Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

Collecting like terms is one of the most fundamental skills in algebra. It means simplifying an expression by adding or subtracting terms that have the same variable and the same power. This skill is essential for solving equations, factorising, and working with formulas.

合并同类项是代数中最基础的技能之一。它通过将具有相同变量和相同幂次的项相加或相减来化简表达式。这项技能对于解方程、因式分解以及处理公式都至关重要。

1. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same variable part. The coefficient (the number in front) can 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。前面的系数（数字）可以不同，但字母及其指数必须完全一致。

3x and 5x are like terms because both contain x¹.

3x 和 5x 是同类项，因为两者都含有 x¹。
4y² and −2y² are like terms because both contain y².

4y² 和 −2y² 是同类项，因为两者都含有 y²。
7 and 3 are like terms (constant terms).

7 和 3 是同类项（常数项）。
2x and 2x² are NOT like terms because the powers of x differ.

2x 和 2x² 不是同类项，因为 x 的幂次不同。

3x + 5x = 8x 4y² − 2y² = 2y²

2. Why Collect Like Terms? | 为什么要合并同类项？

Collecting like terms makes expressions shorter, easier to read, and easier to substitute values into. It also prepares the expression for further operations such as factorisation or solving equations.

合并同类项能使表达式更简洁、更易读，也更容易代入数值。它还为后续操作（如因式分解或解方程）做好准备。

For example, 2a + 3a + 5a looks messy, but after collecting we get 10a, which immediately tells us the total coefficient.

例如，2a + 3a + 5a 看起来杂乱，合并后得到 10a，立刻能看出总系数。

In exams, you are often asked to “simplify” an expression. That is exactly the instruction to collect like terms.

在考试中，你常被要求“化简”一个表达式。这正是合并同类项的指令。

3. The Rule of Signs | 符号法则

Before collecting like terms, you must look at the sign in front of each term. The sign belongs to the term that follows it.

在合并同类项之前，必须先看每个项前面的符号。该符号属于它后面的那个项。

3x − 2x + 4x = (3 − 2 + 4)x = 5x

Here, −2x means we subtract 2x from 3x, then add 4x.

这里，−2x 表示从 3x 中减去 2x，然后再加上 4x。

When a term has no sign, it is positive. For example, x means +1x.

当一个项没有符号时，它就是正数。例如，x 表示 +1x。

4. Combining Positive Terms | 合并正项

When all terms have the same variable and all signs are positive, simply add the coefficients.

当所有项变量相同且符号均为正时，直接把系数相加即可。

2x + 3x + 7x = (2 + 3 + 7)x = 12x

Also, combine constant positive terms: 4 + 6 + 2 = 12.

同时合并正的常数项：4 + 6 + 2 = 12。

Example: Simplify 5m + 8m + 3.

例：化简 5m + 8m + 3。

Combine m terms: 5m + 8m = 13m.

合并 m 项：5m + 8m = 13m。
The constant 3 has no other constant term, so the answer is 13m + 3.

常数项 3 没有其他常数项可合并，所以答案是 13m + 3。

5. Combining Negative Terms | 合并负项

Negative coefficients work exactly the same way. Add the absolute values and keep the negative sign if the sum is negative.

负系数的合并方式完全一样。将绝对值相加，如果和为负则保留负号。

−3a − 5a = −8a

When positive and negative terms are mixed, treat it as addition of signed numbers.

当正项和负项混合时，把它看作有符号数的加法。

7b − 4b = 3b −2c + 5c = 3c

If the coefficients cancel completely, the term disappears.

如果系数完全抵消，该项就消失了。

4d − 4d = 0

6. Mixed Terms and Multiple Variables | 混合项与多个变量

An expression may contain several different variables. Collect like terms for each variable separately.

一个表达式可能包含多个不同的变量。对每个变量分别合并同类项。

3x + 2y − x + 5y = 2x + 7y

Notice that x terms are collected together, and y terms are collected together. The order of the final answer is usually alphabetical, but not strictly required.

注意 x 项合并在一起，y 项合并在一起。最终答案通常按字母顺序排列，但不是严格要求的。

Example: Simplify 4p − 3q + 6p + q.

例：化简 4p − 3q + 6p + q。

p terms: 4p + 6p = 10p.

p 项：4p + 6p = 10p。
q terms: −3q + 1q = −2q.

q 项：−3q + 1q = −2q。
Answer: 10p − 2q.

答案：10p − 2q。

7. Dealing with Brackets | 处理括号

Sometimes you must expand brackets before collecting like terms. Use the distributive law: a(b + c) = ab + ac.

有时需要先展开括号，再合并同类项。使用乘法分配律：a(b + c) = ab + ac。

3(x + 2) = 3x + 6

Then combine any like terms that appear after expanding.

然后在展开后合并出现的同类项。

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

例：化简 2(x + 3) + 4x。

2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6

Be careful with negative signs outside brackets: −2(x − 1) = −2x + 2.

注意括号外的负号：−2(x − 1) = −2x + 2。

8. Squares and Higher Powers | 平方与更高次幂

x² and x are not like terms. You can only combine terms with the same power of the same variable.

x² 和 x 不是同类项。你只能合并相同变量且幂次相同的项。

Example: Simplify x² + 3x + 2x² − x.

例：化简 x² + 3x + 2x² − x。

x² terms: 1x² + 2x² = 3x².

x² 项：1x² + 2x² = 3x²。
x terms: 3x − 1x = 2x.

x 项：3x − 1x = 2x。
Answer: 3x² + 2x.

答案：3x² + 2x。

Also, xy and x² are different terms; xy and yx are the same because multiplication is commutative.

此外，xy 和 x² 是不同的项；xy 和 yx 相同，因为乘法满足交换律。

9. Real Exam-Style Questions | 真题风格练习

The following questions are typical of IGCSE-style algebra simplifying questions.

以下是典型的 IGCSE 风格代数化简题。

Question / 题目	Answer / 答案
Simplify 5a + 2b − 3a + b	2a + 3b
Simplify 4x + 3 − 2x + 7	2x + 10
Simplify 3(x + 2) + 2(x − 1)	5x + 4
Simplify y² + 2y + 3y² − y	4y² + y

Take your time with each step. Write down the terms you are combining before computing the final coefficient.

每一步都要仔细。在计算最终系数之前，先把要合并的项写出来。

10. Common Mistakes to Avoid | 常见错误避免

Many students make the same small errors when collecting like terms. Knowing them helps you avoid them.

许多学生在合并同类项时会犯同样的小错误。了解这些错误有助于避免它们。

Mistake: Combining 3x and 3x² as 6x. Correction: These are not like terms; leave them separate.

错误：将 3x 和 3x² 合并成 6x。 纠正：它们不是同类项；应保持分开。
Mistake: Forgetting the sign before a term, e.g. writing 5 − 2x + 3x as 5 + 5x. Correction: −2x + 3x = x, so the answer is 5 + x.

错误：忘记项前的符号，例如把 5 − 2x + 3x 写成 5 + 5x。 纠正：−2x + 3x = x，所以答案是 5 + x。
Mistake: Treating constants as variable terms. Correction: Only combine numbers with numbers.

错误：将常数项当作变量项。 纠正：只能数字与数字合并。
Mistake: Mixing up x and y terms. Correction: Always group by variable first.

错误：混淆 x 项和 y 项。 纠正：始终先按变量分组。

11. Practice Summary | 练习总结

Complete the following quick-fire questions to test yourself. Look at the expression, decide which terms are like, and simplify.

完成以下快速小测来测试自己。观察表达式，判断哪些项是同类项，然后化简。

1) 7x + 2x 2) 5a − 3a + 4a 3) 2p + 3q − p + q

4) 4(x + 2) + 3x 5) m² + 2m + 3m² − m

Answers: 9x; 6a; p + 4q; 7x + 8; 4m² + m.

答案：9x；6a；p + 4q；7x + 8；4m² + m。

If you got all five correct, you have mastered collecting like terms. If not, review the sections above and try again.

如果你五题全部答对，说明你已经掌握了合并同类项。如果还有错，请回顾上面的章节再试一次。

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Collecting Like Terms | 合并同类项