📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, simplifying an expression usually begins with collecting like terms. This means identifying terms that have exactly the same variable parts, then adding or subtracting their coefficients while keeping the variable part unchanged. Mastering this skill helps you tidy up expressions, solve equations more efficiently, and avoid sign errors in later topics such as solving quadratics or rearranging formulae.
在 IGCSE 代数中，化简表达式通常从合并同类项开始。所谓合并同类项，就是先找出变量部分完全相同的项，再把它们的系数相加或相减，而变量部分保持不变。掌握这项技能可以帮助你整理表达式、更高效地解方程，并避免后续内容（如解二次方程或变换公式）中的符号错误。
This topic looks simple, but it is one of the most frequently tested skills in IGCSE Mathematics. Questions may hide like terms inside longer expressions, fraction coefficients, negative signs, or brackets, so careful reading and a clear method are essential.
这个主题看似简单，但它其实是 IGCSE 数学中最常考查的技能之一。题目可能会把同类项藏在较长的表达式、分数系数、负号或括号中，因此仔细审题和清晰的方法非常重要。
1. What Are Like Terms? | 什么是同类项
A term is a single number, variable, or product of numbers and variables, such as 5, x, 3x, or −2xy. Terms are called like terms when their variable parts are exactly the same, including the same powers.
项是一个单独的数、变量，或数与变量的乘积，例如 5、x、3x 或 −2xy。当两个项的变量部分完全相同，包括相同的指数时，它们就叫做同类项。
For example, 3x and 5x are like terms because both contain x. Also, 4x² and −7x² are like terms because both contain x². However, 3x and 3x² are not like terms because the powers differ.
例如，3x 和 5x 是同类项，因为它们都含有 x。同样，4x² 和 −7x² 是同类项，因为它们都含有 x²。但是，3x 和 3x² 不是同类项，因为指数不同。
Constants such as 2, −9 and 100 are also like terms with each other, because they have no variable part at all. They combine only with other constants.
常数项（如 2、−9 和 100）彼此也是同类项，因为它们完全没有变量部分。常数项只能与其他常数项合并。
2. Identifying Like Terms | 识别同类项
To decide whether two terms are like terms, compare the variable letters and their exponents. The coefficient in front does not matter when identifying like terms, so 2x, −8x and ½x are all like terms.
判断两个项是否为同类项时，要比较变量字母及其指数。识别同类项时，前面的系数并不重要，所以 2x、−8x 和 ½x 都是同类项。
The table below shows several comparisons that IGCSE exam questions often use to test your understanding of like terms.
下表展示了 IGCSE 考试中常用来考查同类项概念的一些对比。
| First term | Second term | Like terms? | Reason |
|---|---|---|---|
| 3x | 9x | Yes | Both contain x |
| 2x² | 5x² | Yes | Both contain x² |
| 7xy | −3xy | Yes | Both contain xy |
| 4x | 2y | No | Different variables |
| 5x² | 5x | No | Different powers of x |
| 3 | 3x | No | One is constant, one contains x |
When terms are unlike, they must stay as separate terms in your answer. You cannot merge 3x and 2y into a single term such as 5xy.
如果两个项不是同类项，它们就必须在答案中保持为单独的项。你不能把 3x 和 2y 合并成一个项，比如 5xy。
3. Combining Like Terms: The Basic Rule | 合并同类项的基本规则
and keep the variable part exactly the same. For example, 3x + 5x = 8x because 3 + 5 = 8.
合并同类项时，将系数相加或相减，变量部分保持不变。例如，3x + 5x = 8x，因为 3 + 5 = 8。
3x + 5x = 8x
cient is understood to be 1. So x + 4x = 5x, and 7x − x = 6x, because x is the same as 1x.
如果项单独写成 x，它的系数默认为 1。因此 x + 4x = 5x，7x − x = 6x，因为 x 就等于 1x。
ng all positive coefficients and subtracting all negative coefficients: 2x + 3x + 4x = 9x.
你可以一步合并两个以上的同类项，把所有正系数相加并减去所有负系数：2x + 3x + 4x = 9x。
The variable part does not change when you collect like terms. In 3x + 4x, you add the coefficients 3 and 4 to get 7, and the x remains as x, not x².
合并同类项时变量部分不会改变。在 3x + 4x 中，你把系数 3 和 4 相加得到 7，而 x 仍然是 x，不会变成 x²。
4. Working with Negative Coefficients | 处理负系数
Negative signs often cause mistakes. Treat the sign in front of a term as part of its coefficient: 7x − 3x means 7x + (−3x), which equals 4x.
负号经常导致错误。要把项前面的符号看作系数的一部分：7x − 3x 表示 7x + (−3x)，结果为 4x。
er examples: −2x + 5x = 3x, because −2 + 5 = 3. Also, −4x − 3x = −7x, because −4 − 3 = −7.
再看几个例子：−2x + 5x = 3x，因为 −2 + 5 = 3。另外，−4x − 3x = −7x，因为 −4 − 3 = −7。
x. If the signs are mixed, use the sign of the coefficient with the larger absolute value.
当所有系数都为负时，结果仍为负：−x − 6x = −7x。如果符号混合，则取绝对值较大的系数的符号作为结果的符号。
rearranging. The expression 5 − 2x + 3x should be combined as −2x + 3x = x, giving 5 + x.
在重排各项时，符号必须始终跟随该项。表达式 5 − 2x + 3x 应合并为 −2x + 3x = x，结果是 5 + x。
5. Constants: Numbers on Their Own | 常数项：单独的数字
Constants are numbers without variables, and they combine only with other constants. For example, in 3x + 4 + 2x + 7, the x terms combine to 5x and the constants combine to 11, giving 5x + 11.
常数项是不含变量的数，只能与其他常数项合并。例如，在 3x + 4 + 2x + 7 中，x 项合并为 5x，常数项合并为 11，结果为 5x + 11。
3x + 4 + 2x + 7 = 5x + 11
Never try to combine a constant with a variable term: 5 + 2x cannot be simplified further because 5 and 2x are not like terms.
千万不要把常数项与含变量的项合并：5 + 2x 不能再化简，因为 5 和 2x 不是同类项。
nts, you can simply add or subtract them using integer rules. For example, 2 − 9 + 6 = −1.
当某一部分中的所有项都是常数项时，你只需按照整数运算法则进行加减。例如，2 − 9 + 6 = −1。
6. Terms with Different Powers | 不同次幂的项
Power or exponent is part of the variable identity. x² and x are different, so 3x² + 4x cannot be combined. Only terms with identical powers can merge.
幂或指数是变量身份的一部分。x² 和 x 是不同的，所以 3x² + 4x 不能合并。只有指数完全相同的项才能合并。
2x² + 5x remains as 2x² + 5x. The same rule applies to x³ and x²; they are not like terms.
例如，2x² + 5x² = 7x²，但 2x² + 5x 仍然是 2x² + 5x。同样的规则也适用于 x³ 和 x²；它们不是同类项。
When simplifying,
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