Collecting Like Terms | IGCSE 数学：合并同类项

📚 Collecting Like Terms | IGCSE 数学：合并同类项

In IGCSE algebra, simplifying an expression often begins with collecting like terms. This process makes expressions shorter, clearer and ready for solving equations. A term is a single number, variable or product of numbers and variables. Like terms have exactly the same variable part, including the same powers.

在 IGCSE 代数中，化简表达式通常从合并同类项开始。这个过程让表达式更短、更清晰，也为解方程做好准备。项是单个数字、变量或数字与变量的乘积。同类项具有完全相同的变量部分（包括相同的幂次）。

1. What Are Like Terms? | 什么是同类项

Like terms are terms whose variable parts are identical. For example, 3x and -7x are like terms because both contain the variable x to the power 1. However, 3x and 3x² are not like terms because the powers of x are different.

同类项是指变量部分完全相同的项。例如，3x 与 -7x 是同类项，因为两者都含有一次变量 x。但是，3x 与 3x² 不是同类项，因为 x 的幂次不同。

Constants such as 5, -2 and 100 are also like terms with each other. They can be combined into a single number because they all have no variable part.

常数项如 5、-2 和 100 也互为同类项。它们可以合并为一个数字，因为它们都没有变量部分。

• 3a, -5a, 12a → 同类项
• 4xy, 18xy, -xy → 同类项
• 2x, 5y → 不是同类项
• x², 4x → 不是同类项

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2. Coefficients, Variables and Constants | 系数、变量与常数

A term such as -6ab has a coefficient -6 and a variable part ab. The coefficient is the numerical factor that multiplies the variables. Identifying coefficients helps you combine terms correctly.

像 -6ab 这样的项具有系数 -6 和变量部分 ab。系数是乘以变量的数字因子。识别系数有助于正确合并项。

In the expression 4x + 3y - 7, the coefficient of x is 4, the coefficient of y is 3, and -7 is a constant term. A constant has no variable attached to it.

在表达式 4x + 3y - 7 中，x 的系数是 4，y 的系数是 3，-7 是常数项。常数项没有附带变量。

Term	Coefficient	Variable part
8x	8	x
-5y²	-5	y²
7	7	none

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3. The Golden Rule of Collecting Like Terms | 合并同类项的核心法则

Only like terms can be added or subtracted. You add or subtract their coefficients while keeping the variable part unchanged. This is the golden rule of collecting like terms.

只有同类项才能相加或相减。合并时，将系数相加或相减，变量部分保持不变。这就是合并同类项的核心法则。

ax + bx = (a + b)x

This rule works because multiplication distributes over addition. The variable part is a common factor, so it can be kept outside the bracket.

这一法则成立的原因是乘法对加法具有分配律。变量部分是公因子，因此可以保留在括号外。

For example, 3x + 5x means three x's plus five x's, giving eight x's. That is why 3x + 5x = 8x.

例如，3x + 5x 表示三个 x 加五个 x，得到八个 x。因此 3x + 5x = 8x。

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4. Adding and Subtracting Like Terms | 加减同类项

To simplify 7x + 4x, add the coefficients 7 and 4 to get 11x. To simplify 9y - 3y, subtract the coefficients to get 6y.

化简 7x + 4x 时，将系数 7 和 4 相加，得到 11x。化简 9y - 3y 时，将系数相减，得到 6y。

signs, be careful with negative coefficients. Example: 5a - 9a + 2a = (5 - 9 + 2)a = -2a.

遇到混合正负号时，要特别注意负系数。例如：5a - 9a + 2a = (5 - 9 + 2)a = -2a。

5a - 9a + 2a = -2a

Longer expressions can be simplified by collecting the like terms step by step. Write down the coefficient sum first if it helps avoid sign errors.

较长的表达式可以通过逐步合并同类项来化简。如果担心符号错误，可以先把系数和写下来。

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5. Terms With Different Variables | 含不同变量的项

Terms with different variable parts cannot be combined into one term. For example, 3x + 4y remains 3x + 4y. They are not like terms because the variables x and y are different.

变量部分不同的项不能合并为一项。例如，3x + 4y 仍然写作 3x + 4y。它们不是同类项，因为变量 x 和 y 不同。

You can still simplify an expression by grouping the existing like terms together. In 3x + 2y + 4x + 7y, collect the x terms and the y terms separately.

你仍然可以通过把已有同类项分组来化简表达式。在 3x + 2y + 4x + 7y 中，分别合并 x 项和 y 项。

3x + 2y + 4x + 7y = 7x + 9y

Here, 3x + 4x = 7x and 2y + 7y = 9y. The final expression has one x term and one y term.

这里，3x + 4x = 7x，2y + 7y = 9y。最终表达式含有一个 x 项和一个 y 项。

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6. Powers and Like Terms | 幂次与同类项

Terms with the same variable but different powers are

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