📚 Combining Like Terms | 合并同类项
In IGCSE mathematics, simplifying an algebraic expression often starts with collecting, or combining, like terms. This skill is essential for solving equations, rearranging formulas, and scoring well on algebra questions. It appears frequently in both core and extended papers, often as the first step in longer problems.
在 IGCSE 数学中，化简代数式通常从合并同类项开始。这项技能对解方程、整理公式以及在代数题中取得高分都至关重要。它在核心和扩展试卷中都很常见，通常是较长问题的第一步。
1. What Are Like Terms? | 什么是同类项？
In algebra, a term is a single number, variable, or product of numbers and variables. Expressions are built from terms separated by plus or minus signs. Identifying which terms can be combined is the first step in simplification.
在代数中，项是单个数字、变量或数字与变量的乘积。表达式由用加号或减号分隔的项组成。识别哪些项可以合并是化简的第一步。
Like terms are terms whose variable parts are exactly the same. The variable letters and their exponents must match; only the numerical coefficient may be different. Constants such as 5 and -3 are also like terms because they contain no variable part.
同类项是指变量部分完全相同的项。变量字母及其指数必须一致，只有数字系数可以不同。常数项如 5 和 -3 也属于同类项，因为它们不含变量部分。
For example, 4x and -7x are like terms, and 3x² and 5x² are like terms. However, 4x and 4x² are not like terms because the powers of x differ, and 3x and 3y are not like terms because the variable letters differ.
例如，4x 和 -7x 是同类项，3x² 和 5x² 也是同类项。然而，4x 和 4x² 不是同类项，因为 x 的幂不同；3x 和 3y 也不是同类项，因为变量字母不同。
4x and -7x → like terms; 4x and 4x² → not like terms
2. Why Combine Like Terms? | 为什么要合并同类项？
Combining like terms makes an expression shorter and easier to understand. It also prepares the expression for further steps such as solving equations, substituting values, or drawing graphs. A simplified expression is less likely to cause mistakes later.
合并同类项可以使表达式更短、更容易理解。它还能为后续步骤做好准备，例如解方程、代入数值或绘制图像。化简后的表达式更不容易在后续步骤中出错。
For example, 3x + 5x can be written as 8x, which is much cleaner. If an equation contains 3x + 5x on one side, writing it as 8x first makes the next step obvious.
例如，3x + 5x 可以写成 8x，这样更简洁。如果方程的一侧含有 3x + 5x，先将它写成 8x 会使下一步变得清晰。
3x + 5x = 8x
This skill is not optional in IGCSE: many questions award marks for simplifying correctly before solving. Even if a final answer is not reached, correct simplification can gain method marks.
这项技能在 IGCSE 中不是可有可无的：许多题目在求解之前会先对正确化简给分。即使最终答案没有得出，正确的化简也能获得方法分。
3. The Golden Rule: Same Variable, Same Power | 黄金法则：同变量、同指数
The golden rule is simple: you can only combine terms if the variable part is identical. The coefficient may be positive, negative, fractional, or decimal; it does not affect whether terms are like.
黄金法则很简单：只有当变量部分完全相同时，才能合并这些项。系数可以是正数、负数、分数或小数，这并不影响它们是否为同类项。
For instance, 6a²b and -2a²b are like terms because both contain a²b, so they combine to 4a²b. In the same way, 0.5xy and 1.5xy are like terms, even though one coefficient is a decimal.
例如，6a²b 和 -2a²b 是同类项，因为它们都含有 a²b，所以合并后得到 4a²b。同样，0.5xy 和 1.5xy 也是同类项，尽管其中一个系数是小数。
6a²b - 2a²b = 4a²b
When checking like terms, compare the variable part exactly: letters must be the same, and each must have the same exponent. If one variable has a different power, the terms cannot be combined.
判断同类项时，要准确比较变量部分：字母必须相同，且每个字母的指数必须相同。如果某个变量的幂不同，这些项就不能合并。
4. Adding and Subtracting Like Terms | 同类项的加减
When adding or subtracting like terms, combine the coefficients only and keep the variable part unchanged. Think of the variable part as a label or object: 7 apples plus 4 apples gives 11 apples, so 7x plus 4x gives 11x.
在加减同类项时，只合并系数，变量部分保持不变。可以把变量部分想象成标签或物品：7 个苹果加 4 个苹果等于 11 个苹果，所以 7x 加 4x 等于 11x。
7x + 4x = 11x
mple, 9y - 3y = 6y because 9 - 3 = 6. In the same way, -5m + 2m = -3m because -5 + 2 = -3.
做减法时要注意每个系数的符号。例如，9y - 3y = 6y，因为 9 - 3 = 6。同样，-5m + 2m = -3m，因为 -5 + 2 = -3。
9y - 3y = 6y; -5m + 2m = -3m
Always calculate the coefficient part first, then rewrite the variable part exactly as it was. This helps avoid accidentally changing the variable or its exponent.
总是先计算系数部分，然后原样重写变量部分。这有助于避免无意中改变变量或其指数。
5. Combining Constants | 合并常数项
Constants are numbers without variables, and they can always be combined with each other. Add or subtract them using normal integer arithmetic. This is one of the easiest parts of simplification but still needs care with negative signs.
常数项是没有变量的数字，它们之间总是可以相互合并。使用普通的整数加减法即可。这是化简中最简单的部分之一，但仍需注意负号。
3 + 5 - 8 = 0
In an expression such as 2x + 3 + 5 - 8, the constants simplify to 0, leaving just 2x. If the result had been different, the constant would remain in the final answer.
在如 2x + 3 + 5 - 8 的表达式中，常数项化简为 0，只剩下 2x。如果结果不同，常数项就会保留在最终答案中。
Remember that a constant on its own is like a term with no variable. It should never be combined with an x term or an x² term.
请记住，单独的常数是同类项，但它不含变量。它绝不能与 x 项或 x² 项合并。
6. Terms with Multiple Variables | 含多个变量的项
Terms with more than one variable are like only if each variable has the same power. The order of multiplication does not matter, so 4ab and 7ba are like terms because both contain a times b.
含有多个变量的项只有在每个变量的幂都相同时才是同类项。乘法顺序不影响，因此 4ab 和 7ba 是同类项，因为两者都含有 a 乘以 b。
4ab + 7ba = 11ab
However, 5x²y and 3xy² are not like terms: the first has x squared, while the second has y squared. Similarly, 2xy and 2x²y are not like terms because the power of x is different.
然而，5x²y 和 3xy² 不是同类项：前者含有 x 的平方，而后者含有 y 的平方。类似地，2xy 和 2x²y 也不是同类项，因为 x 的幂不同。
When writing the final answer, it is common to put variables in alphabetical order, so 7ba is normally written as 7ab. This makes it easier to compare terms.
书写最终答案时，通常按字母顺序排列变量，因此 7ba 通常写成 7ab。这样更容易比较各项。
7. Expressions with Brackets | 含括号的表达式
If brackets appear, expand them first using the distributive law, then collect like terms. This is a very common IGCSE task. Do not try to collect terms before expanding, because terms inside brackets are not yet visible as separate items.
如果出现括号，应先用分配律展开括号，再合并同类项。这是 IGCSE 非常常见的题型。不要试图在展开前合并项，因为括号内的项还未作为独立项出现。
2(x + 3) + 3(x - 1) = 2x + 6 + 3x - 3 = 5x + 3
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