📚 Collecting Like Terms | 合并同类项
Collecting like terms is one of the most important algebra skills in IGCSE Mathematics. It allows you to simplify expressions by adding or subtracting terms that have exactly the same variable part. A clear understanding of this topic will help you with solving equations, expanding brackets, factorising, and graphing linear and quadratic functions.
合并同类项是 IGCSE 数学中最重要的代数技能之一。它允许你把字母部分完全相同的项相加或相减，从而简化表达式。清楚理解这个主题将帮助你解方程、展开括号、因式分解，以及绘制一次函数和二次函数图像。
1. What Are 'Like Terms'? | 什么是同类项
In algebra, an expression is made up of terms. A term can be a single number, a single letter, or a product of numbers and letters. For example, 4, x, 3x, −2y, and 5xy are all terms.
在代数中，表达式由项组成。项可以是一个数、一个字母，也可以是数与字母的乘积。例如，4、x、3x、−2y 和 5xy 都是项。
Like terms are terms that have exactly the same variable part, including the same power. The number in front of the term is called the coefficient, and the coefficient does not need to be the same for terms to be like terms.
同类项是指字母部分完全相同、包括指数也完全相同的项。项前面的数字叫做系数，两个项要成为同类项，系数不一定要相同。
For example, 3x and −5x are like terms because both contain x raised to the power 1. However, 3x and 3x² are not like terms because the exponents of x are different.
例如，3x 和 −5x 是同类项，因为它们都含有 x 的一次方。但是，3x 和 3x² 不是同类项，因为 x 的指数不同。
2. Why Do We Combine Like Terms? | 为什么要合并同类项
Combining like terms makes an algebraic expression shorter and easier to use. Instead of writing 4x + 3x as two separate terms, we can write 7x. This process is called simplifying the expression.
合并同类项可以使代数表达式更短、更易使用。我们可以把 4x + 3x 写成 7x，而不用写成两个单独的项。这个过程叫做化简表达式。
In IGCSE exam questions, you will often be asked to simplify expressions before solving an equation, substituting values, or finding a gradient. A simplified expression reduces the chance of arithmetic mistakes later.
在 IGCSE 考试题中，你经常需要在解方程、代入数值或求斜率之前先化简表达式。化简后的表达式可以减少后续计算出错的机会。
For example, simplifying 2x + 3 + 5x − 1 to 7x + 2 makes it much easier to see how the expression changes when x is given a value.
例如，把 2x + 3 + 5x − 1 化简为 7x + 2 后，当给 x 赋值时，就更容易看出表达式的变化。
3. The Golden Rule: Same Variable and Same Power | 黄金法则：同字母、同指数
You can only combine terms if they have exactly the same variable part and exactly the same exponent. This is the golden rule of collecting like terms. If the letters or the powers are different, the terms cannot be added or subtracted into a single term.
只有当两个项的字母部分和指数完全相同时，你才能把它们合并。这是合并同类项的黄金法则。如果字母或指数不同，就不能把它们加减成一个项。
The following table shows examples of like and unlike terms.
下表列出了一些同类项和不同类项的例子。
| Term 1 | Term 2 | Are they like terms? |
|---|---|---|
| 3x | −8x | Yes, both are x¹ |
| 4x² | 9x² | Yes, both are x² |
| 2xy | −5xy | Yes, both are xy |
| 6a²b | 3a²b | Yes, both are a²b |
| 7x | 7x² | No, powers are different |
| 5ab | 5a | No, different variables |
Remember that x means x¹, so x and x¹ are like terms. However, x and x² are not like terms because their powers are different.
记住，x 表示 x¹，所以 x 和 x¹ 是同类项。但是，x 和 x² 不是同类项，因为它们的指数不同。
4. Combining Positive and Negative Coefficients | 合并正负系数
To combine like terms, add or subtract the coefficients and keep the variable part exactly the same. The sign in front of each term belongs to that term, so you must carry it carefully.
合并同类项时，只把系数相加或相减，字母部分保持不变。每一项前面的符号属于该项，因此必须小心保留符号。
7a − 2a = 5a
Here the coefficient of the first term is 7 and the coefficient of the second term is −2. Adding 7 and −2 gives 5, so the result is 5a.
这里第一项的系数是 7，第二项的系数是 −2。7 加 −2 等于 5，所以结果是 5a。
−4y + 9y = 5y
The coefficients are −4 and 9. Their sum is 5, so the simplified term is 5y.
系数是 −4 和 9。它们的和是 5，所以化简后的项是 5y。
−3m − 5m = −8m
Here both coefficients are negative: −3 and −5. Adding them gives −8, so the answer is −8m.
这里两个系数都是负数：−3 和 −5。它们相加得到 −8，所以答案是 −8m。
5. Constants and Single Variables | 常数项与单个字母
Constants are terms without variables, such as 4, −7, and 12. Constants are like terms with each other, even though they have no letter part. You can combine all constants into one number.
常数项是没有字母的项，例如 4、−7 和 12。常数项彼此是同类项，尽管它们没有字母部分。你可以把所有常数项合并成一个数。
Single variables such as x, y, or p can only be combined with other terms that have exactly the same letter. For example, 5x and −3x can be combined, but 5x and −3 cannot.
单个字母如 x、y 或 p 只能与字母完全相同的其他项合并。例如，5x 和 −3x 可以合并，但 5x 和 −3 不能合并。
Consider the expression 5 + 3x − 2 + 7x. Combine the constants 5 and −2 to get 3. Combine the x terms 3x and 7x to get 10x.
考虑表达式 5 + 3x − 2 + 7x。把常数项 5 和 −2 合并得到 3。把 x 项 3x 和 7x 合并得到 10x。
5 + 3x − 2 + 7x = 3 + 10x
The final answer is usually written with the variable term first and the constant second, so 10x + 3 is preferred, but 3 + 10x is also correct.
最终答案通常先写字母项，再写常数项，所以写成 10x + 3 更常见，但 3 + 10x 也是正确的。
6. Worked Example 1: Linear Expression | 例题1：一次表达式
Simplify 5x + 3y − 2x + 8y.
化简 5x + 3y − 2x + 8y。
Step 1: Identify the like terms. The x terms are 5x and −2x. The y terms are 3y and 8y.
第一步：找出同类项。x 项是 5x 和 −2x，y 项是 3y 和 8y。
Step 2: Combine the x terms: 5x − 2x = 3x.
第二步：合并 x 项：5x − 2x = 3x。
Step 3: Combine the y terms: 3y + 8y = 11y.
第三步：合并 y 项：3y + 8y = 11y。
5x + 3y − 2x + 8y = 3x + 11y
Because 3x and 11y have different letters, they cannot be combined any further. The expression is now fully simplified.
因为 3x 和 11y 字母不同，所以它们不能继续合并。这个表达式现在已经完全化简。
7. Worked Example 2: Mixed Powers | 例题2：混合指数
Simplify 4x² + 3x − 2x² + 5x + 7.
化简 4x² + 3x − 2x² + 5x + 7。
In this expression there are three groups: x² terms, x terms, and constants. You must not combine x² with x, because their powers are different.
这个表达式中有三组项：x² 项、x 项和常数项。你不能把 x² 和 x 合并，因为它们的指数不同。
Combine the x² terms: 4x² − 2x² = 2x².
合并 x² 项：4x² − 2x² = 2x²。
Combine the x terms: 3x + 5x = 8x.
合并 x 项：3x + 5x = 8x。
The constant term is 7, and there are no other constants to combine with it.
常数项是 7，没有其他常数项可以与之合并。
4x² + 3x − 2x² + 5x + 7 = 2x² + 8x + 7
The simplified expression has three different terms: 2x², 8x, and 7. These are not like terms, so no further simplification is possible.
化简后的表达式有三个不同的项：2x²、8x 和 7。它们不是同类项，所以不能再继续化简。
8. Worked Example 3: Expressions with Brackets | 例题3：含括号表达式
When an expression contains brackets, expand the brackets first, then collect like terms. Do not try to combine terms inside and outside the brackets before expanding.
当表达式中含有括号时，要先把括号展开，然后再合并同类项。不要在展开之前就试图合并括号内外的项。
Simplify 3(2x + 4) + 5(x − 1).
化简 3(2x + 4) + 5(x − 1)。
First expand the first bracket: 3(2x + 4) = 6x + 12.
先展开第一个括号：3(2x + 4) = 6x + 12。
Then expand the second bracket: 5(x − 1) = 5x − 5.
再展开第二个括号：5(x − 1) = 5x − 5。
+ 12 + 5x − 5. Now combine the x terms: 6x + 5x = 11x. Combine the constants: 12 − 5 = 7.
表达式变为 6x + 12 + 5x − 5。现在合并 x 项：6x + 5x = 11x。合并常数项：12 − 5 = 7。
3(2x + 4) + 5(x − 1) = 11x + 7
Here is another example with a negative sign in front of a bracket: Simplify 2(3a − b) − (a − 4b).
再看一个括号前有负号的例子：化简 2(3a − b) − (a − 4b)。
b) = −a + 4b, because the negative sign changes the sign of every term inside the bracket.
展开第一个括号：2(3a − b) = 6a − 2b。小心展开第二个括号：−(a − 4b) = −a + 4b，因为负号会改变括号内每一项的符号。
Now collect like terms: 6a − a = 5a, and −2b + 4b = 2b.
现在合并同类项：6a − a = 5a，−2b + 4b = 2b。
2(3a − b) − (a − 4b) = 5a + 2b
9. Common Mistakes | 常见错误
One common mistake is trying to combine terms that are not like terms. For example, x + x² is sometimes incorrectly written as 2x², but this is wrong because x and x² have different powers.
一个常见错误是试图合并不属于同类的项。例如，x + x² 有时被错误地写成 2x²，但这是错误的，因为 x 和 x² 的指数不同。
Another mistake is losing a negative sign. When simplifying 4a − 3 + 2a, the correct result is 6a − 3, not 6a + 3. The sign before the 3 must stay with the constant.
另一个错误是丢失负号。化简 4a − 3 + 2a 时，正确结果是 6a − 3，而不是 6a + 3。3 前面的符号必须和常数项一起保留。
Some students also change the power when adding coefficients. For example, x² + x² is 2x², not x⁴. The power never changes when you add or subtract like terms.
一些学生在系数相加时还会改变指数。例如，x² + x² 等于 2x²，而不是 x⁴。加减同类项时，指数永远不变。
wrong expansion such as 3(x + 2) = 3x + 2 will lead to an incorrect simplified expression.
最后，展开括号时，记住括号外的因数要乘以括号内的每一项。像 3(x + 2) = 3x + 2 这样的错误展开，会导致化简结果出错。
10. Exam-Style Practice | 考试风格练习
Try these IGCSE-style questions. Simplify each expression fully.
尝试以下 IGCSE 风格练习题。请把
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