📚 Constructing and Solving Linear Equations | 构造并求解线性方程
Linear equations are the foundation of algebra. In KS3 Mathematics, you learn to represent problems with variables, construct equations, and solve them using systematic steps. This article guides you through the key concepts and techniques you need to master, with plenty of examples and practice advice.
线性方程是代数的基础。在初中数学中，你将学习用变量表示问题，构建方程，并使用系统步骤求解。本文将带你掌握关键概念与技巧，并提供丰富的示例和练习建议。
1. Introduction to Linear Equations | 线性方程简介
A linear equation is an equation where the highest power of the variable is 1, such as x + 5 = 12 or 3x − 2 = 7. These equations form a straight line when graphed, hence the name ‘linear’. In KS3, you focus on solving equations and applying them to real-world problems.
线性方程是指变量的最高次幂为 1 的方程，例如 x + 5 = 12 或 3x − 2 = 7。由于这类方程在图像上表示为一条直线，因此得名“线性”。在 KS3 阶段，你需要掌握如何求解这些方程并将其应用于实际问题。
Solving a linear equation means finding the value of the variable that makes the equation true. This involves reversing a series of operations while keeping the equation balanced – much like keeping a seesaw level.
解线性方程意味着找出使等式成立的变量值。这需要我们在保持等式平衡的同时逆向操作——就像保持跷跷板水平一样。
2. Understanding Variables and Expressions | 理解变量与表达式
A variable is a symbol, often a letter like x or y, that stands for an unknown number. An expression is a combination of numbers, variables and operations (+, −, ×, ÷) without an equals sign, e.g., 2x + 3. Before forming equations, you must be comfortable translating phrases into algebraic expressions.
变量是一个符号，通常用 x 或 y 等字母表示一个未知数。表达式是由数字、变量和运算符号（+、−、×、÷）组合而成，但没有等号，例如 2x + 3。在建立方程之前，你必须能够熟练地将语句转化为代数表达式。
Common translations include: ‘a number increased by 7’ → x + 7; ‘five less than a number’ → x − 5; ‘twice a number’ → 2x. Always be precise with the order of terms, especially with subtraction and multiplication.
常见的翻译包括：“一个数加 7” → x + 7；“一个数减去 5” → x − 5；“一个数的两倍” → 2x。要特别注意减法和乘法中各项的顺序，确保准确无误。
3. From Words to Equations | 从文字到方程
To form an equation, you write two expressions separated by an equals sign, showing that they are equal. For example, ‘I think of a number, multiply it by 4 and add 7. The result is 31.’ This translates to 4x + 7 = 31, where x represents the number thought of. Always define the variable clearly.
要构建一个方程，你需要写出两个表达式，并用等号将它们连接起来，表示二者相等。例如，“我想一个数，将它乘以 4 再加 7，结果是 31”。这可以转化为方程 4x + 7 = 31，其中 x 代表所想的数。请务必明确定义变量。
4x + 7 = 31
Let’s try another: ‘The sum of three consecutive integers is 36.’ If the smallest integer is n, then the next two are n + 1 and n + 2. The equation becomes n + (n + 1) + (n + 2) = 36, which simplifies to 3n + 3 = 36.
再试一例：“三个连续整数的和为 36。” 若设最小的整数为 n，则后两个分别为 n + 1 和 n + 2。可列出方程 n + (n + 1) + (n + 2) = 36，化简得 3n + 3 = 36。
4. Solving Using Inverse Operations | 使用逆运算求解
The goal when solving an equation is to isolate the variable on one side. You can ‘undo’ operations by using inverse operations: addition ↔ subtraction, multiplication ↔ division. Always perform the same operation on both sides of the equation to keep it balanced.
解方程的目标是将变量单独移到等式的一边。你可以使用逆运算来“撤销”运算：加法 ↔ 减法，乘法 ↔ 除法。必须在等式两边同时进行相同的运算，以保持平衡。
For 4x + 7 = 31: subtract 7 from both sides: 4x + 7 − 7 = 31 − 7 → 4x = 24. Then divide both sides by 4: 4x/4 = 24/4 → x = 6.
对于 4x + 7 = 31：两边同时减去 7：4x + 7 − 7 = 31 − 7 → 4x = 24。然后两边同时除以 4：4x/4 = 24/4 → x = 6。
4x + 7 = 31 → x = 6
Another example: 5y − 3 = 22. Add 3 to both sides: 5y = 25. Divide by 5: y = 5. The sequence is always opposite to the order of operations (undo addition/subtraction first, then multiplication/division).
另一个例子：5y − 3 = 22。两边加 3：5y = 25。再除以 5：y = 5。求解的顺序总是与运算优先级相反（先加减后乘除）。
5. The Balancing Method | 平衡法
The balancing method visualises an equation as a set of scales. What you do to one side, you must do to the other to keep the scales balanced. This method helps you remember to maintain equality at every step. You can subtract terms, add terms, divide or multiply both sides as needed.
平衡法将方程想象成一架天平。你对一边所做的操作，必须同样施加于另一边，以保持天平平衡。这种方法能帮助你记住在每一步都保持等式相等。你可以根据需要同时在两边减去项、加上项、除以或乘以某个数。
Consider 3x + 4 = 19. The step-by-step balance is shown in the table below.
考虑方程 3x + 4 = 19。其逐步平衡过程如下表所示。
| Operation | Equation |
|---|---|
| Start | 3x + 4 = 19 |
| Subtract 4 from both sides | 3x = 15 |
| Divide both sides by 3 | x = 5 |
The table shows each operation and how the equation evolves. You can check by substituting 5 back: 3(5) + 4 = 15 + 4 = 19, which matches the left side.
表格展示了每一步操作以及方程的演变过程。你可以将 5 代回进行检验：3(5) + 4 = 15 + 4 = 19，与等式左边一致。
6. Equations with Brackets | 含括号的方程
When an equation contains brackets, you need to expand them first or divide appropriately. For instance, 2(x + 3) = 10. You can either expand: 2x + 6 = 10, then solve, or divide both sides by 2 right away: x + 3 = 5, leading to x = 2. Both methods are valid; expand and simplify is often safer.
当方程含有括号时，你需要先展开括号，或适当进行除法。例如 2(x + 3) = 10。你可以先展开：2x + 6 = 10，然后求解；也可以立即在两边同时除以 2：x + 3 = 5，得到 x = 2。两种方法都可行，但先展开再化简通常更稳妥。
2(x + 3) = 10 → x = 2
For expressions with a minus sign before the bracket, be extra careful: 5 − 2(x − 1) = 3. First expand: 5 − 2x + 2 = 3 → 7 − 2x = 3. Then subtract 7: −2x = −4, so x = 2.
如果括号前有减号，需要格外小心：5 − 2(x − 1) = 3。先展开：5 − 2x + 2 = 3 → 7 − 2x = 3。然后减去 7：−2x = −4，得 x = 2。
7. Unknown on Both Sides | 未知数在等式两边
Sometimes the variable appears on both sides of the equation, e.g., 5x − 3 = 2x + 9. The strategy is to collect all variable terms on one side and constants on the other. Subtract 2x from both sides: 3x − 3 = 9, then add 3: 3x = 12, so x = 4. Always aim to get the smaller variable term to avoid negatives.
有时变量会出现在等式的两边，例如 5x − 3 = 2x + 9。策略是将所有含变量的项移到等式一边，常数项移到另一边。两边同时减去 2x：3x − 3 = 9，再加 3：3x = 12，得到 x = 4。通常将较小的变量项移走，以避免出现负数系数。
5x − 3 = 2x + 9 → x = 4
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