📚 Combining Like Terms | 合并同类项
In algebra, expressions often contain multiple terms that can be grouped and simplified. Combining like terms is one of the most fundamental skills in algebra, allowing us to transform long, complex expressions into shorter, more manageable forms. This technique is essential for solving equations, factorising expressions, and understanding functions at IGCSE level and beyond.
在代数学中，表达式通常包含多个可以分组和化简的项。合并同类项是代数学中最基本的核心技能之一，它使我们能够将冗长复杂的表达式转化为更简短、更易处理的形式。这一技巧对于解方程、因式分解表达式以及理解函数都至关重要，是 IGCSE 考试中的必备基础。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable parts, with the same variables raised to the same powers. The coefficient (the numerical factor in front) may differ, but the variable portion must be identical for two terms to be considered "like".
同类项是指变量部分完全相同的项，即所含的字母相同，且各字母的指数也分别相同。各项的系数（前面的数字因数）可以不同，但变量部分必须完全一致，这样的项才能被称为“同类项”。
For example, in the expression 3x + 5x - 2x, all three terms are like terms because each contains the variable x raised to the first power. Similarly, 4a²b and -7a²b are like terms because both contain a squared and b to the first power.
例如，在表达式 3x + 5x - 2x 中，三个项都是同类项，因为每一项都包含一次幂的变量 x。类似地，4a²b 和 -7a²b 是同类项，因为它们都包含 a 的平方和 b 的一次幂。
Consider these examples to solidify your understanding:
请通过以下示例加深理解：
- 5a and 3a are like terms.
- 5a 和 3a 是同类项。
- 4x² and 7x² are like terms, but 4x² and 7x are not.
- 4x² 和 7x² 是同类项，但 4x² 和 7x 不是同类项。
- 2ab and 5ab are like terms; note that 2ab and 2ba are identical due to the commutative property of multiplication.
- 2ab 和 5ab 是同类项；注意 2ab 与 2ba 完全相同，因为乘法满足交换律。
- 6x²y and -2x²y are like terms, but 6x²y and 6xy² are not.
- 6x²y 和 -2x²y 是同类项，但 6x²y 和 6xy² 不是同类项。
2. Identifying the Coefficient | 识别系数
The coefficient is the numerical factor that multiplies the variable part of a term. When combining like terms, we add or subtract only the coefficients while keeping the variable part unchanged. The sign of the coefficient must always be preserved.
系数是代数项中与变量部分相乘的数字因数。在合并同类项时，我们只对系数进行加或减，而保持变量部分不变。系数的正负号必须始终保留。
Here is a table showing the coefficient of various terms:
下表展示了不同项中系数的识别方法：
| Term | Coefficient | Variable Part |
| 7x | 7 | x |
| -3y | -3 | y |
| 2a²b | 2 | a²b |
| -x | -1 | x |
| ½ p³ | ½ | p³ |
Notice that when a term appears as -x, the coefficient is -1, not 0. This common misunderstanding can lead to errors in simplification.
请注意，当一项显示为 -x 时，其系数是 -1 而不是 0。这是常见的误解，如果忽略这一点，就会导致化简出错。
3. Combining Terms with One Variable | 合并含单一变量的项
When all terms contain the same variable, we simply add or subtract their coefficients. The variable part stays exactly the same throughout the operation.
当所有项都包含相同变量时，我们只需对它们的系数进行加法或减法运算。变量部分在整个运算过程中保持不变。
Work through this example step by step:
请看下面的逐步示例：
3x + 5x = (3 + 5)x = 8x
9x - 4x = (9 - 4)x = 5x
Now try a longer expression that includes both addition and subtraction:
现在尝试一个同时包含加法和减法的较长表达式：
7x + 2x - 5x + x
7 + 2 - 5 + 1 = 5, so the simplified result is 5x. Always remember that x alone means 1x.
将系数分组计算：7 + 2 - 5 + 1 = 5，因此化简结果为 5x。始终记住，单独的 x 表示 1x。
4. Combining Like Terms with Multiple Variables | 合并含多个变量的同类项
When an expression contains more than one type of variable term, we must identify each group of like terms separately and combine each group independently.
当一个表达式包含不同类型的变量项时，我们必须分别识别每一组同类项，并对每组进行独立的合并运算。
Consider the expression:
考虑以下表达式：
4x + 2y - x + 3y
Step 1: Identify groups of like terms. The x-terms are 4x and -x; the y-terms are 2y and 3y.
第一步：识别同类项组。含 x 的项是 4x 和 -x；含 y 的项是 2y 和 3y。
Step 2: Combine each group. For x: 4 - 1 = 3, giving 3x. For y: 2 + 3 = 5, giving 5y.
第二步：分别合并每组。对于 x：4 - 1 = 3，得到 3x。对于 y：2 + 3 = 5，得到 5y。
4x + 2y - x + 3y = 3x + 5y
A critical rule: you cannot combine 3x and 5y, because they have different variables. They are not like terms.
一个关键规则：不能将 3x 与 5y 合并，因为它们的变量不同，不是同类项。
When variables appear with exponents, the exponents must also match. For example:
当变量带有指数时，指数也必须相同才能合并。例如：
5a² + 3b² - 2a² + b² = (5 - 2)a² + (3 + 1)b² = 3a² + 4b²
5. Combining Constant Terms | 合并常数项
Constant terms are numbers without any variable attached. They are all like terms with each other, so they can always be combined regardless of their position in the expression.
常数项是不带任何变量的数字。所有常数项彼此之间都是同类项，因此无论它们在表达式中处于什么位置，都可以合并。
bine the x-terms (2x + 3x = 5x) and then combine the constants (7 - 4 = 3), giving 5x + 3.
对于表达式 2x + 7 + 3x - 4，我们先合并含 x 的项（2x + 3x = 5x），然后合并常数项（7 - 4 = 3），得到 5x + 3。
Be careful with signs when combining constants:
合并常数项时务必注意符号：
- 8 + 3 - 5 = 6
- 8 + 3 - 5 = 6
- -4 + 9 - 2 = 3
- -4 + 9 - 2 = 3
- 6 - 10 + 4 = 0
- 6 - 10 + 4 = 0
Constants provide a useful check: once all constant terms are combined, the result should be a single number.
常数项提供了一个有用的检验方法：当所有常数项合并完成后，结果应当是一个单独的数字。
6. Using the Distributive Property | 运用乘法分配律
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding the products:
乘法分配律指出，一个数与一个和相乘，等于这个数分别与每个加数相乘后再相加：
a(b + c) = ab + ac
a(b - c) = ab - ac
This property is essential when we need to expand brackets before combining like terms. For example:
在需要先展开括号再合并同类项时，这一定律至关重要。例如：
3(2x + 1) = 3 × 2x + 3 × 1 = 6x + 3
2(x - 4) = 2 × x - 2 × 4 = 2x - 8
When the bracket is multiplied by a negative number, every sign inside the bracket flips:
当括号乘以一个负数时，括号内每一项的符号都必须改变：
-2(x + 5) = -2x - 10
-(3x - 7) = -3x + 7
le that -(3x - 7) is equivalent to -1 × (3x - 7), so the coefficient of the bracket is -1.
注意最后一个例子中，-(3x - 7) 等价于 -1 × (3x - 7)，即括号前的系数是 -1。
7. Simplifying Expressions with Brackets | 化简含括号的表达式
Many exam questions ask you to expand brackets and then combine like terms in a single simplification. The procedure is: first expand all brackets, then combine like terms across the entire expression.
许多考试题目要求你先展开括号，然后再合并同类项，完成一步到位的化简。解题步骤是：第一步展开所有括号，第二步在整个表达式范围内合并同类项。
Here is a typical example:
下面是一道典型例题：
5(2x + 3) + 4(3x - 2)
Step 1: Expand each bracket separately.
第一步：分别展开每个括号。
5(2x + 3) = 10x + 15
4(3x - 2) = 12x - 8
Step 2: Combine the expanded terms.
第二步：合并展开后的各项。
10x + 15 + 12x - 8 = (10 + 12)x + (15 - 8) = 22x + 7
Try another example with both brackets and constants:
再试一道同时含有括号和常数项的例题：
3(2x - 1) - 2(4 - x) + 5
mbining x-terms: 6x + 2x = 8x. Combining constants: -3 - 8 + 5 = -6. Final answer: 8x - 6.
展开得：6x - 3 - 8 + 2x + 5。合并含 x 的项：6x + 2x = 8x。合并常数项：-3 - 8 + 5 = -6。最终答案：8x - 6。
8. Applying Like Terms to Equations | 将合并同类项运用于方程
Combining like terms is not just about simplifying neat expressions; it plays a central role in solving linear equations. Before isolating the variable, we must first simplify each side of the equation by combining like terms.
合并同类项不仅仅用于化简规整的表达式，它在求解线性方程中扮演着核心角色。在分离变量之前，我们必须先通过合并同类项来化简方程的每一侧。
Consider this equation:
来看这个方程：
3x + 5x - 4 = 2x + 16
Step 1: Simplify the left side by combining 3x and 5x.
第一步：通过合并 3x 和 5x 来化简方程左侧。
8x - 4 = 2x + 16
Step 2: Rearrange to bring x-terms to the left and constants to the right.
第二步：移项，使含 x 的项在左侧，常数项在右侧。
8x - 2x = 16 + 4
Step 3: Combine like terms on each side.
第三步：在方程两侧分别合并同类项。
6x = 20
Step 4: Divide both sides by 6.
第四步：方程两边同时除以 6。
x = 20 ÷ 6 = 10⁄3
Without combining like terms at Step 3, the equation would remain messy and difficult to solve. This confirms why the skill is so important.
如果在第三步不合并同类项，方程将会保持冗长难解的状态。这再次证明了这一技能的重要性。
9. Solving Word Problems | 解应用题
Word problems in IGCSE often require you to construct an algebraic expression and then simplify it using the techniques learned above. Combining like terms allows you to turn verbose descriptions into concise algebraic forms.
IGCSE 中的应用题通常要求你先建立代数表达式，然后再用上述技巧对其进行化简。合并同类项使你能够将冗长的文字描述转化为简洁的代数形式。
Example: A rectangle has length (x + 5) cm and width (x - 2) cm. Write a simplified expression for its perimeter.
例题：一个长方形的长为 (x + 5) cm，宽为 (x - 2) cm。请写出其周长的化简表达式。
We know the perimeter is twice the sum of length and width:
我们知道周长是长和宽之和的两倍：
P = 2(x + 5) + 2(x - 2)
Expand: 2x + 10 + 2x - 4. Combine x-terms: 2x + 2x = 4x. Combine constants: 10 - 4 = 6.
展开得：2x + 10 + 2x - 4。合并含 x 的项：2x + 2x = 4x。合并常数项：10 - 4 = 6。
P = (4x + 6) cm
Another example: A shop sells pens at p pence each. A customer buys 3 pens and an extra notebook costing 80 pence. The total cost in terms of p is 3p + 80, which cannot be simplified further because 3p and 80 are not like terms.
另一个例子：一家商店以每支 p 便士的价格出售钢笔。一位顾客购买了 3 支钢笔和一本 80 便士的笔记本。总价用 p 表示为 3p + 80，不能进一步化简，因为 3p 与 80 不是同类项。
10. Common Mistakes and Pitfalls | 常见错误与易错点
Students often make predictable mistakes when combining like terms. Recognising these pitfalls is the first step to avoiding them in exams.
学生在合并同类项时常犯一些可以预见的错误。识别这些易错点是避免在考试中犯错的第一步。
example, writing 3x + 4y = 7xy. This is incorrect; 3x and 4y are different types of terms.
易错点 1：合并非同类项。例如，将 3x + 4y 写成 7xy。这是错误的；3x 和 4y 是不同类型的项。
: Forgetting the coefficient of 1 or -1. For instance, x means 1x, so x + 4x = 5x, not 4x.
易错点 2：忘记系数 1 或 -1。例如，x 表示 1x，所以 x + 4x = 5x，而不是 4x。
Mistake 3: Incorrect sign handling. When subtracting a bracket, every term inside must change sign. The expression 3x - (2x + 5) becomes 3x - 2x - 5, not 3x - 2x + 5.
易错点 3：符号处理不当。当减去一个括号时，括号内每一项都要变号。表达式 3x - (2x + 5) 应化为 3x - 2x - 5，而不是 3x - 2x + 5。
Mistake 4: Confusing x² with x. In 3x² + 2x, the terms cannot be combined because the exponents differ. Always check both the variable and its exponent.
易错点 4：混淆 x² 与 x。在 3x² + 2x 中，因为指数不同，两者不能合并。始终检查变量及其指数是否相同。
Mistake 5: Ignoring the order of operations. Always expand brackets and deal with multiplication before combining terms with addition or subtraction.
易错点 5：忽略运算顺序。必须先展开括号并处理乘法，再进行加法和减法合并。
11. Practice Questions | 练习题目
Apply your knowledge with these exam-style questions. Attempt each one before reading the solution.
请尝试以下模拟考试风格的练习题，先在草稿纸上独立完成，再核对答案。
Question 1: Simplify 6a + 3b - 4a + 2b.
题目 1：化简 6a + 3b - 4a + 2b。
Solution: 6
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图