Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, when we say “collecting like terms,” we mean simplifying an expression by adding or subtracting terms that have exactly the same variable part. This process turns long, messy expressions into shorter, cleaner ones.

在代数中，“合并同类项”是指将变量部分完全相同的项进行加减，从而把冗长、杂乱的表达式化简为更简短、整洁的形式。

1. What is a Term in Algebra? | 代数中的“项”是什么？

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression 3x + 5y – 2, the terms are 3x, 5y and -2.

项是一个单独的数字、一个变量，或数字与变量的乘积。例如，在表达式 3x + 5y – 2 中，各项分别是 3x、5y 和 -2。

Each term has two parts: a coefficient (the number in front) and a variable part (the letters and their powers). In 7x², the coefficient is 7 and the variable part is x².

每一项都包含两部分：系数（前面的数字）和变量部分（字母及其幂）。在 7x² 中，系数是 7，变量部分是 x²。

Constant terms are numbers without variables, such as 4, -3 or ½. They are also considered terms.

常数项是不含变量的数字，如 4、-3 或 ½。它们也被视为项。

2. What are Like Terms? | 什么是同类项？

Like terms are terms whose variable parts are identical, including the same letters raised to the same powers. Only the coefficients may be different.

同类项是指变量部分完全相同的项，包括相同的字母以及相同的幂指数。只有系数可以不同。

For instance, 4x and -7x are like terms because both have the variable part x. Similarly, 3x² and 5x² are like terms because both have x².

例如，4x 和 -7x 是同类项，因为它们的变量部分都是 x。同样，3x² 和 5x² 是同类项，因为它们的变量部分都是 x²。

Terms like 2x and 2y are not like terms because the letters are different. Also, 2x and 2x² are not like terms because the powers of x are different.

像 2x 和 2y 这样的项不是同类项，因为字母不同。同时，2x 和 2x² 也不是同类项，因为 x 的幂不同。

Key idea: the variable part must match exactly.

关键：变量部分必须完全相同。

3. The Basic Rule for Collecting Like Terms | 合并同类项的基本法则

To collect like terms, add or subtract their coefficients. Keep the variable part unchanged.

合并同类项时，将它们的系数相加或相减，变量部分保持不变。

This rule comes from the distributive law: ax + bx = (a + b)x.

这个法则来源于分配律：ax + bx = (a + b)x。

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

For subtraction, the same idea applies: 9y – 4y = (9 – 4)y = 5y.

减法同理：9y – 4y = (9 – 4)y = 5y。

When a coefficient is negative, treat it carefully: -2m + 6m = (-2 + 6)m = 4m.

当系数为负数时要仔细处理：-2m + 6m = (-2 + 6)m = 4m。

4. Collecting Terms with More Than One Variable | 合并含多个变量的项

Expressions often contain different types of like terms. Group them together before adding.

表达式中常常包含不同类型的同类项。先将它们分组，再相加。

Example: Simplify 3a + 4b – a + 2b.

示例：化简 3a + 4b – a + 2b。

Group the a terms: 3a – a = 2a

将 a 项分组：3a – a = 2a
Group the b terms: 4b + 2b = 6b

将 b 项分组：4b + 2b = 6b
Combine the results: 2a + 6b

合并结果：2a + 6b

Notice that 2a and 6b cannot be combined further, because they are not like terms.

注意 2a 和 6b 不能再合并，因为它们不是同类项。

Another example: 5x² + 3x – 2x² + 7x = (5 – 2)x² + (3 + 7)x = 3x² + 10x.

另一个例子：5x² + 3x – 2x² + 7x = (5 – 2)x² + (3 + 7)x = 3x² + 10x。

5. Combining Constant Terms | 合并常数项

Constant terms are all like terms with each other. They can be added or subtracted just like ordinary numbers.

常数项之间彼此都是同类项。它们可以像普通数字一样进行加减。

Example: 2x + 5 + 3x – 4 = (2x + 3x) + (5 – 4) = 5x + 1.

示例：2x + 5 + 3x – 4 = (2x + 3x) + (5 – 4) = 5x + 1。

Be careful with negative constants: 7x – 3 + 8 – 2x = (7x – 2x) + (-3 + 8) = 5x + 5.

注意负常数：7x – 3 + 8 – 2x = (7x – 2x) + (-3 + 8) = 5x + 5。

If all terms are constants, the result is a single number: 3 + 9 – 5 = 7.

如果所有项都是常数项，结果就是单一数字：3 + 9 – 5 = 7。

6. Terms with Fractional Coefficients | 含分数系数的项

Fractional coefficients are handled exactly like integer coefficients. Just add or subtract the fractions.

分数系数的处理方法与整数系数完全相同，只需对分数进行加减。

Example: ½x + ⅓x. First write the fractions with a common denominator: ½ = 3⁄6 and ⅓ = 2⁄6.

示例：½x + ⅓x。先将分数化为公分母：½ = 3⁄6，⅓ = 2⁄6。

½x + ⅓x = (3⁄6 + 2⁄6)x = 5⁄6x

Another example: ¾y – ½y = (3⁄4 – 2⁄4)y = ¼y.

另一个例子：¾y – ½y = (3⁄4 – 2⁄4)y = ¼y。

If the expression is mixed, simplify each type of like term separately.

如果表达式是混合的，则分别化简每一类同类项。

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

When an expression contains brackets, expand the brackets first, then collect like terms.

当表达式含有括号时，先去掉括号，再合并同类项。

Example: Simplify 3(2x + 5) + 4x.

示例：化简 3(2x + 5) + 4x。

First expand: 3(2x + 5) = 6x + 15. Then add 4x:

先去括号：3(2x + 5) = 6x + 15。然后加上 4x：

6x + 15 + 4x = (6x + 4x) + 15 = 10x + 15

Subtraction with brackets needs extra care: simplify 5x – 2(x – 3).

带括号的减法需要特别小心：化简 5x – 2(x – 3)。

Expand: 5x – 2x + 6

去括号：5x – 2x + 6
Collect: (5x – 2x) + 6 = 3x + 6

合并：(5x – 2x) + 6 = 3x + 6

Remember: a negative sign outside the bracket changes the sign of every term inside.

记住：括号外的负号会改变括号内每一项的符号。

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

Mistakes happen when terms look similar but are not actually like terms.

当项看起来相似但实际上不是同类项时，就容易出错。

Wrong: x + x² = 2x³. Right: x + x² cannot be simplified further.

错误：x + x² = 2x³。正确：x + x² 不能再化简。

Wrong: 3a + 3b = 6ab. Right: 3a + 3b stays the same.

错误：3a + 3b = 6ab。正确：3a + 3b 保持不变。

Wrong: 2x – (x + 4) = 2x – x + 4. Right: 2x – x – 4 = x – 4.

错误：2x – (x + 4) = 2x – x + 4。正确：2x – x – 4 = x – 4。

Another common issue is forgetting the invisible coefficient 1. For example, x + 3x = 1x + 3x = 4x.

另一个常见问题是忘记系数 1。例如，x + 3x = 1x + 3x = 4x。

Always check that the variable part remains unchanged during combination.

始终检查变量部分在合并过程中是否保持不变。

9. Real-World Applications | 实际应用

Collecting like terms is used when calculating areas, perimeters, and total costs in everyday life.

合并同类项在计算面积、周长和日常总花费时非常有用。

For example, a rectangle has side lengths 2x and 3x + 1. Its perimeter is 2(2x) + 2(3x + 1).

例如，一个长方形的边长为 2x 和 3x + 1。它的周长为 2(2x) + 2(3x + 1)。

Simplify: 4x + 6x + 2 = 10x + 2.

化简：4x + 6x + 2 = 10x + 2。

If a shop sells apples at $a per apple and bananas at $b per banana, buying 3 apples and 5 apples plus 2 bananas and 4 bananas gives a total cost of 3a + 5a + 2b + 4b = 8a + 6b.

如果商店里苹果每个 $a 元，香蕉每根 $b 元，买 3 个苹果和 5 个苹果，再加 2 根香蕉和 4 根香蕉，总花费为 3a + 5a + 2b + 4b = 8a + 6b。

10. Practice Strategies | 练习策略

To avoid mistakes, use marks to identify like terms before combining them.

为避免出错，在合并前先用标记标出同类项。

Underline terms with the same variable part in different ways. For example, draw one line under all x terms, two lines under all y terms, and no line under constants.

用不同方式下划线标记变量部分相同的项。例如，在 x 项下画一条线，在 y 项下画两条线，常数项不画线。

Write the expression in a standard order: variables in alphabetical order, constants at the end. This helps you see which terms can combine.

将表达式按标准顺序排列：变量按字母顺序，常数放在最后。这样有助于看清哪些项可以合并。

Check your final answer by substituting a simple number, such as x = 1, into both the original and simplified expressions.

通过代入一个简单数字（如 x = 1）来检查答案，比较原表达式和化简后的表达式的值是否一致。

11. Summary | 总结

Collecting like terms is a core skill in algebra. It allows us to simplify expressions and prepare them for solving equations.

合并同类项是代数中的核心技能。它帮助我们化简表达式，并为解方程做好准备。

Remember the three steps: identify like terms, group them together, then add or subtract their coefficients.

记住三个步骤：识别同类项，将它们分组，然后对系数进行加减。

Never try to combine terms with different variable parts. Keep the variable part exactly the same.

绝不尝试合并变量部分不同的项。保持变量部分完全一致。

With regular practice, collecting like terms becomes automatic and error-free.

通过经常练习，合并同类项会变得得心应手，不再出错。

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Collecting Like Terms | 合并同类项