Solving Linear Equations | 解一次方程

📚 Solving Linear Equations | 解一次方程

Linear equations are the foundation of algebra. In KS3 Mathematics, learning to solve equations like 2x + 3 = 11 is a critical skill. It helps you understand how to find unknown values by keeping both sides of an equation balanced. This topic appears in many real-life situations, from working out prices to predicting patterns. Mastering solving techniques now will make more advanced algebra, such as simultaneous equations and graphs, much easier later on. Let’s explore step-by-step methods, common mistakes, and plenty of examples to build your confidence.

一次方程是代数的基础。在 KS3 数学中，学会解像 2x + 3 = 11 这样的方程是一项关键技能。它帮助你理解如何通过保持等式两边平衡来求出未知数的值。这个知识点在许多现实生活场景中都会出现，从计算价格到预测规律。现在掌握解题技巧，会让以后学习更高级的代数（比如联立方程和图像）变得容易得多。让我们一起探讨分步解法、常见错误，并通过大量例题来增强你的信心。

1. What Is a Linear Equation? | 什么是一次方程？

A linear equation is an equation where the unknown variable appears only to the power of 1. There are no squares, cubes, or roots directly on the variable. For example, 2x + 3 = 11 is linear because x is raised to the power 1. Equations like x² + 2 = 6 or 1/x = 4 are not linear. The graph of a linear equation is always a straight line. In KS3, we usually solve equations with one variable, often x, and balance operations on both sides.

一次方程是指未知数的最高次数为 1 的方程。变量上不会直接出现平方、立方或根号。例如，2x + 3 = 11 就是一次方程，因为 x 的指数是 1。像 x² + 2 = 6 或 1/x = 4 这样的方程就不是一次方程。一次方程的图像总是一条直线。在 KS3 阶段，我们通常解含有一个变量（经常用 x 表示）的方程，并在等式两边进行平衡运算。

2. Understanding the Balancing Method | 理解平衡法

Think of an equation as a balanced seesaw. Whatever you do to one side, you must do exactly the same to the other side to keep it balanced. If you add 5 to the left, add 5 to the right. If you divide the whole left side by 2, divide the whole right side by 2. This idea is the core of solving any linear equation. It is sometimes called the ‘balance method’ or ‘inverse operations method’.

把方程想象成一个平衡的跷跷板。你对一边做的任何操作，都必须对另一边做完全相同的操作，以保持平衡。如果你在左边加 5，就在右边加 5。如果你把整个左边除以 2，就把整个右边除以 2。这个思想是解任何一次方程的核心。它有时被称作“平衡法”或“逆运算法”。

3. Key Inverse Operations | 关键逆运算

To isolate the variable, you undo what has been done to it. Adding and subtracting are inverse operations. Multiplication and division are also inverse operations. If the equation has x + 7, subtract 7 from both sides. If it has 3x, divide both sides by 3. If x is divided by 5, multiply both sides by 5. Always perform the inverse operation to both sides.

为了把变量单独分离出来，你需要撤销对它所做的运算。加法和减法是互逆运算。乘法和除法也是互逆运算。如果方程中有 x + 7，就在两边都减去 7。如果有 3x，就把两边都除以 3。如果 x 被除以 5，就在两边都乘 5。一定要对两边执行相同的逆运算。

4. Solving Equations in One Step | 解一步方程

Some equations only need one inverse operation. For example, x – 4 = 10. Since 4 is subtracted from x, the inverse is to add 4. Add 4 to both sides: x – 4 + 4 = 10 + 4, so x = 14. Another example is 6x = 42. Here x is multiplied by 6. Divide both sides by 6: 6x ÷ 6 = 42 ÷ 6, giving x = 7. Always check your answer by substituting it back. 14 – 4 = 10, and 6 × 7 = 42, both correct.

有些方程只需要一步逆运算。例如，x – 4 = 10。因为 x 被减去了 4，逆运算是加 4。在两边都加 4：x – 4 + 4 = 10 + 4，所以 x = 14。另一个例子是 6x = 42。这里 x 被乘了 6。两边都除以 6：6x ÷ 6 = 42 ÷ 6，得到 x = 7。每次都要用代入法检验答案。14 – 4 = 10，6 × 7 = 42，都是正确的。

5. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. For instance, 2x + 3 = 11. The order of undoing matters: start with the addition/subtraction before the multiplication/division. Here x is first multiplied by 2, then 3 is added. To solve, undo the addition first: subtract 3 from both sides: 2x + 3 – 3 = 11 – 3, so 2x = 8. Then undo the multiplication: divide by 2: 2x ÷ 2 = 8 ÷ 2, giving x = 4. Always follow this order—reverse of BIDMAS/PEMDAS.

两步方程包含两种运算。例如 2x + 3 = 11。撤销运算的顺序很重要：要先处理加/减法，再处理乘/除法。这里 x 首先被乘以 2，然后加上 3。解方程时，先撤销加法：两边减 3：2x + 3 – 3 = 11 – 3，得到 2x = 8。然后撤销乘法：除以 2：2x ÷ 2 = 8 ÷ 2，得到 x = 4。要一直遵循这个顺序——与 BIDMAS/PEMDAS 的运算顺序相反。

6. Equations with Brackets | 含有括号的方程

If the equation contains brackets, such as 3(x + 2) = 18, you have two strategies. You can expand the brackets first: 3 × x + 3 × 2 = 3x + 6, so 3x + 6 = 18. Then solve as a two-step equation: subtract 6, then divide by 3, x = 4. Alternatively, you can treat the bracket as a package and divide both sides by 3 first: (x + 2) = 6, then subtract 2 to get x = 4. Both methods work; choose the one you find simpler.

如果方程含有括号，例如 3(x + 2) = 18，你有两种解题策略。可以先把括号展开：3 × x + 3 × 2 = 3x + 6，于是 3x + 6 = 18。然后按两步方程来解：减 6，再除以 3，得到 x = 4。另一种方法是，把括号看作一个整体，先把两边都除以 3：(x + 2) = 6，然后再减 2 得到 x = 4。两种方法都可行，选择你觉得更简单的那种。

7. Equations with the Variable on Both Sides | 变量在等式两边的方程

Sometimes the unknown appears on both sides, like 5x – 2 = 2x + 7. The goal is to collect all the variable terms on one side and the constants on the other. First, eliminate the smaller x term. Subtract 2x from both sides: 5x – 2 – 2x = 2x + 7 – 2x, resulting in 3x – 2 = 7. Then add 2 to both sides: 3x = 9, x = 3. Always keep the balance. Check: left side 5(3) – 2 = 13, right side 2(3) + 7 = 13; they match.

有时未知数会同时出现在两边，比如 5x – 2 = 2x + 7。目标是把所有含变量的项移到一边，常数项移到另一边。首先，消去较小的 x 项。两边都减 2x：5x – 2 – 2x = 2x + 7 – 2x，得到 3x – 2 = 7。然后两边加 2：3x = 9，x = 3。始终要保持平衡。检验：左边 5(3) – 2 = 13，右边 2(3) + 7 = 13；它们相等。

8. Equations Involving Fractions | 涉及分数的方程

When fractions are present, such as x/3 + 1 = 5, clear the fraction by performing the inverse operation. First, subtract 1 from both sides: x/3 = 4. Then multiply both sides by 3: x = 12. If the equation is more complex, like (2x)/5 = 8, multiply both sides by 5, then divide by 2, or recognize that you can multiply by 5/2 in one step. Always treat numerators as grouped if needed.

当方程中出现分数时，比如 x/3 + 1 = 5，可以通过逆运算来去掉分数。首先，两边减 1：x/3 = 4。然后两边乘 3：x = 12。如果方程更复杂，比如 (2x)/5 = 8，先两边乘 5，再除以 2，或者认识到你可以一步乘上 5/2。必要时，要把分子当作一个整体来处理。

9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

One frequent error is forgetting to apply the operation to every term on both sides. For example, when solving 2x + 3 = 11, some students divide only the 2x by 2 and leave the 3 untouched. Another mistake is combining unlike terms: 2x + 5 cannot be simplified to 7x. Also, misreading subtraction signs: in 10 – 2x = 4, add 2x to both sides first. Always write each step neatly and check your solution mentally.

一个常见的错误是忘记把运算施加到两边的每一项上。例如，解 2x + 3 = 11 时，有些学生只把 2x 除以 2，而没动 3。另一个错误是合并不同的项：2x + 5 不能化简成 7x。还有，误读减号：在 10 – 2x = 4 中，应先在两边加 2x。要始终把每一步写整齐，并在脑中验算答案。

10. Word Problems Leading to Linear Equations | 导出一次方程的文字题

Many KS3 problems describe a situation in words. Translate the words into an equation. Look for key phrases: ‘sum’ means addition, ‘difference’ means subtraction, ‘product’ means multiplication, and ‘quotient’ means division. For example: “I think of a number, double it, add 5, and the result is 19. What is the number?” Let x be the number. Then 2x + 5 = 19. Solve to get x = 7. Always define your variable clearly.

许多 KS3 的题目会用文字描述一个情景。要把这些文字转换成方程。寻找关键词：’和’ 表示加法，’差’ 表示减法，’积’ 表示乘法，’商’ 表示除法。例如：“我想了一个数，把它加倍，再加 5，结果是 19。这个数是多少？”设这个数为 x，那么方程就是 2x + 5 = 19。解出 x = 7。始终要明确地设定变量。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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