Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In IGCSE Mathematics, combining like terms is one of the first essential algebra skills. It allows you to simplify expressions, solve equations and work with formulas in a clear and efficient way. A term is like another term only when the variable part, including all powers, is exactly the same. Once you master this idea, many harder algebra topics become much easier.

在 IGCSE 数学中，合并同类项是最基础的代数技能之一。它能帮助你化简表达式、解方程以及清晰地使用公式。只有当两个项的变量部分（包括所有指数）完全相同时，它们才是同类项。掌握这个概念后，许多更难的代数主题都会变得简单很多。

1. What Are Like Terms? | 什么是同类项

In algebra, a term is a number, a variable, or a product of numbers and variables. Like terms have exactly the same variable part, including the same powers. For example, 3x and −5x are like terms because both contain x to the power 1. The coefficients 3 and −5 may be different, but the variable part x is identical. In 4x² and 6x², the variable part is x², so they are also like terms.

在代数中，项是一个数、一个变量或数与变量的乘积。同类项具有完全相同的变量部分，包括相同的指数。例如，3x 和 −5x 是同类项，因为两者都含有 x 的一次方。系数 3 和 −5 可以不同，但变量部分 x 是相同的。在 4x² 和 6x² 中，变量部分是 x²，所以它们也是同类项。

Terms such as 4xy and 9xy are like terms because both have the variable part xy. However, 4xy and 4x are not like terms, because one contains y and the other does not. The same rule applies to products with several variables: every variable and every exponent must match.

像 4xy 和 9xy 这样的项是同类项，因为两者的变量部分都是 xy。但是，4xy 和 4x 不是同类项，因为一个含有 y，另一个不含 y。同样的规则也适用于含多个变量的乘积：每个变量及其指数都必须一致。

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2. Why Combining Matters | 为什么需要合并

When expressions become long, combining like terms makes them shorter and easier to use. For example, x + 2x + 3x can be simplified to 6x. This is important because simplified expressions are less likely to cause mistakes in later working, especially when solving equations or substituting values.

当表达式很长时，合并同类项可以让它更短、更容易使用。例如，x + 2x + 3x 可以化简为 6x。这一点很重要，因为化简后的表达式在后续计算中更不容易出错，尤其是在解方程或代入数值时。

Combining like terms is also a foundation for solving linear equations, graphing straight lines and expanding products. In IGCSE examinations, clear simplification often earns method marks even if a small arithmetic slip appears later. Therefore, it is worth practising until the process becomes automatic.

合并同类项也是解一元一次方程、绘制直线图像和展开乘积的基础。在 IGCSE 考试中，即使后面出现小的算术失误，清晰的化简过程通常也能获得过程分。因此，值得反复练习，直到这个过程变得自动化。

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3. Basic Single Variable Collecting | 基础单变量合并

ilarly, 10m − 3m = (10 − 3)m = 7m. Do not multiply the variable part: 4a + 7a is not 11a².

合并同类项时，只对系数进行加减。变量部分保持完全不变。例如，4a + 7a = (4 + 7)a = 11a。同样，10m − 3m = (10 − 3)m = 7m。不要将变量部分相乘：4a + 7a 不等于 11a²。

4a + 7a = 11a

10m − 3m = 7m

fied answer is 5p. This grouping method is especially helpful when negative signs are involved.

当一个表达式包含多个同类项时，可以按变量部分进行分组。在 6p + 2p − 3p 中，所有项都含有 p，因此计算 6 + 2 − 3 = 5。化简结果是 5p。这种分组方法在含有负号时尤其有用。

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4. Multiple Variables and Powers | 多变量与幂

Terms with different variables are not like terms. For instance, 3x and 4y cannot be combined into 7xy. Terms with the same variables but different powers are also unlike: 5x² and 2x are different because the exponents of x are not the same.

变量不同的项不是同类项。例如，3x 和 4y 不能合并为 7xy。变量相同但指数不同的项也不是同类项：5x² 和 2x 是不同的，因为 x 的指数不相同。

For terms with two variables, every variable and every exponent must match. 3xy² and −7xy² are like terms because both contain x¹ and y². However, 3xy² and −7x²y are not like terms, since the exponents attached to x and y are swapped.

对于含有两个变量的项，每个变量及其指数都必须相同。3xy² 和 −7xy² 是同类项，因为两者都含有 x¹ 和 y²。但是，3xy² 和 −7x²y 不是同类项，因为 x 和 y 的指数对调了。

3xy² − 7xy² = −4xy²

7x³, but 2x³ + 5x² cannot be simplified further. Always check the powers before combining.

这一思想也适用于更高次幂。例如，2x³ + 5x³ = 7x³，但 2x³ + 5x² 不能进一步化简。合并之前一定要先检查指数。

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5. Collecting With Brackets | 去括号后合并

2(x − 2y) = 6x + 3y + 2x − 4y. Then collect: 6x + 2x = 8x and 3y − 4y = −y, giving 8x − y.

如果有括号，先去括号。乘以正数时，括号内各项符号不变。乘以负数时，括号内每一项符号都要变号。例如：3(2x + y) + 2(x − 2y) = 6x + 3y + 2x − 4y。然后合并：6x + 2x = 8x，3y − 4y = −y，结果为 8x − y。

3(2x + y) + 2(x − 2y) = 8x − y

Another common case is subtracting a bracket, such as 5(a + 2b) − 3(a − b). Expand to 5a + 10b − 3a + 3b. Notice that −3(−b) becomes +3b. Combining gives 2a + 13b. The sign change is often where marks are lost, so work carefully.

另一种常见情况是减去一个括号，例如 5(a + 2b) − 3(a − b)。展开得到 5a + 10b − 3a + 3b。注意 −3(−b) 变为 +3b。合并后得到 2a + 13b。符号变化往往是失分的地方，因此要仔细计算。

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6. Signs and Negative Coefficients | 符号与负系数

, so the answer is 3x + 3y. A common error is writing −3y + 6y as −9y, which is incorrect.

要小心负系数。使用每一项前面的符号。对于 5x − 3y − 2x + 6y，仔细分组：5x − 2x = 3x，−3y + 6y = 3y，所以答案是 3x + 3y。常见错误是把 −3y + 6y 算成 −9y，这是不正确的。

5x − 3y − 2x + 6y = 3x + 3y

If an expression feels confusing, rewrite it as a sum of signed terms. For example, 5x − 3y − 2x + 6y can be written as 5x + (−3y) + (−2x) + 6y. This makes it easier to see the separate coefficients and reduces sign mistakes.

如果表达式令人困惑，可以将其改写为带符号项的和。例如，5x − 3y − 2x + 6y 可以写成 5x + (−3y) + (−2x) + 6y。这样更容易看清各项的系数，从而减少符号错误。

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7. Expressions With Fractions and Decimals | 含分数与小数的表达式

½x + ⅓x can be combined using a common denominator: ½ + ⅓ = 3/6 + 2/6 = 5/6, so the sum is ⅚x. With decimals, 0.4y + 0.1y = 0.5y.

同样的规则适用于分数和小数系数。只加减系数，保持变量部分不变。例如，½x + ⅓x 可以用公分母合并：½ + ⅓ = 3/6 + 2/6 = 5/6，因此和为 ⅚x。对于小数，0.4y + 0.1y = 0.5y。

½x + ⅓x = ⅚x

0.4y + 0.1y = 0.5y

, which is simply written as m. Avoid leaving unnecessary 1 coefficients in a final answer.

IGCSE 试卷可能要求分数保留最简形式。如果可能，最后要将系数中的公因数约掉。例如，¼m + ¾m = 1m，简写为 m。最终答案中要避免保留不必要的系数 1。

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8. Collecting in Equations | 方程中的合并

This gives 5x − 6 = 30. Then add 6 to both sides and divide by 5 to find x = 36/5 or 7.2.

合并同类项通常是解一元一次方程的第一步。对于 3x + 4 + 2x − 10 = 30，先合并左边的同类项：3x + 2x = 5x，4 − 10 = −6。得到 5x − 6 = 30。然后两边加 6，再除以 5，得到 x = 36/5 或 7.2。

3x + 4 + 2x − 10 = 30

5x − 6 = 30

x = 36/5

tead, collect like terms on each side first or move terms across using inverse operations.

切勿把等号两边的项当作同一边

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