Mastering Linear Equations | 精通一元一次方程

📚 Mastering Linear Equations | 精通一元一次方程

Linear equations are one of the most important building blocks in KS3 mathematics. They appear in almost every algebra topic, from solving simple one-step equations to forming equations from real-life word problems. Mastering this skill will help you succeed in Cambridge Checkpoint tests and prepare you for IGCSE.

一元一次方程是 KS3 数学中最重要的基础模块之一。从解简单的一步方程到根据实际文字题建立方程，几乎每个代数主题都会出现它们。掌握这一技能将帮助你在剑桥 Checkpoint 考试中取得成功，并为 IGCSE 做好准备。

1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation in which the unknown value, usually written as x, is raised only to the power of 1. It can be written in the form ax + b = c, where a, b and c are numbers. For example, 2x + 3 = 11 is a linear equation.

一元一次方程是指未知数（通常写成 x）的指数只为 1 的方程。它可以写成 ax + b = c 的形式，其中 a、b、c 都是数字。例如，2x + 3 = 11 就是一个一元一次方程。

The word ‘linear’ tells us that the graph of the equation is a straight line. When we solve an equation, we find the value of x that makes both sides equal. This value is called the solution or the root of the equation.

“线性”一词告诉我们该方程的图像是一条直线。当我们解方程时，就是在寻找使等式两边相等的 x 值。这个值被称为方程的解或根。

Example: x + 5 = 12 is linear. | 示例：x + 5 = 12 是一元一次方程。
Non-example: x² + 5 = 12 is not linear because x has power 2. | 反例：x² + 5 = 12 不是一元一次方程，因为 x 的次数为 2。

2. The Balance Method | 天平法

To solve a linear equation, we use the balance method. Think of an equation as a balance scale: the left side and the right side must always have the same value. If we add, subtract, multiply or divide one side by a number, we must do exactly the same to the other side.

解一元一次方程时，我们使用天平法。把方程想象成一个天平：左边和右边必须始终保持相同的值。如果我们对一边进行加、减、乘或除一个数，就必须对另一边做完全相同的运算。

This idea is called ‘keeping the equation balanced’. Whatever operation you perform on one side, perform it on the other side as well. This keeps the equality true while we isolate the unknown x.

这个思想叫做“保持方程平衡”。无论你对一边进行什么运算，都要对另一边进行同样的运算。这样可以在保持等式成立的同时，将未知数 x 分离出来。

x + 3 = 7 → x + 3 − 3 = 7 − 3 → x = 4

In the example above, we subtracted 3 from both sides to remove the + 3 on the left, leaving x by itself.

在上面的例子中，我们从两边减去 3，以消去左边的 + 3，使 x 单独留在等式一边。

3. Solving One-Step Equations | 解一步方程

A one-step equation needs only one operation to find the solution. For example, in x − 4 = 9, we add 4 to both sides to get x = 13. In 5x = 35, we divide both sides by 5 to get x = 7.

一步方程只需要一次运算就能求出解。例如，在 x − 4 = 9 中，我们给两边同时加上 4，得到 x = 13。在 5x = 35 中，我们把两边同时除以 5，得到 x = 7。

The key is to perform the inverse operation. If the equation has addition, use subtraction; if it has multiplication, use division. Always apply the inverse operation to both sides.

关键是要进行逆运算。如果方程中有加法，就用减法；如果有乘法，就用除法。始终对两边同时进行逆运算。

Equation \| 方程	Inverse operation \| 逆运算	Solution \| 解
x + 8 = 15	Subtract 8 \| 减去 8	x = 7
x − 6 = 11	Add 6 \| 加上 6	x = 17
4x = 28	Divide by 4 \| 除以 4	x = 7
x ÷ 3 = 5	Multiply by 3 \| 乘以 3	x = 15

Practise these basic cases until the inverse operation becomes automatic. This foundation makes two-step equations much easier to handle.

练习这些基本情形，直到逆运算变得自动化。这个基础会让两步方程容易处理得多。

4. Solving Two-Step Equations | 解两步方程

A two-step equation involves two operations. For example, 2x + 3 = 11 contains multiplication by 2 and addition of 3. To solve it, we undo the operations in reverse order: first remove the constant term, then remove the coefficient of x.

两步方程包含两种运算。例如，2x + 3 = 11 包含乘以 2 和加上 3。解这类方程时，我们要按相反顺序撤销运算：先消去常数项，再消去 x 的系数。

So for 2x + 3 = 11, subtract 3 from both sides to get 2x = 8. Then divide both sides by 2 to get x = 4. This reverse order follows the same logic as undressing: you take off your coat before your shirt.

因此对于 2x + 3 = 11，先从两边减去 3 得到 2x = 8。然后把两边除以 2 得到 x = 4。这种相反顺序就像脱衣服：先脱外套再脱衬衫。

2x + 3 = 11 → 2x = 8 → x = 4

Always check that your solution works by substituting it back into the original equation. If 2 × 4 + 3 equals 11, then x = 4 is correct.

一定要把解代回原方程检验是否正确。如果 2 × 4 + 3 等于 11，那么 x = 4 就是正确的。

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, the usual first step is to expand them. For example, 3(x + 2) = 21 becomes 3x + 6 = 21 after expanding. Then solve the resulting two-step equation: subtract 6 from both sides to get 3x = 15, then divide by 3 to get x = 5.

当方程中含有括号时，通常第一步是去括号。例如，3(x + 2) = 21 去括号后变为 3x + 6 = 21。然后解得到的两步方程：从两边减去 6 得到 3x = 15，再除以 3 得到 x = 5。

Remember to multiply every term inside the bracket by the number outside. A common error is to write 3(x + 2) = 3x + 2, forgetting to multiply the 2 by 3. The correct expansion is 3x + 6.

记住用括号外面的数乘以括号内的每一项。常见错误是写成 3(x + 2) = 3x + 2，忘记了将 2 也乘以 3。正确的展开式是 3x + 6。

Correct: 2(x − 4) = 2x − 8 | 正确：2(x − 4) = 2x − 8
Wrong: 2(x − 4) = 2x − 4 | 错误：2(x − 4) = 2x − 4

If the bracket is on one side only, expand it first and then solve. If there are brackets on both sides, expand both before collecting like terms.

如果括号只出现在一边，先去括号再求解。如果两边都有括号，先展开两边，再合并同类项。

6. Equations with Unknowns on Both Sides | 未知数在两边

Some equations have x on both sides, such as 5x + 2 = 2x + 14. The strategy is to collect all x terms on one side and all number terms on the other side. Usually we subtract the smaller x term from both sides.

有些方程的两边都有 x，例如 5x + 2 = 2x + 14。策略是把所有含 x 的项移到一边，把所有数字项移到另一边。通常我们从两边减去较小的 x 项。

5x + 2 = 2x + 14 → 5x − 2x + 2 = 14 → 3x + 2 = 14 → 3x = 12 → x = 4

Here we subtracted 2x from both sides. This removes the unknown from the right-hand side and makes the equation easier to solve. Then we solved the two-step equation as before.

这里我们从两边减去了 2x。这样就去掉了右边的未知数，使方程更容易求解。然后我们像前面一样解两步方程。

Always check by substituting your answer into both sides of the original equation. For x = 4, the left side is 5 × 4 + 2 = 22, and the right side is 2 × 4 + 14 = 22, so the solution is correct.

始终通过把答案代入原方程的两边进行检查。当 x = 4 时，左边是 5 × 4 + 2 = 22，右边是 2 × 4 + 14 = 22，因此解是正确的。

7. Equations with Fractions | 含分数的方程

Equations with fractions can look difficult, but they become simple if you multiply every term by the common denominator. For example, in x/3 + 2 = 6, multiply every term by 3 to remove the fraction: x + 6 = 18. Then subtract 6 from both sides to get x = 12.

含有分数的方程看起来很难，但如果你把每一项都乘以公分母，它们就会变得简单。例如，在 x/3 + 2 = 6 中，把每一项乘以 3 来消去分数：x + 6 = 18。然后从两边减去 6 得到 x = 12。

If the equation is (x + 4)/2 = 9, multiply both sides by 2 to get x + 4 = 18. Then subtract 4 from both sides to find x = 14. The common denominator acts as a tool to clear all fractions in one step.

如果方程是 (x + 4)/2 = 9，把两边乘以 2 得到 x + 4 = 18。然后从两边减去 4 得到 x = 14。公分母是同时清除所有分数的工具。

Be careful when multiplying a fraction with a numerator that contains more than one term. Always put the numerator in a bracket if it is an expression: (x + 4)/2, not x + 4/2. This prevents mistakes.

当分子含有不止一项时，乘以公分母时要小心。如果分子是一个表达式，一定要用括号括起来：(x + 4)/2，而不是 x + 4/2。这可以防止错误。

8. Forming Equations from Word Problems | 从文字题建立方程

Many KS3 exam questions ask you to form an equation from a written situation. The first step is to identify the unknown and give it a letter, usually x. Then translate words into mathematical operations: ‘more than’ means add, ‘less than’ means subtract, ‘times’ means multiply, and ‘divided by’ means divide.

许多 KS3 考试题目要求你根据文字描述建立方程。第一步是找出未知数并用字母表示，通常用 x。然后把文字转换成数学运算：“比……多”表示加，“比……少”表示减，“倍”表示乘，“除以”表示除。

For example, ‘I think of a number, multiply it by 4, add 7, and the result is 31.’ Let the number be x. The equation is 4x + 7 = 31. Solve it: subtract 7 from both sides to get 4x = 24, then divide by 4 to get x = 6.

例如，“我想一个数，把它乘以 4，再加上 7，结果是 31。”设这个数为 x。方程是 4x + 7 = 31。解方程：两边减去 7 得到 4x = 24，然后除以 4 得到 x = 6。

Always define your variable clearly. Writing ‘let x be the unknown number’ helps you and the examiner understand your working. Then check that your solution matches the original problem conditions.

一定要清楚地定义变量。写“设 x 为未知数”有助于你和阅卷老师理解你的解题过程。然后检查你的解是否符合原题条件。

9. Checking Your Solution | 检验你的解

Checking your answer is an essential part of solving equations. Substitute your value of x into the left side and the right side of the original equation, then compare them. If both sides give the same number, your solution is correct.

检验答案是解方程必不可少的一部分。把你的 x 值代入原方程的左边和右边，然后比较结果。如果两边得到相同的数值，你的解就是正确的。

For example, if you solve 7x − 5 = 16 and get x = 3, check by calculating 7 × 3 − 5 = 21 − 5 = 16. Since the left side equals 16, the solution is correct.

例如，如果你解 7x − 5 = 16 得到 x = 3，代入计算 7 × 3 − 5 = 21 − 5 = 16。因为左边等于 16，所以解是正确的。

Checking is especially useful in exams because it can catch small arithmetic errors. If your check does not work, go back and look for where you made a mistake.

检验在考试中特别有用，因为它可以发现小的计算错误。如果检验不成立，就回头检查哪里出错了。

10. Common Mistakes | 常见错误

Many students lose marks in algebra because of avoidable mistakes. One common error is forgetting to apply an operation to both sides. For example, in x + 4 = 10, writing x = 10 without subtracting 4 from the left side is wrong.

许多学生在代数中因为本可避免的错误而丢分。一个常见错误是忘记对两边同时进行运算。例如，在 x + 4 = 10 中，写出 x = 10 而不从左边的 10 中减去 4 就是错误的。

Another common error is mishandling negative signs. For example, in −2x = 8, dividing both sides by −2 gives x = −4, not x = 4. Always pay attention to whether the coefficient of x is positive or negative.

另一个常见错误是处理负号不当。例如，在 −2x = 8 中，两边除以 −2 得到 x = −4，而不是 x = 4。始终要注意 x 的系数是正还是负。

Forgot to balance both sides | 忘记保持两边平衡
Incorrect expansion of brackets | 去括号时出错
Sign errors with negative numbers | 负数符号错误
Not checking the final answer | 没有检验最终答案

11. Quick Practice Questions | 快速练习

Try these questions to test your understanding. Solve each equation and check your answer by substitution.

尝试以下问题来测试你的理解。解每个方程并通过代入检验答案。

Question \| 题目	Answer \| 答案
x + 12 = 25	x = 13
3x − 7 = 14	x = 7
2(x + 5) = 28	x = 9
6x + 3 = 2x + 19	x = 4
(x + 6)/4 = 3	x = 6

Work through each solution step by step, showing all your working. This is the best way to build confidence and speed for Checkpoint assessments.

一步一步地写出每道题的解，展示所有步骤。这是为 Checkpoint 评估建立信心和速度的最佳方法。

12. Summary | 小结

Linear equations are a core part of KS3 mathematics and appear in many forms. The balance method is the foundation: whatever you do to one side, do to the other. Always undo operations in reverse order and expand brackets before collecting like terms.

一元一次方程是 KS3 数学的核心内容，以多种形式出现。天平法是基础：你对一边做什么，另一边也要做同样的事。始终按逆序撤销运算，并在合并同类项之前先去括号。

With regular practice, forming and solving linear equations becomes straightforward. Focus on accuracy, check your answers, and manage negative signs carefully. These habits will help you in every future algebra topic.

通过经常练习，建立和求解一元一次方程会变得轻松。重视准确性、检验答案并仔细处理负号。这些习惯将帮助你学好未来的每一个代数主题。

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Mastering Linear Equations | 精通一元一次方程