分保持不变。例如，4x + 3x = (4 + 3)x = 7x。变量x完全不变，只是系数进行了合并。

When subtracting, be careful with signs. The expression 9x – 5x equals 4x, because 9 minus 5 is 4. Similarly, -7x + 3x equals -4x, because -7 plus 3 equals -4.

做减法时要小心符号。表达式9x – 5x等于4x，因为9减5等于4。同样，-7x + 3x等于-4x，因为-7加3等于-4。

Here is a step-by-step approach: First, identify all like terms. Second, group them together. Third, combine their coefficients according to the rules of integer arithmetic. Finally, write the simplified expression.

这是一个逐步方法：首先，找出所有同类项；其次，将它们分组；然后，按照整数运算法则合并它们的系数；最后，写出简化后的表达式。

6a + 2a – 3a = (6 + 2 – 3)a = 5a

4. Collecting Terms with Different Powers | 合并不同指数的项

Terms with different powers must be treated separately. For instance, in the expression 3x² + 2x + 5x² – 4x, the x² terms (3x² and 5x²) are combined to give 8x², and the x terms (2x and -4x) are combined to give -2x. The result is 8x² – 2x.

不同指数的项必须分开处理。例如，在表达式3x² + 2x + 5x² – 4x中，x²项（3x²和5x²）合并得到8x²，x项（2x和-4x）合并得到-2x。结果是8x² – 2x。

It is a common error to combine x² with x. Remember that x² represents x multiplied by itself, which is fundamentally different from x. They cannot be added together to form a single term.

将x²与x合并是一个常见错误。请记住，x²表示x乘以自身，与x有本质区别。它们不能合并成单一的一项。

When simplifying expressions with multiple powers, keep the terms organised. It is often helpful to write the simplified expression in descending order of powers, starting with the highest exponent.

当简化含有多个指数的表达式时，请保持各项条理清晰。通常按指数降序排列简化后的表达式，从最高指数开始写。

4x² + 7x – 2x² + 3x = (4 – 2)x² + (7 + 3)x = 2x² + 10x

5. Like Terms with Multiple Variables | 多变量同类项

When terms contain more than one variable, all variables and their powers must match for the terms to be considered like terms. The terms 2ab and 5ab are like terms because both contain the product a × b. However, 2ab and 2a²b are not like terms, because the power of a differs.

当项含有多个变量时，所有变量及其指数都必须匹配才能称为同类项。2ab和5ab是同类项，因为两者都包含a × b的乘积。但2ab和2a²b不是同类项，因为a的指数不同。

Consider the expression 3xy + 2x + 4xy – x. The xy terms (3xy and 4xy) combine to give 7xy. The x terms (2x and -x) combine to give x. The simplified expression is 7xy + x.

考虑表达式3xy + 2x + 4xy – x。xy项（3xy和4xy）合并得到7xy。x项（2x和-x）合并得到x。简化后的表达式为7xy + x。

For triangular terms like xyz, all three variables must match. The terms 6xyz and -2xyz can be combined to give 4xyz, but 6xyz and 6xy cannot be combined.

对于像xyz这样的三项变量，三个变量都必须匹配。6xyz和-2xyz可以合并得到4xyz，但6xyz和6xy不能合并。

5pq – 3p + 2pq – p = 7pq – 4p

6. Collecting Constant Terms | 合并常数项

Constant terms are numbers without any variables. All constant terms are considered like terms with each other. In the expression 4x + 7 + 3x – 2, the constants 7 and -2 can be combined to give 5, and the x terms combine to give 7x.

常数项是不含任何变量的数字。所有常数项互视为同类项。在表达式4x + 7 + 3x – 2中，常数7和-2可以合并得到5，x项合并得到7x。

Remember that constants can be positive or negative. When collecting them, apply the rules of signed numbers. The expression 5 – 9 simplifies to -4, and -3 + 8 simplifies to 5.

记住，常数可以是正数或负数。合并时，应用带符号数的运算法则。表达式5 – 9简化为-4，-3 + 8简化为5。

Always include the constant term in your final simplified expression. It is easy to forget, especially when the constant is negative. The expression 3x + 5 – 2x + 1 simplifies to x + 6.

在最终简化表达式中始终包含常数项。尤其当常数是负数时很容易遗漏。表达式3x + 5 – 2x + 1简化为x + 6。

2x + 9 + 5x – 3 = (2 + 5)x + (9 – 3) = 7x + 6

7. Simplifying with Brackets | 含括号的化简

When an expression contains brackets, you must first expand the brackets using the distributive law, and then collect like terms. The distributive law states that a(b + c) = ab + ac. Every term inside the bracket must be multiplied by the term outside.

当表达式含有括号时，必须先使用分配律展开括号，然后再合并同类项。分配律指出a(b + c) = ab + ac。括号内的每一项都必须与括号外的项相乘。

For example, simplify 3(2x + 1) + 4x. First, expand the bracket: 3 × 2x = 6x and 3 × 1 = 3. The expression becomes 6x + 3 + 4x. Now collect like terms: 6x + 4x = 10x. The final answer is 10x + 3.

例如，化简3(2x + 1) + 4x。首先，展开括号：3 × 2x = 6x，3 × 1 = 3。表达式变为6x + 3 + 4x。然后合并同类项：6x + 4x = 10x。最终答案是10x + 3。

Be careful with negative signs outside brackets. The expression -(2x – 3) means -1 × (2x – 3), which expands to -2x + 3. Notice how every sign inside the bracket changes when subtracting the entire bracket. This is a common source of errors.

要特别注意括号外的负号。表达式-(2x – 3)表示-1 × (2x – 3)，展开为-2x + 3。注意当减去整个括号时，括号内每一项的符号都会改变。这是常见错误来源。

2(x + 4) – 3(x – 1) = 2x + 8 – 3x + 3 = -x + 11

8. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One of the most frequent mistakes is combining non-like terms, such as adding x to x². Always check that the variables AND their exponents match before combining. Writing out each step carefully helps reduce this type of error.

最常见的错误之一是合并非同类项，例如将x与x²相加。在合并之前，务必检查变量及其指数是否完全匹配。仔细写出每一步有助于减少此类错误。

Another common mistake is mishandling negative signs. When collecting terms with subtraction, remember that the sign belongs to the term that follows it. In 5x – 3x, the minus sign applies to 3x, so the result is 2x, not 8x.

另一个常见错误是负号处理不当。合并含减法的项时，记住符号属于它后面的项。在5x – 3x中，减号作用于3x，所以结果是2x，而不是8x。

Students also frequently forget to combine all like terms, especially when the expression contains many terms. A systematic approach is to underline or highlight like terms with the same pattern before combining them in a single step.

学生也经常忘记合并所有同类项，尤其是当表达式包含许多项时。系统的方法是先给同类项划线或做标记，然后再一次性合并。

Finally, make sure to write the coefficient correctly when it is 1 or -1. The term 1x is normally written simply as x, and -1x is written as -x. Writing 1x is not incorrect, but it is not the standard convention.

最后，确保正确书写系数为1或-1的情况。1x通常简写为x，-1x简写为-x。写1x不算错，但不符合标准约定。

3x + 5x² – 2x + x² = x + 6x²

9. Word Problems and Real-World Applications | 应用题与实际应用

Simplifying algebraic expressions has many real-world applications. For example, suppose a rectangle has length (3x + 2) cm and width (x + 5) cm. The perimeter is found by adding twice the length and twice the width: P = 2(3x + 2) + 2(x + 5). Expanding and collecting like terms gives P = 6x + 4 + 2x + 10 = 8x + 14 cm.

简化代数表达式在实际生活中有许多应用。例如，假设一个矩形的长为(3x + 2)厘米，宽为(x + 5)厘米。周长等于两倍长加两倍宽：P = 2(3x + 2) + 2(x + 5)。展开并合并同类项得P = 6x + 4 + 2x + 10 = 8x + 14厘米。

Another example: a shop sells pens for x pounds each and notebooks for (x + 2) pounds each. If a customer buys 3 pens and 2 notebooks, the total cost is 3x + 2(x + 2). Simplifying: 3x + 2x + 4 = 5x + 4 pounds.

另一个例子：商店以每支x英镑出售钢笔，每本(x + 2)英镑出售笔记本。如果顾客购买3支钢笔和2本笔记本，总费用为3x + 2(x + 2)。化简：3x + 2x + 4 = 5x + 4英镑。

In geometry and physics, collecting like terms allows us to combine measurements and derive simpler formulas. The ability to simplify expressions quickly and accurately is a foundation for calculus, where manipulating algebraic expressions is essential.

在几何学和物理学中，合并同类项使我们可以组合测量值并推导更简洁的公式。快速准确地简化表达式的技能是微积分的基础，在微积分中操作代数表达式至关重要。

10. Practice Exercises | 练习巩固

Here are some practice exercises to help you master collecting like terms. Work through each question carefully, showing all your steps. The answers are provided below for self-checking.

以下是一些练习，帮助你掌握合并同类项。请仔细完成每道题，写出所有步骤。答案在下方供自检。

Question \| 题目	Answer \| 答案
1. Simplify 7x + 3x + 2x 化简 7x + 3x + 2x	12x
2. Simplify 5a + 3b – 2a + b 化简 5a + 3b – 2a + b	3a + 4b
3. Simplify 4x² + 6x – 2x² + 3x 化简 4x² + 6x – 2x² + 3x	2x² + 9x
4. Simplify 2(3x + 1) – 4x 化简 2(3x + 1) – 4x	2x + 2
5. Simplify 3xy + 2x – xy + 5x 化简 3xy + 2x – xy + 5x	2xy + 7x

For question 4, the full working is: 2(3x + 1) – 4x = 6x + 2 – 4x = 2x + 2. Take your time with each question and check that you understand every step before moving on.

对于第4题，完整步骤为：2(3x + 1) – 4x = 6x + 2 – 4x = 2x + 2。请仔细完成每道题，确保理解每一步后再继续。

Collecting like terms is a gateway skill that unlocks all further algebraic manipulation. Once you master this technique, solving equations, expanding brackets, and factorising expressions become significantly more manageable. Keep practising until it becomes second nature, and always check that you have combined every possible like term in your final answer.

合并同类项是打开后续所有代数操作之门的技能。一旦掌握了这个技巧，解方程、展开括号和因式分解表达式都会变得容易得多。持续练习，直到它成为你的本能，并始终检查最终答案中是否已合并所有可能的同类项。

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Collecting Like Terms | 合并同类项