Substituting Numbers into Algebraic Expressions | 代入数值求代数式的值

📚 Substituting Numbers into Algebraic Expressions | 代入数值求代数式的值

Algebra is a powerful tool in mathematics that uses letters to stand for numbers. When we replace those letters with given values, we are performing substitution. This topic is a cornerstone of the Cambridge KS3 Mathematics curriculum, appearing in everything from solving equations to using formulas in science. Mastering substitution builds a strong foundation for higher-level algebra and real-world problem solving.

代数是数学中一种强大的工具，它用字母代表数字。当我们用给定的数值替换这些字母时，就是在进行代入。这个主题是剑桥 KS3 数学课程的基础，从解方程到科学中的公式运用都会用到。掌握代入法能为更高阶的代数以及现实问题求解打下扎实的基础。

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1. Understanding Algebraic Expressions | 理解代数式

An algebraic expression is a combination of numbers, letters (variables), and operation symbols such as +, −, ×, and ÷. For example, 3x + 5, a² − 2b, and 4(x + y) are all algebraic expressions. The letters represent unknown or changeable quantities. In an expression, multiplication is often written without the ‘×’ sign, so 3x means 3 × x.

代数式是由数字、字母（变量）以及 +、−、×、÷ 等运算符号组合而成的式子。例如 3x + 5、a² − 2b 和 4(x + y) 都是代数式。字母代表未知或可变的量。在代数式中，乘法常常省略乘号，因此 3x 表示 3 × x。

The value of an expression depends on the numbers we place into the letter positions. Until substitution happens, the expression is just a general rule waiting for specific input. Being able to read and interpret these expressions without numbers is the first step to fluent algebraic work.

代数式的值取决于我们放入字母位置的数字。在代入之前，代数式只是一个等待具体输入的通用规则。能够不带数字地阅读和解释这些式子，是熟练进行代数运算的第一步。

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2. What Is Substitution? | 什么是代入法？

Substitution means replacing each variable in an expression with a given number. Once all letters are replaced, we simplify the result using arithmetic. For instance, if we are told that x = 4 in the expression 2x + 3, we substitute x with 4 to get 2(4) + 3, and then calculate 8 + 3 = 11.

代入法就是将代数式中的每个变量用给定的数字替换。一旦所有字母都被替换，我们就用算术运算化简结果。例如，如果已知在表达式 2x + 3 中 x = 4，我们用 4 替换 x，得到 2(4) + 3，然后计算出 8 + 3 = 11。

Substitution is not just about plugging in numbers; it is a way of evaluating a formula or rule. Whenever you use a formula — for area, speed, or temperature conversion — you are performing substitution. Understanding this process helps you see why algebra is used to describe patterns and relationships in a concise way.

代入法不仅仅是套数字；它是一种求公式或规则值的方式。每当你使用公式——无论是面积、速度还是温度转换——你都在进行代入。理解这一过程有助于你明白为什么代数能简洁地描述模式和关系。

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3. The Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)

When evaluating an expression after substitution, we must follow the order of operations: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). This is often remembered as BIDMAS or PEMDAS. A common mistake is to work from left to right without respecting the hierarchy.

代入后计算代数式的值时，我们必须遵循运算顺序：先算括号，再算指数（幂），然后按从左到右的顺序计算除法和乘法，最后按从左到右的顺序计算加法和减法。这通常记作 BIDMAS 或 PEMDAS。一个常见错误是不顾优先级单纯从左往右计算。

For example, if we substitute x = 3 into 5x², the index applies to x first, so we square 3 to get 9, then multiply by 5: 5 × 9 = 45. If we mistakenly did 5 × 3 first and then squared, we would get 15² = 225, which is wrong. Always apply BIDMAS carefully.

例如，如果我们把 x = 3 代入 5x²，指数优先作用于 x，因此先平方 3 得到 9，再乘以 5：5 × 9 = 45。如果我们错误地先做 5 × 3 再平方，就会得到 15² = 225，这显然是错误的。务必仔细运用 BIDMAS。

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4. Substituting Positive Integers | 代入正整数

Positive integers are the simplest numbers to substitute. Consider the expression 4a + 7b − 3. If a = 5 and b = 2, we replace the letters directly: 4 × 5 + 7 × 2 − 3. Following BIDMAS, multiplication comes first: 20 + 14 − 3, then addition and subtraction from left to right: 34 − 3 = 31.

正整数是最简单的代入数字。考虑表达式 4a + 7b − 3。如果 a = 5 且 b = 2，我们直接替换字母：4 × 5 + 7 × 2 − 3。遵循 BIDMAS，乘法优先：20 + 14 − 3，然后从左到右算加减：34 − 3 = 31。

Always use brackets around the substituted number when there is a coefficient and a letter to avoid sign errors. Writing 4(5) + 7(2) − 3 makes the multiplication explicit and reduces mistakes. This is even more important when negatives appear later.

当数字与字母相乘时，建议把代入的数字用括号括起来，例如写成 4(5) + 7(2) − 3，这能让乘法更加清晰并减少错误。这在后面出现负数时尤其重要。

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5. Substituting Zero and Negative Numbers | 代入零和负数

Zero can cause surprising results if not handled correctly. Substituting x = 0 into 3x² + 5x − 7 gives 3(0)² + 5(0) − 7 = 0 + 0 − 7 = −7. Any term with a variable becomes zero, leaving only the constant term. Always write zero explicitly and multiply carefully.

如果处理不当，零会导致令人意外的结果。将 x = 0 代入 3x² + 5x − 7 得到 3(0)² + 5(0) − 7 = 0 + 0 − 7 = −7。任何包含变量的项都变成零，只剩下常数项。务必明确写出零并仔细计算乘法。

Negative numbers require extra care with signs. If p = −3, and we evaluate 2p² − p, we first substitute: 2(−3)² − (−3). The index applies to −3 first: (−3)² = 9. Then 2 × 9 = 18, and − (−3) becomes +3, so the total is 18 + 3 = 21. Using brackets around negative numbers prevents losing the negative sign.

负数需要格外注意符号。如果 p = −3，计算 2p² − p，我们首先代入：2(−3)² − (−3)。指数优先作用于 −3：(−3)² = 9。然后 2 × 9 = 18，而 − (−3) 变为 +3，所以总和是 18 + 3 = 21。用括号括起负数可以防止丢掉负号。

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6. Substituting Fractions | 代入分数

Fractions can be put into expressions just like integers. Suppose m = ½, and we want to evaluate 4m + 1. Substituting gives 4 × ½ + 1 = 2 + 1 = 3. Because 4 and the denominator 2 of ½ cancel neatly, the calculation is straightforward. When denominators do not cancel, keep the answer as a simplified fraction or mixed number.

分数可以像整数一样代入代数式。假设 m = ½，我们要求 4m + 1 的值。代入后得 4 × ½ + 1 = 2 + 1 = 3。由于 4 与 ½ 的分母 2 能约分，计算就很简单。当分母不能约分时，可将答案保留为最简分数或带分数。

When substituting fractions into expressions with indices, be careful to square both numerator and denominator. For x = ⅔, the term 3x² becomes 3 × (⅔)² = 3 × (⁴⁄₉) = ¹²⁄₉ = ⁴⁄₃ or 1⅓. Always apply the index to the whole fraction by using brackets.

将分数代入含有指数的表达式时，要小心同时平方分子和分母。对于 x = ⅔，3x² 变为 3 × (⅔)² = 3 × (⁴⁄₉) = ¹²⁄₉ = ⁴⁄₃ 或 1⅓。务必用括号将整个分数括起来再应用指数。

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7. Using Formulas in Context | 在实际公式中代入

Formulas are just algebraic expressions that describe real-world relationships. The area of a triangle A = ½ bh is a formula. If base b = 8 cm and height h = 5 cm, we substitute: A = ½ × 8 × 5. Multiply in any order: ½ × 8 = 4, then 4 × 5 = 20 cm². Always include the correct units in your final answer.

公式就是描述现实世界关系的代数式。三角形面积公式 A = ½ bh 就是一个例子。如果底 b = 8 cm，高 h = 5 cm，我们代入：A = ½ × 8 × 5。按任意顺序相乘：½ × 8 = 4，然后 4 × 5 = 20 cm²。最后答案一定要带上正确的单位。

Another common formula is converting Celsius to Fahrenheit: F = ⅗ C + 32. If C = 20, then F = ⅗ × 20 + 32 = 36 + 32 = 68°F. Notice how the fraction multiplies the variable first, then the constant is added. Substitution shows the link between mathematics and daily life.

另一个常见公式是摄氏度转华氏度：F = ⅗ C + 32。如果 C = 20，那么 F = ⅗ × 20 + 32 = 36 + 32 = 68°F。注意这里分数先乘变量，再加上常数。代入体现了数学与日常生活的联系。

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8. Common Mistakes to Avoid | 常见错误及避免方法

One of the most frequent errors is forgetting to use brackets around negative numbers. Writing 3 × −5² without brackets might be interpreted as 3 × (−5)² or 3 × −(5²). The correct interpretation when x = −5 in 3x² is 3(−5)² = 3 × 25 = 75. Always place the substituted value in parentheses: 3(−5)².

最常见的错误之一是忘记在负数周围加括号。把 3 × −5² 写成不带括号的形式可能被理解为 3 × (−5)² 或 3 × −(5²)。当 x = −5 在 3x² 中时，正确的计算是 3(−5)² = 3 × 25 = 75。务必把代入的值放在括号里：3(−5)²。

Another mistake is adding and subtracting before multiplying. In 2 + 3k for k = 4, some mistakenly add 2 + 3 first to get 5, then multiply by 4 to get 20. The correct sequence is 2 + 3(4) = 2 + 12 = 14. Always let BIDMAS guide you.

另一个错误是先加减后乘除。当 k = 4 时计算 2 + 3k，有人错误地先算 2 + 3 得 5，再乘以 4 得 20。正确的顺序是 2 + 3(4) = 2 + 12 = 14。始终让 BIDMAS 为你指路。

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9. Practice Example: Area of a Trapezium | 练习示例：梯形面积

Let’s apply substitution to the formula for the area of a trapezium: A = ½ (a + b)h, where a and b are the parallel sides and h is the height. Suppose a = 7 cm, b = 9 cm, and h = 4 cm. Substitute: A = ½ (7 + 9) × 4. Work inside the brackets first: 7 + 9 = 16. Then ½ × 16 × 4 = 8 × 4 = 32 cm².

让我们把代入法应用到梯形面积公式中：A = ½ (a + b)h，其中 a 和 b 是平行边，h 是高。假设 a = 7 cm，b = 9 cm，h = 4 cm。代入：A = ½ (7 + 9) × 4。先算括号内：7 + 9 = 16。然后 ½ × 16 × 4 = 8 × 4 = 32 cm²。

Notice how the formula uses brackets to ensure we add the two parallel sides before multiplying. This reinforces the importance of structured substitution and the order of operations. Always write down the substituted expression step by step to track your work.

请注意公式中用括号来确保先加再乘。这进一步说明了有结构地进行代入以及遵守运算顺序的重要性。逐步写下代入后的表达式以便跟踪计算过程。

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10. Checking Your Answers | 检查答案

After you evaluate an expression, it is wise to check your work with another method or by estimation. For example, if you substitute x = 2 into x² + 3x − 4 and get 6, mentally ask: 2² = 4, 3 × 2 = 6, so 4 + 6 − 4 = 6. An estimate based on rounded numbers can confirm your answer is reasonable.

求出代数式的值之后，最好用另一种方法或通过估算来检查。例如，把 x = 2 代入 x² + 3x − 4 得到 6，可在心中默算：2² = 4，3 × 2 = 6，因此 4 + 6 − 4 = 6。基于四舍五入数字的估算可以确认答案的合理性。

You can also substitute the result into the original relationship if it is a formula. For area questions, check that the number of significant figures matches the input data and that the units make sense. For pure expressions, revisiting the BIDMAS steps can quickly uncover a misplaced operation.

如果是公式，也可以将结果代回原关系验证。对于面积题，检查有效数字位数是否与输入数据相符，单位是否合理。对于纯代数式，回顾 BIDMAS 步骤能快速发现运算错误。

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11. Real-life Applications | 实际应用

Substitution is used widely in science, engineering, and finance. When a car travels at a speed v for time t, the distance s = v × t is found by substituting values. If v = 60 km/h and t = 2.5 h, then s = 60 × 2.5 = 150 km. Without substitution, such calculations would be expressed awkwardly in words.

代入法广泛应用于科学、工程和金融领域。汽车以速度 v 行驶时间 t，距离 s = v × t 就是通过代入值得出的。如果 v = 60 km/h，t = 2.5 h，那么 s = 60 × 2.5 = 150 km。没有代入法，这类计算就只能用笨拙的文字描述。

In money matters, simple interest I = P × r × t uses substitution. With principal P = £200, rate r = 0.05 per year, and time t = 3 years, I = 200 × 0.05 × 3 = £30. By mastering substitution, you gain the ability to use any formula you encounter confidently.

在财务中，单利 I = P × r × t 也要用到代入。本金 P = £200，年利率 r = 0.05，时间 t = 3 年，I = 200 × 0.05 × 3 = £30。掌握代入法后，你就能自信地使用任何遇到的公式。

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12. Summary and Key Points | 总结与要点

Substitution is the process of replacing variables with given numbers and then calculating the value of the resulting arithmetic expression. Always follow the order of operations (BIDMAS), use brackets around negative numbers and fractions, and double-check your work. Practice with a variety of expressions — including those with indices, fractions, and real-life formulas — to build fluency.

代入法是用给定数字替换变量然后计算出算术表达式的值的过程。始终遵循运算顺序（BIDMAS），在负数和分数周围加上括号，并仔细检查。通过练习各种表达式——包括含指数、分数和实际公式的式子——来达到熟练。

Whether you are working on page 86 of your Cambridge Mathematics book or tackling a science problem, substitution is the bridge between abstract algebra and concrete answers. Keep a calm, step-by-step approach, and you will find that even the most complex-looking formulas become manageable.

无论你是在做剑桥数学教材第 86 页的练习，还是在解决一个科学问题，代入法都是连接抽象代数与具体答案的桥梁。保持冷静、逐步推进，你会发觉即使是最复杂的公式也会变得易于掌控。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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