Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, simplifying expressions by collecting like terms is a fundamental skill. It allows you to rewrite long, complicated expressions into concise forms that are easier to evaluate and use.

在代数中，通过合并同类项来化简表达式是一项基本技能。它能将冗长复杂的表达式改写为简洁的形式，便于求值和使用。

1. What is a Term? | 什么是“项”？

A term is a single mathematical expression that may contain numbers, variables, or both. In an algebraic expression, terms are separated by ‘+’ or ‘−’ signs. For example, in the expression 3x + 2y − 5, there are three terms: 3x, 2y, and −5.

“项”是一个单一的数学表达式，可以包含数字、变量或两者兼有。在代数表达式中，各项由“+”或“−”符号分隔。例如，在表达式 3x + 2y − 5 中，有三个项：3x、2y 和 −5。

Each term has two parts: a coefficient (the number in front of the variable) and a variable part (the letter or letters with any exponents). For instance, in the term 4x², 4 is the coefficient and x² is the variable part.

每一个项都有两部分：系数（变量前面的数字）和变量部分（字母及其指数）。例如，在项 4x² 中，4 是系数，x² 是变量部分。

2. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part, including the same powers or exponents. The coefficients may be different, but the letters and their exponents must match exactly.

同类项是指变量部分完全相同的项，包括相同的幂或指数。系数可以不同，但字母及其指数必须完全相同。

For example, 5x and −3x are like terms because both contain x. Likewise, 2y² and 7y² are like terms because both contain y². However, 5x and 5x² are not like terms, because the exponents of x are different.

例如，5x 和 −3x 是同类项，因为两者都含有 x。同样地，2y² 和 7y² 是同类项，因为两者都含有 y²。然而，5x 和 5x² 不是同类项，因为 x 的指数不同。

3. Why Combine Like Terms? | 为什么要合并同类项？

Combining like terms makes an expression simpler and easier to work with. A shorter expression is less likely to lead to mistakes in subsequent calculations, and it often reveals the structure of a problem more clearly.

合并同类项能使表达式更简洁、更易处理。较短的表达式在后续计算中更不容易出错，而且通常能更清晰地揭示问题的结构。

For instance, the expression 7a + 3a − 2a + 6 + 1 can be simplified to 8a + 7. This simplified form is much easier to evaluate when a specific value of a is substituted.

例如，表达式 7a + 3a − 2a + 6 + 1 可以化简为 8a + 7。当代入 a 的具体值时，这种简化形式更容易求值。

4. The Golden Rule | 合并的黄金法则

When combining like terms, only add or subtract the coefficients. The variable part stays exactly the same. Do not change the letters or their exponents.

合并同类项时，只对系数进行加或减，变量部分保持不变。不要改变字母或其指数。

ax + bx = (a + b)x

For example, 4m + 9m = (4 + 9)m = 13m. The variable m remains unchanged; only the numbers 4 and 9 are added.

例如，4m + 9m = (4 + 9)m = 13m。变量 m 保持不变，只有数字 4 和 9 相加。

This rule applies to any number of terms. If you have 3p − 5p + 8p, you combine the coefficients: 3 − 5 + 8 = 6, so the result is 6p.

这条法则适用于任意数量的项。如果有 3p − 5p + 8p，合并系数：3 − 5 + 8 = 6，所以结果是 6p。

5. Step-by-Step Examples | 分步示例

Let us simplify the expression 6x + 5 − 2x + 3.

我们来化简表达式 6x + 5 − 2x + 3。

Identify like terms: 6x and −2x are like terms; 5 and 3 are constant terms. 找出同类项：6x 和 −2x 是同类项；5 和 3 是常数项。
Combine the x-terms: 6x − 2x = 4x. 合并带 x 的项：6x − 2x = 4x。
Combine the constants: 5 + 3 = 8. 合并常数项：5 + 3 = 8。
Write the final answer: 4x + 8. 写出最终答案：4x + 8。

Another example: 3a² + 2a + 5a² − a = 8a² + a. Here a² terms are combined separately from a terms.

另一个例子：3a² + 2a + 5a² − a = 8a² + a。这里 a² 项和 a 项分别合并。

6. Dealing with Negative Signs | 处理负号

Negative signs must be treated as part of the term that follows them. For example, in the expression 7x − 3x + 2x − 4x, the signs of the coefficients are +7, −3, +2, and −4.

负号必须视为其后项的一部分。例如，在表达式 7x − 3x + 2x − 4x 中，系数的符号分别是 +7、−3、+2 和 −4。

Combining them gives 7 − 3 + 2 − 4 = 2, so the full expression simplifies to 2x.

合并后得到 7 − 3 + 2 − 4 = 2，所以整个表达式化简为 2x。

Be careful when subtracting a negative coefficient. For example, 5y − (−3y) = 5y + 3y = 8y. The two negative signs become a positive sign.

当减去一个负数系数时要小心。例如，5y − (−3y) = 5y + 3y = 8y。两个负号变成正号。

7. Different Variables and Powers | 不同的变量与幂

Terms that have different variables or different powers cannot be combined. For example, x and y are different, so x + y cannot be simplified to a single term.

变量不同或幂不同的项不能合并。例如，x 和 y 不同，所以 x + y 不能化简为单个项。

Similarly, x and x² are different powers, so x + x² cannot be combined. The expression must remain as x + x².

类似地，x 和 x² 是不同幂次，所以 x + x² 不能合并，表达式必须保留为 x + x²。

However, you can collect groups separately: 4x + 3y + 2x + 5y = (4x + 2x) + (3y + 5y) = 6x + 8y.

但可以分组分别合并：4x + 3y + 2x + 5y = (4x + 2x) + (3y + 5y) = 6x + 8y。

If a term has no coefficient written, the coefficient is understood to be 1. For example, x means 1x and −x means −1x.

如果一项没有写出系数，则默认系数为 1。例如，x 表示 1x，−x 表示 −1x。

8. Expanding First | 先展开再合并

Sometimes you need to use the distributive law before collecting like terms. In IGCSE algebra, expressions often contain brackets that must be expanded first.

有时需要先使用分配律，再进行合并同类项。在 IGCSE 代数中，表达式常含括号，必须先展开括号。

Example: Simplify 3(x + 2) + 4x.

示例：化简 3(x + 2) + 4x。

Expand: 3(x + 2) = 3x + 6. 展开：3(x + 2) = 3x + 6。
The expression becomes 3x + 6 + 4x. 原式变为 3x + 6 + 4x。
Collect like terms: 3x + 4x = 7x, so the answer is 7x + 6. 合并同类项：3x + 4x = 7x，所以答案是 7x + 6。

Always expand brackets carefully: 2(3x − 1) = 6x − 2, not 6x − 1. The number outside must multiply every term inside the bracket.

展开括号时务必仔细：2(3x − 1) = 6x − 2，而不是 6x − 1。括号外的数字必须乘以括号内的每一项。

9. Real-World Application | 实际应用

Collecting like terms is frequently used when finding perimeters and areas of shapes. Suppose a rectangle has a length of (3x + 4) cm and a width of (2x − 1) cm. The perimeter is twice the sum of length and width.

合并同类项经常用于求图形的周长和面积。假设一个长方形的长为 (3x + 4) cm，宽为 (2x − 1) cm，则周长是长与宽之和的两倍。

Perimeter = 2[(3x + 4) + (2x − 1)] = 2[3x + 4 + 2x − 1] = 2[5x + 3] = 10x + 6.

周长 = 2[(3x + 4) + (2x − 1)] = 2[3x + 4 + 2x − 1] = 2[5x + 3] = 10x + 6。

This simplified formula allows you to quickly find the perimeter for any value of x without redoing all the algebra.

这个简化后的公式使你在任意 x 值下都能快速求出周长，而不需要重新做所有代数运算。

10. Common Mistakes and Tips | 常见错误与提示

Mistake: Combining 2x and 3x² as 5x³. Error code: x and x² are not like terms. 错误：把 2x 和 3x² 合并成 5x³。注意：x 和 x² 不是同类项。
Mistake: Forgetting the invisible 1: x + 2x should be 3x, not 2x. 错误：忘记隐藏的 1：x + 2x 应为 3x，而不是 2x。
Mistake: Changing the variable part: 5a + 3a = 8a, not 8a². 错误：改变变量部分：5a + 3a = 8a，而不是 8a²。
Tip: Underline or circle like terms in different colors before combining. 提示：在合并前用不同颜色或符号标出各类同类项。
Tip: Always write the final expression with the terms in descending order of powers, for example 4x² + 3x + 5. 提示：最终表达式通常按幂的降序书写，例如 4x² + 3x + 5。

11. Practice Questions | 自测练习

Try to simplify the following expressions. Check your answers below.

尝试化简下列表达式，并在下方核对答案。

Question 题目	Answer 答案
1. 5x + 3x − 2x	6x
2. 7a + a − 4a	4a
3. 2y² + 3y + y² − y	3y² + 2y
4. 4m + 2 − 3m + 6	m + 8
5. 2(x + 5) + 3x	5x + 10

If you got all five correct, you have mastered collecting like terms at the IGCSE level.

如果你五题全对，说明你已经掌握了 IGCSE 水平的合并同类项。

12. Summary | 总结

Collecting like terms is the process of simplifying algebraic expressions by adding or subtracting the coefficients of terms that have the same variable part. The key is to keep the variable part unchanged and only operate on the numbers in front of the variables.

合并同类项是通过对具有相同变量部分的项进行系数加或减来化简代数表达式的过程。关键在于保持变量部分不变，只对变量前面的数字进行运算。

Remember to identify like terms accurately, handle negative signs with care, expand brackets first when necessary, and never combine terms with different variables or powers. With practice, this skill becomes fast and automatic.

请记住：准确识别同类项、小心处理负号、必要时先展开括号，并且绝不合并变量或幂不同的项。通过练习，这项技能会变得又快又自然。

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Collecting Like Terms | 合并同类项