Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are a central part of the KS3 Cambridge Mathematics course. They appear in many contexts, from simple number puzzles to real-world problems involving speed, cost and measurement. This revision article explains how to solve linear equations step by step, using balancing, brackets, fractions and unknowns on both sides. It also includes common mistakes and practice questions that follow the style of Cambridge exercise pages.

一元一次方程是 KS3 剑桥数学课程的核心内容。它们出现在许多情境中，从简单的数字谜题到涉及速度、成本和测量的实际问题。这篇复习文章将逐步讲解如何解一元一次方程，包括平衡法、括号、分数和两边都有未知数的情形，并总结常见错误和与剑桥教材练习风格一致的练习题。

1. What is a linear equation? | 什么是一元一次方程？

A linear equation is an equation in which the highest power of the variable is 1. It usually has one unknown, often written as x, but it can use other letters such as a, n, or t. The graph of a linear equation is a straight line, which is why it is called “linear”.

一元一次方程是变量最高次数为 1 的方程。它通常有一个未知数，常写作 x，但也可以使用其他字母，如 a、n 或 t。一元一次方程的图像是一条直线，因此称为 “线性” 方程。

x + 3 = 7

2x − 5 = 9

4(x + 1) = 20

2. The balancing method | 方程的天平法则

To solve an equation, think of it as a balance scale. The equals sign means the left side and the right side have the same value. If you add, subtract, multiply or divide one side, you must do exactly the same to the other side. This keeps the equation balanced and allows you to isolate the unknown.

解方程时，可以把它想像成天平。等号意味着左边和右边的值相等。如果你在一侧加、减、乘或除以某个数，就必须在另一侧做完全相同的运算。这样可以保持方程平衡，并逐步把未知数单独留在一边。

If a = b, then a + c = b + c, a − c = b − c, a × c = b × c, a ÷ c = b ÷ c (c ≠ 0)

This rule is the foundation for every method shown in this article. Always perform the same operation on both sides, not just on the side with the variable.

这条规则是本文所有解法的基础。一定要在等式两边同时进行相同的运算，而不能只在含有未知数的一边进行运算。

3. Solving one-step equations | 解一步方程

One-step equations require only one operation to solve. For x + 3 = 7, subtract 3 from both sides to get x = 4. For x − 5 = 9, add 5 to both sides to get x = 14. For 3x = 12, divide both sides by 3 to get x = 4.

一步方程只需要一步运算就能解出答案。例如 x + 3 = 7，两边同时减去 3，得到 x = 4。对于 x − 5 = 9，两边同时加上 5，得到 x = 14。对于 3x = 12，两边同时除以 3，得到 x = 4。

x + 3 = 7 → x = 7 − 3 → x = 4

x − 5 = 9 → x = 9 + 5 → x = 14

3x = 12 → x = 12 ÷ 3 → x = 4

Always check that the operation you choose is the inverse of the operation in the equation. Addition is reversed by subtraction, subtraction by addition, multiplication by division, and division by multiplication.

要始终确保所选运算是方程中运算的逆运算。加法用减法来反算，减法用加法来反算，乘法用除法来反算，除法用乘法来反算。

4. Solving two-step equations | 解两步方程

Two-step equations need two operations. Follow the reverse order of operations: undo addition or subtraction first, then undo multiplication or division. For 2x + 3 = 11, subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.

两步方程需要两步运算。按照运算顺序的逆序进行：先处理加减，再处理乘除。例如 2x + 3 = 11，先将两边都减去 3，得到 2x = 8；再将两边都除以 2，得到 x = 4。

Step \| 步骤	Equation \| 方程
Start \| 开始	2x + 3 = 11
Subtract 3 \| 两边减 3	2x = 8
Divide by 2 \| 两边除以 2	x = 4

Remember that the number in front of x means multiplication. So 2x means 2 × x, and the inverse of multiplying by 2 is dividing by 2.

记住 x 前面的数字表示相乘。所以 2x 表示 2 × x，而乘以 2 的逆运算是除以 2。

5. Equations with brackets | 含括号的方程

If an equation contains brackets, expand them first or divide by the factor outside the bracket. For 3(x + 2) = 18, you can expand to 3x + 6 = 18, subtract 6 from both sides to get 3x = 12, then divide by 3 to get x = 4. Alternatively, divide both sides by 3 first: x + 2 = 6, then subtract 2 to get x = 4.

如果方程含有括号，可以先展开括号，或者先除以括号外的系数。对于 3(x + 2) = 18，可以先展开为 3x + 6 = 18，两边减去 6 得到 3x = 12，再除以 3 得到 x = 4。也可以先把两边都除以 3，得到 x + 2 = 6，再减去 2 得到 x = 4。

3(x + 2) = 18 → 3x + 6 = 18 → 3x = 12 → x = 4

Both methods are correct, but dividing first can be quicker when the coefficient outside the bracket divides both sides exactly. Always expand brackets carefully by multiplying every term inside.

两种方法都正确，但当括号外的系数能整除两边时，先除以这个系数通常会更快。展开括号时一定要仔细，括号里的每一项都要乘以括号外的数。

6. Unknowns on both sides | 两边都有未知数的方程

To solve equations like 5x + 2 = 3x + 10, collect the variable terms on one side and the number terms on the other. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8. Finally divide by 2 to get x = 4.

要解像 5x + 2 = 3x + 10 这样的方程，需要把含未知数的项集中到一边，把常数项集中到另一边。两边同时减去 3x，得到 2x + 2 = 10；两边同时减去 2，得到 2x = 8；最后两边除以 2，得到 x = 4。

5x + 2 = 3x + 10 → 2x + 2 = 10 → 2x = 8 → x = 4

A useful tip is to remove the smaller unknown term first. In the example, 3x is smaller than 5x, so subtracting 3x keeps the coefficient of x positive. This reduces negative-number mistakes.

一个有用的技巧是先消去较小的未知数项。在上例中，3x 小于 5x，所以减去 3x 可以保持 x 的系数为正，从而减少负数带来的错误。

7. Equations with fractions | 含分数的方程

Equations with fractions can be made simpler by clearing the denominators. For x/4 + 1 = 3, subtract 1 from both sides to get x/4 = 2, then multiply both sides by 4 to get x = 8. If there are multiple denominators, multiply every term by the least common multiple.

含分数的方程可以通过去分母来简化。例如 x/4 + 1 = 3，先将两边都减去 1，得到 x/4 = 2；再将两边都乘以 4，得到 x = 8。如果有多个分母，可以把每一项都乘以最小公倍数。

x/4 + 1 = 3 → x/4 = 2 → x = 2 × 4 → x = 8

When you multiply every term by the least common multiple, make sure you multiply both the fractions and the whole numbers. This removes denominators and turns the equation into a simpler one without fractions.

把每一项都乘以最小公倍数时，要确保分数项和整数项都乘到。这样可以去分母，把方程转化为不含分数的更简单方程。

8. Checking your solution | 检验答案

Always substitute your final answer back into the original equation to check it works. If x = 4 in 2x + 3 = 11, then the left side is 2 × 4 + 3 = 11, which matches the right side. This habit helps you catch arithmetic errors.

一定要把最终答案代回原方程检验是否正确。例如 x = 4 代入 2x + 3 = 11，左边为 2 × 4 + 3 = 11，与右边相等。这个习惯有助于发现计算错误。

Left side: 2 × 4 + 3 = 8 + 3 = 11

Right side: 11

If the two sides are not equal, go back and check each step carefully. Substitution is also useful for word problems because it confirms that the answer makes sense in the original context.

如果两边不相等，就要回到每一步仔细检查。代入法对文字题也很有用，因为它可以确认答案在原问题情境中是否合理。

9. Common mistakes to avoid | 常见错误

Common mistakes include forgetting to operate on both sides, mixing up the order of inverse operations, losing a negative sign, and failing to expand brackets correctly. When moving terms, remember that the sign travels with the number.

常见错误包括忘记在等式两边同时运算、搞混逆运算的顺序、漏掉负号，以及括号展开不正确。移项时，要记住符号跟着数字一起移动。

Forgetting to subtract from both sides in x + 5 = 12
Dividing before subtracting in 3x + 6 = 15
Writing 2(x + 3) as 2x + 3 instead of 2x + 6
Losing the minus sign when solving 8 − x = 3

在 x + 5 = 12 中忘记两边同时减去 5
在 3x + 6 = 15 中先除以 3 而不是先减去 6
把 2(x + 3) 写成 2x + 3，而不是 2x + 6
在解 8 − x = 3 时漏掉负号

Being aware of these pitfalls will help you avoid them under exam pressure. Slow down, write each line neatly, and check the sign of every term.

了解这些易错点有助于你在考试压力下避免犯错。放慢速度，书写工整，并检查每一项的符号。

10. Worked example in Cambridge style | 剑桥风格例题

Let us solve 4(2x − 1) = 3x + 11. Expand brackets: 8x − 4 = 3x + 11. Subtract 3x from both sides: 5x − 4 = 11. Add 4 to both sides: 5x = 15. Divide by 5: x = 3. Check: left side = 4(2 × 3 − 1) = 4 × 5 = 20; right side = 3 × 3 + 11 = 20.

我们来解 4(2x − 1) = 3x + 11。展开括号：8x − 4 = 3x + 11。两边减去 3x：5x − 4 = 11。两边加上 4：5x = 15。两边除以 5：x = 3。检验：左边 = 4(2 × 3 − 1) = 4 × 5 = 20；右边 = 3 × 3 + 11 = 20。

4(2x − 1) = 3x + 11 → 8x − 4 = 3x + 11 → 5x = 15 → x = 3

This example combines brackets, collecting like terms, and checking. It is typical of the multi-step questions found in Cambridge Lower Secondary exercises, where you must show clear working at every stage.

这道例题综合了括号、合并同类项和检验。它是剑桥初中练习中常见的多步题，这类题目要求你在每一步都展示清晰的运算过程。

11. Practice questions | 练习题

Try the following questions without looking at the answers. Write down every step and then check by substituting your answer back into the original equation.

请在不看答案的情况下完成以下练习题。写出每一步，并通过把答案代回原方程进行检验。

x + 9 = 15
2a − 7 = 5
5(y + 2) = 35
7n + 3 = 4n + 15
m/3 + 2 = 6

Answers: 1. x = 6; 2. a = 6; 3. y = 5; 4. n = 4; 5. m = 12

答案：1. x = 6；2. a = 6；3. y = 5；4. n = 4；5. m = 12

If you made a mistake, do not just correct the final answer. Go back through your steps, find where the error occurred, and rewrite the whole solution clearly.

如果你做错了，不要只改正最终答案。要回到你的步骤中，找出错误发生的位置，并重新清楚地写出完整解答。

12. Summary | 总结

To solve linear equations confidently, keep the equation balanced, undo operations in reverse order, deal with brackets and fractions carefully, and always check your answer.

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Step \| 步骤	Equation \| 方程
Start \| 开始	2x + 3 = 11
Subtract 3 \| 两边减 3	2x = 8
Divide by 2 \| 两边除以 2	x = 4

Solving Linear Equations | 解一元一次方程