📚 Collecting Like Terms | 合并同类项
In IGCSE algebra, one of the first skills you need is simplifying expressions by collecting like terms. This process turns long or messy expressions into shorter, cleaner forms, making problems easier to solve.
在 IGCSE 代数中，你需要掌握的第一批技能之一就是通过合并同类项来化简表达式。这个过程可以把冗长或杂乱的表达式变成更短、更清晰的形式，从而让问题更容易解决。
1. What is a Term? | 什么是项？
A term is a single number, a single variable, or numbers and variables multiplied together. In the expression 4x + 7y - 3, there are three terms: 4x, 7y and -3.
项是一个单独的数、一个单独的变量，或者数与变量相乘的结果。在表达式 4x + 7y - 3 中，有三个项：4x、7y 和 -3。
Terms are separated by plus or minus signs. When you look at an expression, first split it into terms using the + and - signs as boundaries. In an expression such as 3x² + 2xy - 5, the terms are 3x², 2xy and -5. The minus sign belongs to the constant that follows it.
项由加号或减号隔开。看到一个表达式时，先把加号和减号当作边界，把它拆成若干个项。在 3x² + 2xy - 5 这样的表达式中，各项是 3x²、2xy 和 -5。负号属于它后面的常数。
2. Variables, Coefficients and Constants | 变量、系数与常数
A variable is a letter that stands for an unknown or changing value, such as x, y or n. A coefficient is the number in front of a variable. In 5x, 5 is the coefficient and x is the variable.
变量是表示未知或变化值的字母，如 x、y 或 n。系数是变量前面的数。在 5x 中，5 是系数，x 是变量。
A constant is a term with no variable part, like 6 or -2. Constants do not change within a given expression. A term like x has an implied coefficient of 1, and a term like -x has an implied coefficient of -1.
常数是不含变量部分的项，如 6 或 -2。常数在给定表达式中不会改变。像 x 这样的项隐含系数为 1，像 -x 这样的项隐含系数为 -1。
3. What Are Like Terms? | 什么是同类项？
Like terms have exactly the same variable part, including the same variable letters and the same powers. For example, 3x and -7x are like terms because both contain x to the power 1.
同类项具有完全相同的变量部分，包括相同的变量字母和相同的指数。例如，3x 和 -7x 是同类项，因为它们都包含 x 的一次方。
Unlike terms have different variable parts. 3x and 3x² are not like terms because the powers are different. 3x and 3y are also not like terms because the variables are different.
非同类项的变量部分不同。3x 和 3x² 不是同类项，因为指数不同。3x 和 3y 也不是同类项，因为变量不同。
| Term 1 | 项 1 | Term 2 | 项 2 | Like terms? | 是否同类项 |
|---|---|---|
| 3x | -7x | Yes | 是 |
| 2a | 5a | Yes | 是 |
| -4x² | 9x² | Yes | 是 |
| 7xy | -2xy | Yes | 是 |
| 3x | 3y | No | 否 |
| 3x | 3x² | No | 否 |
4. Why Combine Like Terms? | 为什么要合并同类项？
Combining like terms simplifies expressions, which makes them easier to evaluate and solve. For example, 2x + 5x + 3 becomes 7x + 3, and from there you can substitute values or solve equations more quickly.
合并同类项可以化简表达式，使其更容易求值和求解。例如，2x + 5x + 3 化简为 7x + 3，这样你就能更快地代入数值或解方程。
In IGCSE exams, simplifying an expression is often a required step before solving equations, expanding brackets, or factorising. A simplified answer is also easier to check and less likely to contain errors.
在 IGCSE 考试中，化简表达式通常是解方程、展开括号或因式分解之前的必要步骤。化简后的答案也更容易检查，不容易出错。
5. The Golden Rule of Combining | 合并的黄金法则
You can only combine terms that have exactly the same variable part. To combine them, add or subtract their coefficients and keep the variable part unchanged.
你只能合并变量部分完全相同的项。合并时，将它们的系数相加或相减，变量部分保持不变。
ax + bx = (a + b)x
3x + 5x = (3 + 5)x = 8x. The variable x stays as x; only the numbers 3 and 5 are combined.
这一法则意味着 3x + 5x = (3 + 5)x = 8x。变量 x 仍然是 x；只有数字 3 和 5 被合并。
4a = (10 - 4)a = 6a. Always combine the coefficients first, then write the variable once.
同样的法则也适用于减法：10a - 4a = (10 - 4)a = 6a。始终先合并系数，再将变量写一次。
6. Combining Positive and Negative Terms | 合并正负项
, use the rules for adding and subtracting integers. For example, 7x - 4x = (7 - 4)x = 3x.
当项的系数有正有负时，使用整数加减法则。例如，7x - 4x = (7 - 4)x = 3x。
the sum of the coefficients is negative, keep the negative sign: 2x - 9x = (2 - 9)x = -7x.
如果系数之和为负，保留负号：2x - 9x = (2 - 9)x = -7x。
m. In -5y + 3y, the coefficient of the first term is -5, so the result is (-5 + 3)y = -2y.
始终保留每项前面的符号。在 -5y + 3y 中，第一项的系数是 -5，所以结果是 (-5 + 3)y = -2y。
eep the sign with each term. For example, 4x - 3y + 2x + 5y = 4x + 2x - 3y + 5y = 6x + 2y.
重新排列各项时，符号要跟随各项。例如，4x - 3y + 2x + 5y = 4x + 2x - 3y + 5y = 6x + 2y。
7. Combining Terms with Powers | 合并含幂的项
d the exponent match exactly. For example, 3x² and 5x² are like terms, so 3x² + 5x² = 8x².
含幂的项只有在变量和指数完全相同时才能合并。例如，3x² 和 5x² 是同类项，所以 3x² + 5x² = 8x²。
Do not combine x² and x³: they are unlike terms. The expression 4x² + 2x³ cannot be simplified further.
不要把 x² 和 x³ 合并：它们不是同类项。表达式 4x² + 2x³ 不能再化简。
Recognise that x is the same as x¹. Therefore 2x and 3x² are not like terms, because the exponents 1 and 2 are different.
要认识到 x 等同于 x¹。因此 2x 和 3x² 不是同类项，因为指数 1 和 2 不同。
Also, terms like x²y and xy² are not like terms, because the powers on x and y are not the same. You can only combine terms such as 3x²y and -5x²y.
此外，像 x²y 和 xy² 这样的项不是同类项，因为 x 和 y 上的指数不同。你只能合并像 3x²y 和 -5x²y 这样的项。
8. Expanding Brackets Before Combining | 先展开括号再合并
tributive law, then collect like terms. For example, 3(x + 2) + 2x = 3x + 6 + 2x = 5x + 6.
当有括号时
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