Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain many terms. Learning how to simplify them by collecting like terms is a fundamental skill for IGCSE Mathematics. It helps you solve equations faster and understand how variables behave.

在代数中，表达式通常包含许多项。学会通过合并同类项来化简它们是 IGCSE 数学的基本技能。它可以帮助你更快地解方程，并理解变量的运算规律。

1. What Is a Term? | 什么是“项”？

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 4y – 5, the terms are 3x, 4y and -5.

项是单个数字、变量或数字与变量的乘积。例如，在表达式 3x + 4y – 5 中，各项分别是 3x、4y 和 -5。

Each term has a sign in front of it. The sign belongs to that term when we move it around. So in 3x + 4y – 5, the term -5 has a negative sign.

每个项前面都有符号。当我们移动项时，符号属于该项。因此在 3x + 4y – 5 中，项 -5 带有负号。

2. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part. The same variable(s) with the same powers. For example, 5x and -3x are like terms because both have x to the power 1.

同类项是指变量部分完全相同的项，即相同的变量并且变量的次数相同。例如，5x 和 -3x 是同类项，因为它们都含有 x 的一次方。

Similarly, 2x² and 7x² are like terms because both have x squared. But 4x² and 4x are not like terms because the powers of x are different.

同样，2x² 和 7x² 是同类项，因为它们都含有 x 的平方。但 4x² 和 4x 不是同类项，因为 x 的次数不同。

Constant numbers are also like terms. For instance, 3 and -8 are like terms because they are both plain numbers.

常数项也是同类项。例如，3 和 -8 是同类项，因为它们都是纯数字。

3. How to Collect Like Terms | 如何合并同类项

To collect like terms, add or subtract their coefficients while keeping the variable part unchanged. A coefficient is the number in front of a variable.

合并同类项时，将它们的系数相加或相减，而变量部分保持不变。系数是变量前面的数字。

For example, combine 3x and 5x. Both have the variable x, so add the coefficients: 3 + 5 = 8. The result is 8x.

例如，合并 3x 和 5x。两者都有变量 x，所以将系数相加：3 + 5 = 8。结果是 8x。

3x + 5x = (3 + 5)x = 8x

4. Working with Subtraction | 处理减法

When subtracting terms, treat the negative sign as part of the coefficient. For example, 9x – 4x means the coefficients are 9 and -4.

做减法时，将负号视为系数的一部分。例如，9x – 4x 中系数分别是 9 和 -4。

9x – 4x = (9 – 4)x = 5x

Be careful with expressions like -2x – 3x. The coefficients are -2 and -3, so the sum is -5x.

注意像 -2x – 3x 这样的表达式。系数是 -2 和 -3，所以和为 -5x。

For constants, we use the same method: 12 – 5 = 7, or -8 + 3 = -5.

对于常数项，我们使用同样的方法：12 – 5 = 7，或 -8 + 3 = -5。

5. Mixing Different Variables | 混合不同变量

Only like terms can be combined. Different variables, such as x and y, are not like terms, so they must remain separate in the final expression.

只有同类项才能合并。不同的变量，例如 x 和 y，不是同类项，因此它们必须在最终表达式中保持分开。

Simplify 4x + 3y – 2x + y. First combine the x terms: 4x – 2x = 2x. Then combine the y terms: 3y + y = 4y. The answer is 2x + 4y.

化简 4x + 3y – 2x + y。先合并含 x 的项：4x – 2x = 2x。再合并含 y 的项：3y + y = 4y。答案为 2x + 4y。

4x + 3y – 2x + y = (4 – 2)x + (3 + 1)y = 2x + 4y

6. Powers and Indices | 幂与指数

When variables have different powers, they are not like terms. For example, x and x² cannot be combined because their indices are different.

当变量的幂不同时，它们不是同类项。例如，x 和 x² 不能合并，因为它们的指数不同。

Simplify 5x² + 2x – x² – 3x. Combine x² terms: 5x² – x² = 4x². Combine x terms: 2x – 3x = -x. The final answer is 4x² – x.

化简 5x² + 2x – x² – 3x。合并 x² 项：5x² – x² = 4x²。合并 x 项：2x – 3x = -x。最终答案为 4x² – x。

Remember that x² is different from x, and xy is different from x²y. The variable part must match exactly, including all powers.

记住，x² 不同于 x，xy 也不同于 x²y。变量部分必须完全匹配，包括所有的幂。

7. Using the Distributive Rule | 使用乘法分配律

Sometimes an expression contains brackets. The distributive rule says a(b + c) = ab + ac. This lets us remove brackets before collecting like terms.

有时表达式含有括号。乘法分配律指出 a(b + c) = ab + ac。这让我们可以在合并同类项之前先去括号。

Simplify 2(3x + 4) + 5x. First expand: 2 × 3x + 2 × 4 = 6x + 8. Then add 5x: 6x + 5x + 8 = 11x + 8.

化简 2(3x + 4) + 5x。先去括号：2 × 3x + 2 × 4 = 6x + 8。然后加上 5x：6x + 5x + 8 = 11x + 8。

2(3x + 4) + 5x = 6x + 8 + 5x = 11x + 8

Be careful when a negative sign appears before brackets, such as 3(x – 2) – (x – 1). The second bracket means -1 × (x – 1), so it becomes -x + 1.

当括号前有负号时要小心，例如 3(x – 2) – (x – 1)。第二个括号相当于 -1 × (x – 1)，所以它会变成 -x + 1。

8. Common Mistakes | 常见错误

One common mistake is combining non-like terms, such as 2x + 3y = 5xy. This is wrong because x and y are different variables.

一个常见错误是合并非同类项，例如 2x + 3y = 5xy。这是错误的，因为 x 和 y 是不同的变量。

Another mistake is forgetting to include the variable part when adding coefficients. For example, 4x + 5x is 9x, not 9.

另一个错误是忘记在系数相加后保留变量部分。例如，4x + 5x 应该是 9x，而不是 9。

Also, watch the sign of a term. For example, 6 – 2x + x means 6 + (-2x + x) = 6 – x. Many students write 6 – 2x + x = 6 – x, which is correct, but they may accidentally drop the negative sign.

同时要注意项的符号。例如，6 – 2x + x 表示 6 + (-2x + x) = 6 – x。许多学生写成 6 – 2x + x = 6 – x，这是正确的，但他们可能会不小心丢掉负号。

9. Worked Examples | 综合例题

Example 1: Simplify 7a + 2b – 3a – 5b.

例 1：化简 7a + 2b – 3a – 5b。

Combine a terms: 7a – 3a = 4a. Combine b terms: 2b – 5b = -3b. Answer: 4a – 3b.

合并 a 项：7a – 3a = 4a。合并 b 项：2b – 5b = -3b。答案：4a – 3b。

Example 2: Simplify 4(x + 2) – 2(2x – 1).

例 2：化简 4(x + 2) – 2(2x – 1)。

Expand: 4x + 8 – 4x + 2. Now combine like terms: 4x – 4x = 0, and 8 + 2 = 10. So the answer is 10.

去括号：4x + 8 – 4x + 2。合并同类项：4x – 4x = 0，8 + 2 = 10。所以答案是 10。

Example 3: Simplify 3x² – 2x + 4x – x² + 5.

例 3：化简 3x² – 2x + 4x – x² + 5。

Combine x² terms: 3x² – x² = 2x². Combine x terms: -2x + 4x = 2x. Constants: +5. Final answer: 2x² + 2x + 5.

合并 x² 项：3x² – x² = 2x²。合并 x 项：-2x + 4x = 2x。常数项：+5。最终答案：2x² + 2x + 5。

10. Practice Questions | 练习题目

Try these by yourself. They get slightly harder as you go.

请自行尝试以下练习。难度会逐渐增加。

1. Simplify 8x + 3x – 5x.
1. 化简 8x + 3x – 5x。
2. Simplify 2a + 3b + 4a – b.
2. 化简 2a + 3b + 4a – b。
3. Simplify 5p – 2p² + p – p².
3. 化简 5p – 2p² + p – p²。
4. Simplify 3(2x – 1) + 2(4x + 5).
4. 化简 3(2x – 1) + 2(4x + 5)。
5. Simplify 4y + 3x – (x – 2y).
5. 化简 4y + 3x – (x – 2y)。

Answers: 1) 6x 2) 6a + 2b 3) 6p – 3p² 4) 14x + 7 5) 6y + 2x.

答案：1) 6x 2) 6a + 2b 3) 6p – 3p² 4) 14x + 7 5) 6y + 2x。

11. Summary | 总结

Collecting like terms is the process of simplifying expressions by adding or subtracting the coefficients of terms that have the same variable part. Always keep the variable and its power unchanged during the operation.

合并同类项是通过对具有相同变量部分的项的系数进行加法或减法来化简表达式的过程。在运算过程中，要始终保持变量及其幂不变。

Make sure to identify like terms carefully, respect the signs of each term, and expand brackets first when needed. With practice, this skill will become quick and natural.

务必仔细识别同类项，注意每个项的符号，必要时先去括号。经过练习，这项技能会变得快速而自然。

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Collecting Like Terms | 合并同类项