Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

When we simplify algebraic expressions, one of the most important skills is collecting like terms. This process combines terms that have exactly the same variables and powers, making the expression shorter and easier to work with.

在化简代数表达式时，最重要的技能之一就是合并同类项。这一过程将具有完全相同变量和指数的项合并在一起，使表达式更简洁、更易处理。

1. What Are ‘Like Terms’? | 什么是“同类项”？

Like terms are terms that contain the same variable(s) raised to the same power. The coefficient (the number in front) can be different, but the variable part must match exactly.

同类项是指包含相同变量且变量指数相同的项。前面的系数（数字部分）可以不同，但变量部分必须完全一致。

For example, \(3x\) and \(5x\) are like terms because both contain \(x\). However, \(3x\) and \(3x^2\) are not like terms because the powers of \(x\) are different.

例如，\(3x\) 和 \(5x\) 是同类项，因为它们都包含 \(x\)。但 \(3x\) 和 \(3x^2\) 不是同类项，因为 \(x\) 的指数不同。

Constants (numbers without variables) are also like terms with each other. For instance, \(2\), \(-7\) and \(0.5\) are all like terms.

常数项（不含变量的数字）之间也都是同类项。例如，\(2\)、\(-7\) 和 \(0.5\) 都是同类项。

2. Recognising Like Terms | 识别同类项

Look at the variable part of each term. Ask yourself: are the variables the same? Are their exponents the same? If both answers are yes, the terms are like terms.

查看每一项的变量部分。问自己：变量是否相同？指数是否相同？如果两个答案都是“是”，那么这些项就是同类项。

\(4ab\) and \(-2ab\) are like terms because both have \(ab\).
\(4ab\) and \(4ba\) are also like terms because multiplication is commutative: \(ab = ba\).
\(4ab\) and \(4a^2b\) are not like terms because the exponent of \(a\) is different.

注意：\(4ab\) 与 \(-2ab\) 是同类项，因为都包含 \(ab\)；\(4ab\) 与 \(4ba\) 也是同类项，因为乘法可交换：\(ab = ba\)；但 \(4ab\) 与 \(4a^2b\) 不是同类项，因为 \(a\) 的指数不同。

Variables written without a power are considered to have power 1. Thus \(y\) means \(y^1\).

没有写出指数的变量视为指数为 1。因此 \(y\) 即 \(y^1\)。

3. The Rule for Adding and Subtracting Like Terms | 合并同类项的规则

To collect like terms, add or subtract their coefficients while keeping the variable part unchanged. This is similar to counting objects: \(3\) apples plus \(5\) apples gives \(8\) apples.

合并同类项时，只需将系数相加或相减，变量部分保持不变。这类似于数物体：\(3\) 个苹果加 \(5\) 个苹果得到 \(8\) 个苹果。

\(ax + bx = (a+b)x\)

For example, \(2x + 3x = 5x\). The variable \(x\) does not change.

例如，\(2x + 3x = 5x\)。变量 \(x\) 不变。

If the coefficients are decimals or fractions, the rule is exactly the same: \(0.5y + 0.25y = 0.75y\).

如果系数是小数或分数，规则完全相同：\(0.5y + 0.25y = 0.75y\)。

4. Collecting Terms with Addition | 含加法的合并

Simplify \(4x + 7x – 2x\). First identify all \(x\)-terms. Then combine their coefficients: \(4 + 7 – 2 = 9\). The result is \(9x\).

化简 \(4x + 7x – 2x\)。首先找出所有含 \(x\) 的项，然后合并系数：\(4 + 7 – 2 = 9\)。结果为 \(9x\)。

Simplify \(3p + 5q – p + 2q\). Collect \(p\)-terms: \(3p – p = 2p\). Collect \(q\)-terms: \(5q + 2q = 7q\). The simplified expression is \(2p + 7q\).

化简 \(3p + 5q – p + 2q\)。合并 \(p\) 项：\(3p – p = 2p\)；合并 \(q\) 项：\(5q + 2q = 7q\)。化简结果为 \(2p + 7q\)。

Always reorder the terms so that like terms are next to each other before adding coefficients.

在相加系数之前，务必重新排列各项，使同类项相邻。

5. Dealing with Subtraction | 处理减法

When subtracting like terms, change the sign of the term being subtracted, then add. For example, \(6a – 2a = 4a\).

合并含减法的同类项时，将减去的项变为相反数再相加。例如，\(6a – 2a = 4a\)。

Simplify \(8x – 3y – 5x + y\). Combine \(x\)-terms: \(8x – 5x = 3x\). Combine \(y\)-terms: \(-3y + y = -2y\). Thus the answer is \(3x – 2y\).

化简 \(8x – 3y – 5x + y\)。合并 \(x\) 项：\(8x – 5x = 3x\)；合并 \(y\) 项：\(-3y + y = -2y\)。因此答案为 \(3x – 2y\)。

Do not forget the negative sign in front of a term. The term \(-4y\) has coefficient \(-4\).

不要忘记项前面的负号。项 \(-4y\) 的系数是 \(-4\)。

6. Terms with Multiple Variables | 含多个变量的项

Terms with different variable combinations are not like terms. For example, \(2xy\) and \(3x\) are not like terms; \(2xy\) and \(3y\) are also not like terms.

变量组合不同的项不是同类项。例如，\(2xy\) 与 \(3x\) 不是同类项；\(2xy\) 与 \(3y\) 也不是同类项。

Only terms with exactly the same variables and powers can be combined. For instance, \(5mn – 2mn = 3mn\).

只有变量和指数完全相同的项才能合并。例如，\(5mn – 2mn = 3mn\)。

Simplify \(3ab + 4b – 2ab + a\). Combine \(ab\)-terms: \(3ab – 2ab = ab\). The \(b\)-term is \(4b\), and the \(a\)-term is \(a\). So the answer is \(ab + 4b + a\).

化简 \(3ab + 4b – 2ab + a\)。合并 \(ab\) 项：\(3ab – 2ab = ab\)。\(b\) 项为 \(4b\)，\(a\) 项为 \(a\)。所以答案为 \(ab + 4b + a\)。

7. Collecting Terms after Expanding Brackets | 去括号后合并同类项

Sometimes we need to expand brackets first, then collect like terms. For example, simplify \(2(x+3) + 4(x-1)\).

有时我们需要先去括号，再合并同类项。例如，化简 \(2(x+3) + 4(x-1)\)。

Expand: \(2x + 6 + 4x – 4\). Now collect like terms: \(2x + 4x = 6x\), and \(6 – 4 = 2\). The simplified expression is \(6x + 2\).

展开得：\(2x + 6 + 4x – 4\)。现在合并同类项：\(2x + 4x = 6x\)，\(6 – 4 = 2\)。化简结果为 \(6x + 2\)。

Be careful with negative signs when expanding. Simplify \(3(a – 2) – 2(a + 1)\): expand to \(3a – 6 – 2a – 2\), then combine to \(a – 8\).

去括号时注意负号。化简 \(3(a – 2) – 2(a + 1)\)：展开为 \(3a – 6 – 2a – 2\)，合并得 \(a – 8\)。

8. Common Mistakes to Avoid | 应避免的常见错误

One common mistake is adding coefficients of unlike terms. For example, \(2x + 3y\) cannot be simplified to \(5xy\). They are not like terms.

一个常见错误是将非同类项的系数相加。例如，\(2x + 3y\) 不能化简为 \(5xy\)。它们不是同类项。

Another mistake is forgetting to include the sign of a term. When moving terms, keep the sign that appears to the left of the term.

另一个错误是忘记项的符号。移动项时，要保持该项左侧的符号。

Also, do not change the exponent when combining. \(x^2 + x^2 = 2x^2\), not \(2x^4\).

此外，合并时不要改变指数。\(x^2 + x^2 = 2x^2\)，而不是 \(2x^4\)。

Expression	Correct	Incorrect
\(5x + 2x\)	\(7x\)	\(7x^2\)
\(3a + 4b\)	\(3a + 4b\)	\(7ab\)
\(4y – y\)	\(3y\)	\(4\)

表格中列出了常见表达式的正确与错误化简方式。

9. Collecting Like Terms with Fractions and Decimals | 分数与小数的同类项合并

When coefficients are fractions, combine them using normal fraction arithmetic. For example, \(\frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x\).

当系数为分数时，使用普通分数运算来合并。例如，\(\frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x\)。

With decimals, align the digits and add or subtract as usual. For instance, \(0.3m + 0.45m = 0.75m\).

对于小数，按通常方式对齐数字后相加或相减。例如，\(0.3m + 0.45m = 0.75m\)。

If an expression contains both fractions and decimals, you may convert them to the same form before combining.

如果表达式中同时含有分数和小数，可以先将其转换成相同形式再合并。

10. Real-World Application | 实际应用

Suppose you buy \(2\) pencils and \(3\) pens, then later buy \(4\) pencils and \(1\) pen. The total number of pencils is \(2p + 4p = 6p\), and pens is \(3q + q = 4q\). Collecting like terms helps organise such totals.

假设你买了 \(2\) 支铅笔和 \(3\) 支钢笔，后来又买了 \(4\) 支铅笔和 \(1\) 支钢笔。铅笔总数为 \(2p + 4p = 6p\)，钢笔总数为 \(3q + q = 4q\)。合并同类项有助于整理这类总数。

Collecting like terms also appears in geometry, such as finding the perimeter of a shape with variables. If a rectangle has side lengths \(x + 2\) and \(2x – 1\), its perimeter is \(2(x+2) + 2(2x-1) = 2x+4+4x-2 = 6x+2\).

合并同类项也出现在几何中，例如求含变量的图形周长。若一个矩形的边长为 \(x + 2\) 和 \(2x – 1\)，其周长为 \(2(x+2) + 2(2x-1) = 2x+4+4x-2 = 6x+2\)。

11. Practice Problems | 练习题

Try these problems to test your understanding:

尝试以下题目来检验你的理解：

Simplify \(7x + 3x – 5x\).
Simplify \(4a + 2b – a + 3b\).
Simplify \(5m^2 – 2m^2 + m\).
Simplify \(2(3p – 4) + 3(p + 2)\).
Simplify \(\frac{2}{3}y + \frac{1}{6}y – y\).

Answers: 1. \(5x\)　2. \(3a + 5b\)　3. \(3m^2 + m\)　4. \(9p – 2\)　5. \(-\frac{1}{6}y\).

答案：1. \(5x\)　2. \(3a + 5b\)　3. \(3m^2 + m\)　4. \(9p – 2\)　5. \(-\frac{1}{6}y\)。

12. Summary | 总结

Collecting like terms is a fundamental algebra skill. Remember to check that terms have identical variable parts and identical exponents before combining. Then add or subtract only the coefficients.

合并同类项是代数的基础技能。记住，合并前要检查项的变量部分和指数是否完全相同，然后只对系数进行加或减。

Mastering this technique makes solving equations, expanding brackets, and manipulating formulas much easier. Keep practising, and soon it will become second nature.

掌握这一技巧会使解方程、去括号和变换公式变得更加容易。坚持练习，很快它就会成为你的自然反应。

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Collecting Like Terms | 合并同类项