📚 Collecting Like Terms | 合并同类项
In IGCSE Maths, simplifying algebraic expressions often begins with one key skill: collecting like terms. This skill allows you to rewrite expressions in a shorter and clearer form without changing their value.
在 IGCSE 数学中，化简代数式通常从一项核心技能开始：合并同类项。掌握这一技能后，你可以在不改变式子数值的前提下，把表达式改写得更短、更清晰。
Collecting like terms is essential for solving equations, working with formulas and simplifying expressions in topics such as linear graphs, brackets and algebraic fractions.
合并同类项是解方程、处理公式以及化简线性图像、去括号和代数分式等主题的基础技能。
1. What Are Like Terms? | 什么是同类项
In algebra, like terms are terms that contain exactly the same variable or variables raised to exactly the same powers. The numerical coefficients can be different.
在代数中，同类项是指含有完全相同的变量，并且这些变量的指数也完全相同的项。它们的数字系数可以不同。
For example, 3x, -7x and 12x are like terms because they all contain x to the power of 1.
例如，3x、-7x 和 12x 是同类项，因为它们都含有 x 的一次方。
On the other hand, 4x² and 4x are not like terms because the powers of x are different.
另一方面，4x² 和 4x 不是同类项，因为 x 的指数不同。
2. Why Do We Collect Like Terms? | 为什么要合并同类项
Algebraic expressions often contain several terms that can be grouped together. Collecting like terms makes the expression simpler and easier to use in later calculations.
代数式常常包含多个可以归为一组的项。合并同类项可以让表达式更简单，也便于后续计算。
For example, the expression 2x + 5y + 3x - 2y can be simplified to 5x + 3y. This shorter form is easier to substitute into formulas and solve.
例如，2x + 5y + 3x - 2y 可以化简为 5x + 3y。这种更短的形式在代入公式和求解时更加方便。
Examiners often expect you to simplify fully, so collecting like terms is a necessary step for gaining method marks.
考试中通常要求完全化简，因此合并同类项是获得过程分的重要一步。
3. Identifying Like Terms | 识别同类项
To identify like terms, look at the variable part of each term and ignore the coefficient at first. Terms must have the same combination of variables and powers.
要识别同类项，先看每一项的变量部分，暂时忽略系数。项之间必须具有相同的变量组合和相同的指数。
Examples of like terms include 6ab, -2ab and 9ab. They all contain a and b, each raised to the power of 1.
同类项的例子包括 6ab、-2ab 和 9ab。它们都含有 a 和 b，且两者的指数都是 1。
However, 5a²b and 3ab² are not like terms because the powers of a and b are swapped.
但是，5a²b 和 3ab² 不是同类项，因为 a 和 b 的指数位置不同。
4. Adding and Subtracting Like Terms | 合并同类项：加法与减法
The rule for combining like terms is simple: add or subtract the coefficients and keep the variable part unchanged.
合并同类项的规则很简单：把系数相加或相减，变量部分保持不变。
5x + 3x = (5 + 3)x = 8x
5x + 3x = (5 + 3)x = 8x
For subtraction, the same principle applies:
对于减法，原理相同：
9y - 4y = (9 - 4)y = 5y
9y - 4y = (9 - 4)y = 5y
When terms have different variables or different powers, they cannot be combined in this way.
当项具有不同的变量或不同的指数时，就不能这样合并。
5. Combining Negative Coefficients | 含负系数的合并
Negative terms must be handled carefully. A common method is to rewrite the expression with addition only, using positive and negative coefficients.
含有负号的项需要小心处理。一种常用方法是把表达式改写成只含加法的形式，用正负系数表示。
7m - 3m can be written as 7m + (-3m). Then add the coefficients: 7 + (-3) = 4, giving 4m.
例如，7m - 3m 可以写成 7m + (-3m)。然后将系数相加：7 + (-3) = 4，得到 4m。
When an expression mixes addition and subtraction, always focus on the sign in front of each term.
当表达式中同时出现加法和减法时，要始终关注每一项前面的符号。
Example: 6k - 8 + 2k + 3 = 6k + 2k - 8 + 3 = 8k - 5.
例如：6k - 8 + 2k + 3 = 6k + 2k - 8 + 3 = 8k - 5。
6. Simplifying Expressions with Several Variables | 含多个变量的化简
When an expression contains more than one variable, group the like terms by their variable combination. It often helps to underline or circle each group before combining.
当表达式含有多个变量时，按变量组合将同类项分组。在合并之前，先给每组画线或画圈通常会很有帮助。
Group the a terms: 4a - 2a = 2a. Group the b terms: 3b + 7b = 10b. The answer is 2a + 10b.
化简 4a + 3b - 2a + 7b。将含 a 的项分为一组：4a - 2a = 2a。将含 b 的项分为一组：3b + 7b = 10b。答案是 2a + 10b。
This grouping method reduces mistakes and helps you see the structure of the expression clearly.
这种分组方法可以减少错误，并帮助你更清楚地看到表达式的结构。
7. Combining Like Terms in Real-Life Formulas | 在实际公式中合并同类项
Collecting like terms is not just an abstract exercise. It is used in perimeter, cost and travel problems where algebraic quantities are added together.
合并同类项并不只是抽象练习。它在周长、成本和行程等问题中都会用到，这些问题需要把代数量相加。
A rectangle has length 3x + 2 and width x + 4. Its perimeter is 2(3x + 2) + 2(x + 4). After expanding and collecting like terms, the perimeter is 8x + 12.
一个长方形的长为 3x + 2，宽为 x + 4。它的周长为 2(3x + 2) + 2(x + 4)。展开并合并同类项后，周长为 8x + 12。
In word problems, you may need to write several expressions, expand brackets first, and then collect like terms to reach the final simplified answer.
在应用题中，你可能需要先写出几个表达式，先去括号，再合并同类项，才能得到最终的化简结果。
8. Common Mistakes to Avoid | 常见错误提醒
Many students try to combine terms that are not truly like terms. For example, 2x + 3y cannot be simplified further unless additional information is given.
许多学生试图合并那些并不是真正同类项的项。例如，2x + 3y 在没有额外信息时无法进一步化简。
ember that x + x = 2x, not x². The power only changes when you multiply, not when you add.
另一个常见错误是合并时改变了指数。请记住，x + x = 2x，而不是 x²。指数只有在乘法时才会改变，加法时不会。
A third error is forgetting the sign of the coefficient. Always bring the sign in front of each term along with the number.
第三个错误是忽略系数的符号。要始终把每一项前面的符号和数字一起处理。
9. Worked Example | 例题详解
Let us simplify the expression 5p² + 3p - 2p² + 6p - 4 + 9.
我们来化简表达式 5p² + 3p - 2p² + 6p - 4 + 9。
Step 1: Group the p² terms: 5p² - 2p² = 3p².
第一步：合并 p² 项：5p² - 2p² = 3p²。
Step 2: Group the p terms: 3p + 6p = 9p.
第二步：合并 p 项：3p + 6p = 9p。
Step 3: Group the constant terms: -4 + 9 = 5.
第三步：合并常数项：-4 + 9 = 5。
The simplified expression is 3p² + 9p + 5.
化简后的表达式为 3p² + 9p + 5。
10. Practice Questions | 练习题
Try simplifying the following expressions by collecting like terms.
请通过合并同类项化简下列表达式。
- 8x + 3x - 2x
- 6a + 4b - 2a + 5b
- 7m² + 3m + 2m² - 5m
- 10 - 3k + 2k - 6
- 4xy + 3x - 2xy + x
Check your answers by substituting a simple value for each variable and comparing the original and simplified expressions.
你可以给每个变量代入一个简单数值，比较原式和化简后的结果，以检查答案是否正确。
- 8x + 3x - 2x = 9x
- 6a + 4b - 2a + 5b = 4a + 9b
- 7m² + 3m + 2m² - 5m = 9m² - 2m
- 10 - 3k + 2k - 6 = 4 - k
- 4xy + 3x - 2xy + x = 2xy + 4x
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