Linear Equations in One Variable | 一元一次方程

📚 Linear Equations in One Variable | 一元一次方程

Welcome to this comprehensive revision guide on solving linear equations in one variable, a core topic in Key Stage 3 Cambridge Mathematics. Whether you are just starting algebra or preparing for a checkpoint test, understanding how to solve equations like 3x + 5 = 20 is essential. This article will take you from basic concepts right through to equations with brackets and fractions, giving you clear methods and plenty of practice tips.

欢迎来到这篇关于解一元一次方程的综合复习指南，这是KS3剑桥数学的核心主题。无论你是刚刚开始学习代数，还是正在为Checkpoint考试做准备，理解如何解像 3x + 5 = 20 这样的方程都至关重要。本文将从基本概念开始，一直延伸到含有括号和分数的方程，为你提供清晰的方法和大量练习题提示。

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

A linear equation in one variable is an equation that can be written in the form ax + b = c, where x is the variable (unknown number), and a, b, c are constants, and a is not zero. The word ‘linear’ means the variable appears only to the power of 1, so there are no x², x³, or other powers. The goal is to find the value of x that makes the equation true.

一元一次方程是可以写成 ax + b = c 形式的方程，其中 x 是变量（未知数），a、b、c 是常数，且 a 不等于零。“一次”是指变量只出现一次方（幂指数为1），因此没有 x²、x³ 或其他高次幂。我们的目标是找出使方程成立的 x 值。

2. Equivalent Equations and the Balance Method | 等式性质与天平法

Think of an equation as a pair of perfectly balanced scales. Any operation you perform on one side must also be performed on the other to keep the scales balanced. This is the golden rule: do the same to both sides. By adding, subtracting, multiplying or dividing both sides by the same non-zero number, you create equivalent equations that get you closer to isolating x.

把方程想象成一架完全平衡的天平。对一边进行的任何操作，也必须在另一边进行，才能保持天平平衡。这是黄金法则：两边做同样的运算。通过同时加减同一个数，或者同时乘除同一个非零数，你可以得到等价方程，从而逐步求出 x。

Operation	Example
Add 4 to both sides	x − 4 = 10 → x = 14
Subtract 2 from both sides	x + 2 = 9 → x = 7
Multiply both sides by 3	x/3 = 5 → x = 15
Divide both sides by 2	2x = 12 → x = 6

操作例子：给等号两边加4，给等号两边减2，给等号两边乘3，给等号两边除以2。

3. Solving One-Step Equations by Adding or Subtracting | 用加减法解一步方程

If the equation is of the form x + a = b, subtract a from both sides. For x − a = b, add a to both sides. Always perform the inverse (opposite) operation to eliminate the constant term next to x.

如果方程是 x + a = b 的形式，两边同时减去 a。如果是 x − a = b，两边同时加上 a。记住，总是用逆运算（相反运算）消去 x 旁边的常数项。

Example 1: x + 7 = 15. Subtract 7 from both sides: x + 7 − 7 = 15 − 7, so x = 8.
例1：x + 7 = 15，两边同时减7：x + 7 − 7 = 15 − 7，得 x = 8。
Example 2: x − 3 = 9. Add 3 to both sides: x − 3 + 3 = 9 + 3, so x = 12.
例2：x − 3 = 9，两边同时加3：x − 3 + 3 = 9 + 3，得 x = 12。

4. Solving One-Step Equations by Multiplying or Dividing | 用乘除法解一步方程

For equations of the type ax = b (where a is multiplied by x), divide both sides by a. For equations like x/a = b, multiply both sides by a. This isolates x by undoing the multiplication or division.

对于 ax = b 这类方程（a 乘以 x），两边同时除以 a。对于 x/a = b 这类方程，两边同时乘以 a。这样就能撤销乘法或除法，求出 x。

Example: 4x = 28. Divide both sides by 4: 4x ÷ 4 = 28 ÷ 4, giving x = 7.
例子：4x = 28，两边同时除以4：4x ÷ 4 = 28 ÷ 4，得 x = 7。
Example: x/5 = 6. Multiply both sides by 5: (x/5) × 5 = 6 × 5, so x = 30.
例子：x/5 = 6，两边同时乘以5：(x/5) × 5 = 6 × 5，得 x = 30。

5. Solving Two-Step Equations | 解两步方程

Two-step equations involve both a multiplication/division and an addition/subtraction, such as 2x + 3 = 11. The strategy is to undo the addition or subtraction first, then deal with the multiplication or division. This order is the reverse of the order of operations (BIDMAS/BODMAS in reverse).

两步方程同时包含乘除和加减，例如 2x + 3 = 11。解题策略是先撤销加减法，再处理乘除法。这个顺序与运算顺序（BIDMAS/BODMAS）相反，即逆序操作。

Step-by-step: Solve 2x + 3 = 11.
分步：解 2x + 3 = 11。
Subtract 3 from both sides: 2x + 3 − 3 = 11 − 3 → 2x = 8.
两边同时减3：2x + 3 − 3 = 11 − 3 → 2x = 8。
Divide both sides by 2: 2x ÷ 2 = 8 ÷ 2 → x = 4.
两边同时除以2：2x ÷ 2 = 8 ÷ 2 → x = 4。

6. Equations with Brackets | 含有括号的方程

When an equation contains brackets, expand them first using the distributive property: a(b + c) = ab + ac. Once the brackets are removed, you will usually get a two-step equation. Be careful with negative signs outside the brackets.

当方程中含有括号时，首先用乘法分配律展开：a(b + c) = ab + ac。去掉括号后，通常会得到一个两步方程。注意括号外面的负号。

Example: 3(2x − 5) = 21. Expand to get 6x − 15 = 21. Then add 15 to both sides: 6x = 36. Divide by 6: x = 6.

例题：3(2x − 5) = 21。展开得 6x − 15 = 21。然后两边加15：6x = 36，再除以6：x = 6。

If you have a negative multiplier like –2(x + 4), remember –2 × x = –2x and –2 × 4 = –8, giving –2x − 8.

如果乘数是负数，如 –2(x + 4)，记住 –2 × x = –2x， –2 × 4 = –8，得到 –2x − 8。

7. Equations with Variables on Both Sides | 两边都含有未知数的方程

Equations can have x on both sides, e.g., 5x + 2 = 3x + 10. The first step is to collect all x-terms on one side and constant numbers on the other. Choose the side that keeps the coefficient of x positive if possible.

方程中未知数可能出现在两边，如 5x + 2 = 3x + 10。第一步是把所有含 x 的项移到一边，常数项移到另一边。尽量选择能使 x 的系数保持为正数的一边。

Method: Subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 10 → 2x + 2 = 10. Then subtract 2: 2x = 8. Divide by 2: x = 4.

方法：两边同时减去 3x：5x − 3x + 2 = 3x − 3x + 10 → 2x + 2 = 10。再减2：2x = 8，除以2：x = 4。

8. Equations Involving Fractions | 含有分数的方程

To solve equations like x/4 + 1 = 3 or (2x)/3 = 10, it is often easiest to eliminate the fraction early. Multiply every term on both sides by the denominator (the lowest common multiple if there are several fractions). This technique clears the fractions and gives a simpler equation.

要解像 x/4 + 1 = 3 或 (2x)/3 = 10 这样的方程，通常最好先消去分数。将等号两边每一项都乘以分母（如果有多个分数，乘以最小公倍数）。这一方法可以清除分数，得到更简单的方程。

Example 1: x/4 + 1 = 3. Multiply all terms by 4: (x/4)×4 + 1×4 = 3×4 → x + 4 = 12. Then x = 8.

例1：x/4 + 1 = 3。每一项乘以4：(x/4)×4 + 1×4 = 3×4 → x + 4 = 12，得 x = 8。

Example 2: (3x + 2)/5 = 4. Multiply both sides by 5: 3x + 2 = 20. Then 3x = 18, so x = 6.

例2：(3x + 2)/5 = 4。两边乘以5：3x + 2 = 20，然后 3x = 18，x = 6。

9. Forming Equations from Word Problems | 根据文字题列方程

Many questions ask you to set up an equation from a description. Identify the unknown quantity and represent it with a letter (often x). Translate words into algebraic expressions: ‘more than’ means +, ‘less than’ means −, ‘times’ means ×, ‘divided by’ means ÷, and ‘is’ or ‘equals’ means =.

许多题目会要求你根据描述列出方程。先确定未知量并用字母（通常用 x）表示。把文字翻译成代数表达式：“比……多”用 +，“比……少”用 −，“的几倍”用 ×，“除以”用 ÷，“是”或“等于”用 =。

Example: “I think of a number, multiply it by 6, then add 11. The result is 47.” Let the number be x: 6x + 11 = 47. Solve: 6x = 36, x = 6.

例题：“我想了一个数，把它乘以6，再加上11，结果是47。”设这个数为 x：6x + 11 = 47。解：6x = 36，x = 6。

10. Checking Your Solution | 检验你的解

Always substitute your answer back into the original equation to make sure both sides are equal. This is a quick way to catch mistakes, especially when dealing with negative numbers or fractions. Write down the check clearly to show your working.

一定要把你的答案代回原方程，验证等号两边是否相等。这是快速发现错误的方法，尤其是在涉及负数或分数时。清晰地写下检验过程，展示你的步骤。

Checking Example: For 2x + 3 = 11 with solution x = 4, left side = 2(4) + 3 = 8 + 3 = 11, right side = 11. They match, so the solution is correct.

检验示例：对方程 2x + 3 = 11，解为 x = 4，左边 = 2(4) + 3 = 8 + 3 = 11，右边 = 11，一致，所以解正确。

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

Many students lose marks because of careless errors. The most common mistakes include forgetting to perform an operation on both sides, incorrectly expanding brackets with a negative sign outside, dividing instead of subtracting (and vice versa), and missing that a negative fraction is different from a positive one. Slow down and write each step, especially when dealing with negatives.

许多学生因为粗心而丢分。最常见的错误包括：忘记对两边同时进行运算、展开括号时外面有负号时出错、该减却除了（或混淆逆运算）、以及忽略了负分数与正分数的区别。请放慢速度，写出每一步，尤其当涉及负数时。

Mistake: 2x + 3 = 11 → subtract 2 from both sides. (Incorrect: you must subtract 3, not 2.)
错误：2x + 3 = 11 → 两边减去2。（错误：应该减去3，而不是2。）
Mistake: –(2x − 5) expanded as –2x − 5. (Incorrect: should be –2x + 5)
错误：–(2x − 5) 展开为 –2x − 5。（正确应为 –2x + 5）

12. Practice Progression and Summary | 练习进阶与总结

Mastering linear equations comes from practicing a variety of question types. Start with simple one-step equations, move to two-step, then brackets, variables on both sides, and finally word problems. Always use the balance method and check your answer. With consistent practice, you will build speed and accuracy, which are crucial for Cambridge Checkpoint success.

要掌握一元一次方程，需要练习各种类型的题目。从简单的一步方程开始，然后到两步方程，再到带括号的、两边有未知数的，最后解决应用题。始终使用天平法并检验答案。通过持续练习，你会提高速度和准确性，这对剑桥Checkpoint考试的成功至关重要。

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Linear Equations in One Variable | 一元一次方程