📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression often starts with collecting like terms. This skill allows you to rewrite expressions such as 5x + 3 - 2x + 7 as one simplified expression. Mastering it is essential for solving equations, expanding brackets and working with formulas.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。这项技能能让你把 5x + 3 - 2x + 7 这样的表达式改写为一个最简表达式。掌握它对于解方程、去括号和运用公式都至关重要。
1. What Are Like Terms? | 什么是同类项
Like terms are algebraic terms that have exactly the same variable part, including the same power. For example, 3x and 5x are like terms because both contain x to the power of 1. The number in front, called the coefficient, can be different.
同类项是指变量部分完全相同（包括相同次幂）的代数项。例如，3x 和 5x 是同类项，因为两者都含有 x 的一次幂。前面的数字，也就是系数，可以不同。
Unlike terms cannot be combined by simple addition or subtraction. For instance, 3x and 3x² are not like terms, because the powers of x are different. You cannot write 3x + 3x² as 6x or 6x²; it must stay as 3x + 3x².
不同类项不能通过简单的加法或减法合并。例如，3x 和 3x² 不是同类项，因为 x 的次幂不同。你不能把 3x + 3x² 写成 6x 或 6x²，它必须保持为 3x + 3x²。
2. Key Vocabulary | 关键术语
Before practising, make sure you know these key words.
练习之前，先确保你理解以下关键术语。
| English term | 中文术语 | Example |
|---|---|---|
| Term | 项 | 4x, -3y, 7 |
| Coefficient | 系数 | In 4x, the coefficient is 4 |
| Variable | 变量 | x, y, a, b |
| Constant | 常数 | 5, -2, 9 |
| Like terms | 同类项 | 3x and -8x |
The coefficient is the number multiplying the variable. For example, in -7y, the coefficient is -7, including the minus sign.
系数是乘以变量的数字。例如，在 -7y 中，系数是 -7，包括负号。
3. Identifying Like Terms | 识别同类项
To decide whether terms are like terms, ignore the coefficient and look at the variables and their powers. For example, 2ab and -7ab are like terms, while 2ab and 3a are not.
判断两项是否为同类项时，先忽略系数，只看变量及其次幂。例如，2ab 和 -7ab 是同类项，而 2ab 和 3a 不是。
ive, so ab = ba. This is important in IGCSE exam questions that test rearranged variables.
此外，ab 和 ba 是同类项，因为乘法满足交换律，所以 ab = ba。这一点在 IGCSE 考试中考查变量顺序变换时很重要。
Examples of like terms include: 4x and -9x, 2x² and 3x², 5xy and -6xy. Examples of unlike terms include: x and y, x and x², 3a and 3b, 2xy and 2x.
同类项的例子包括：4x 和 -9x、2x² 和 3x²、5xy 和 -6xy。不同类项的例子包括：x 和 y、x 和 x²、3a 和 3b、2xy 和 2x。
4. The Golden Rule of Combining | 合并的黄金法则
To combine like terms, add the coefficients and keep the variable part exactly the same. The variable part and its power do not change during addition or subtraction.
合并同类项时，把系数相加，并保持变量部分完全不变。在加法或减法过程中，变量部分及其次幂不会改变。
The coefficient of the result is the sum of the coefficients. For example:
合并后结果的系数是各系数的和。例如：
3x + 5x = (3 + 5)x = 8x
Similarly, when subtracting, add the coefficients with their signs:
同样，做减法时，带着符号把系数相加：
7y - 2y = 7y + (-2y) = (7 - 2)y = 5y
This rule works for all like terms, no matter how many terms are present. Just total all coefficients and attach the common variable part.
无论有多少项，这个法则都适用。只要把所有系数加起来，再附上共同的变量部分即可。
5. Combining Positive and Negative Terms | 合并正项与负项
When you have a mixture of positive and negative terms, keep the sign in front of each term attached to its coefficient. For example, 6a - 4a + 2a means 6a + (-4a) + 2a, which gives 4a.
当你遇到正项和负项混合时，要把每项前面的符号和它的系数看作一个整体。例如，6a - 4a + 2a 表示 6a + (-4a) + 2a，结果是 4a。
6a - 4a + 2a = (6 - 4 + 2)a = 4a
It is often helpful to circle or underline the sign with each term before rearranging. This reduces sign errors in longer expressions.
在重新排列之前，把每一项的符号圈出或划线通常会很有帮助。这可以减少较长表达式中的符号错误。
ans the term is subtracted. For example, -3x means subtract 3x, so 10x - 3x = 7x, not 13x.
负系数表示该项被减去。例如，-3x 表示减去 3x，所以 10x - 3x = 7x，而不是 13x。
6. Constants vs Variable Terms | 常数项与含变量项
out variables. They can only be combined with other constants. For example, 4 + 7 - 2 = 9.
常数项是不含变量的数字。它们只能与其他常数项合并。例如，4 + 7 - 2 = 9。
Never combine a constant with a variable term. For example, 5x + 3 cannot be simplified further because one term contains x and the other is a constant.
千万不要把常数项与含变量的项合并。例如，5x + 3 不能再化简，因为一项含有 x，另一项是常数。
When simplifying an expression such as 2x + 8 + 5x - 3, collect x terms and constants separately:
化简 2x + 8 + 5x - 3 这样的表达式时，要分别合并 x 项和常数项：
2x + 5x = 7x
8 - 3 = 5
Final answer: 7x + 5
This step-by-step grouping prevents mixing unlike categories.
这种分步分组可以避免把不同类别混在一起。
7. Terms with Powers | 含有幂的项
Terms with the same variable but different powers are not like terms. For example, x and x² cannot be combined; x + x² remains x + x².
变量相同但次幂不同的项不是同类项。例如，x 和 x² 不能合并；x + x² 仍然写作 x + x²。
ecause all terms have x². The coefficients combine as 3 + 5 - 2 = 6, so the result is 6x².
但是，3x² + 5x² - 2x² 可以化简，因为所有项都含有 x²。系数合并为 3 + 5 - 2 = 6，所以结果是 6x²。
powers: 2x³ + 4x³ = 6x³, but 2x³ + 4x² cannot be simplified. The power must match exactly.
处理更高次幂时要小心：2x³ + 4x³ = 6x³，但 2x³ + 4x² 不能化简。次幂必须完全一致。
8. Expressions with Multiple Variables | 含多个变量的表达式
For terms with more than one variable, all variables and their powers must match. For example, 3xy and 5xy are like terms, but 3xy and 5x are not.
对于含有多个变量的项，所有变量及其次幂都必须一致。例如，3xy 和 5xy 是同类项，但 3xy 和 5x 不是。
xy + 7yx - xy = (2 + 7 - 1)xy = 8xy. Here xy and yx are treated as the same variable part.
例如：2xy + 7yx - xy = (2 + 7 - 1)xy = 8xy。这里 xy 和 yx 被视为相同的变量部分。
Example with powers: 4a²b + 3ab² cannot be simplified because the powers on a and b do not match in both terms. The first has a²b, while the second has ab².
含次幂的例子：4a²b + 3ab² 不能化简，因为两项中 a 和 b 的次幂不相同。第一项是 a²b，而第二项是 ab²。
9. Common Mistakes | 常见错误
IGCSE examiners frequently see the same errors when students simplify expressions. Watch out for these mistakes.
IGCSE 考官在批改化简表达式题目时，经常会看到一些相同的错误。请留意以下问题。
| Mistake | Wrong answer | Correct answer |
|---|---|---|
| Combining unlike terms | x + x² = 2x² | x + x² stays x + x² |
| Changing the power when adding | 2x + 3x = 5x² | 2x + 3x = 5x |
| Forgetting the minus sign | 5a - 3a = 8a | 5a - 3a = 2a |
| Adding a constant to a variable term | 4x + 2 = 6x | 4
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