Mastering Linear Equations | 掌握一元一次方程

📚 Mastering Linear Equations | 掌握一元一次方程

Linear equations are the foundation of algebra and appear throughout the Cambridge KS3 Mathematics course. This revision guide is based on practice questions from p60_2.pdf and will help you solve one-step, two-step, bracketed, and variable-both-sides equations confidently.

一元一次方程是代数的基础，贯穿剑桥 KS3 数学课程。本复习指南基于 p60_2.pdf 中的练习题，将帮助你自信地解一步方程、两步方程、含括号方程以及变量在等式两边的方程。

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 can usually be written in the form ax + b = c, where a, b and c are constants and x is the unknown.

一元一次方程是指未知数的最高次数为 1 的方程。它通常可以写成 ax + b = c 的形式，其中 a、b、c 是常数，x 是未知数。

For example, 3x − 2 = 10, x/4 + 1 = 3, and 2(x − 1) = 6 are all linear equations because the variable x is never raised to a power higher than 1.

例如，3x − 2 = 10、x/4 + 1 = 3 和 2(x − 1) = 6 都是一元一次方程，因为变量 x 的指数从未超过 1。

2. Key Vocabulary | 关键术语

Before solving equations, you need to understand the main terms. A variable is a symbol, usually a letter, that stands for an unknown number. A constant is a fixed number. A coefficient is the number multiplied by the variable. An equation is a statement that two expressions are equal, and the solution is the value of the variable that makes the equation true.

在解方程之前，你需要理解主要术语。变量是一个符号，通常是字母，代表未知数。常数是一个固定的数。系数是与变量相乘的数。方程是表示两个表达式相等的语句，而解是使方程成立的变量值。

For example, in 5x + 2 = 17, the coefficient is 5, the constants are 2 and 17, and the solution is x = 3 because 5 × 3 + 2 = 17.

例如，在 5x + 2 = 17 中，系数是 5，常数是 2 和 17，解是 x = 3，因为 5 × 3 + 2 = 17。

3. Balancing Method | 平衡法

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep the scale balanced. This is called the balancing method.

把方程想象成一台天平。你对一边做的任何操作，都必须对另一边做同样的操作，以保持天平平衡。这就是平衡法。

For example, to solve x + 8 = 15, subtract 8 from both sides to isolate x.

例如，要解 x + 8 = 15，两边同时减去 8 以分离出 x。

x + 8 = 15 → x = 15 − 8 → x = 7

You can check by substituting 7 back into the original equation: 7 + 8 = 15, which is true.

你可以把 7 代回原方程进行检验：7 + 8 = 15，等式成立。

4. Solving One-Step Equations | 解一步方程

One-step equations require only one operation to isolate the variable. The operation depends on the equation type: addition, subtraction, multiplication, or division.

一步方程只需一步运算就能分离出变量。运算类型取决于方程的类型：加法、减法、乘法或除法。

Here are the four basic forms and how to solve them:

以下是四种基本形式及其解法：

x + a = b → x = b − a
x − a = b → x = b + a
ax = b → x = b ÷ a
x ÷ a = b → x = b × a

For example, solve 4x = 28. Divide both sides by 4 to get x = 7. Solve x ÷ 5 = 9 by multiplying both sides by 5 to get x = 45.

例如，解 4x = 28。两边同时除以 4，得到 x = 7。解 x ÷ 5 = 9，两边同时乘以 5，得到 x = 45。

5. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. Always undo addition or subtraction first, then undo multiplication or division. This is the reverse of the order of operations.

两步方程涉及两种运算。总是先撤销加法或减法，再撤销乘法或除法。这与运算顺序相反。

For example, solve 3x − 4 = 8. First add 4 to both sides to get 3x = 12. Then divide both sides by 3 to find x = 4.

例如，解 3x − 4 = 8。首先两边加 4，得到 3x = 12。然后两边除以 3，得到 x = 4。

3x − 4 = 8 → 3x = 8 + 4 → 3x = 12 → x = 12 ÷ 3 → x = 4

Another example: solve x/2 + 6 = 11. Subtract 6 from both sides to get x/2 = 5. Then multiply both sides by 2 to get x = 10.

另一个例子：解 x/2 + 6 = 11。两边减去 6，得到 x/2 = 5。然后两边乘以 2，得到 x = 10。

6. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law. Then solve the resulting two-step equation as usual.

当方程含有括号时，先用分配律展开括号。然后像往常一样解得到的两步方程。

For example, solve 2(x + 3) = 14. Expand the bracket to get 2x + 6 = 14. Subtract 6 from both sides: 2x = 8. Divide by 2: x = 4.

例如，解 2(x + 3) = 14。展开括号得到 2x + 6 = 14。两边减去 6：2x = 8。除以 2：x = 4。

2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4

If there is a negative sign before the bracket, multiply every term inside by the negative. For example, solve 3(x − 2) = 15. Expand: 3x − 6 = 15. Add 6: 3x = 21. Divide by 3: x = 7.

如果括号前有负号，则将括号内每一项都乘以该负数。例如，解 3(x − 2) = 15。展开：3x − 6 = 15。加 6：3x = 21。除以 3：x = 7。

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

When the variable appears on both sides of the equation, collect all variable terms on one side and all constant terms on the other side. Use inverse operations to move terms across the equals sign.

当变量出现在等式的两边时，将所有变量项移到一边，将所有常数项移到另一边。使用逆运算将项移过等号。

For example, solve 5x + 2 = 2x + 11. Subtract 2x from both sides to get 3x + 2 = 11. Then subtract 2 from both sides to get 3x = 9. Divide by 3 to find x = 3.

例如，解 5x + 2 = 2x + 11。两边减去 2x，得到 3x + 2 = 11。然后两边减去 2，得到 3x = 9。除以 3，得到 x = 3。

5x + 2 = 2x + 11 → 5x − 2x = 11 − 2 → 3x = 9 → x = 3

Another example: solve 7 − 2x = x + 1. Add 2x to both sides: 7 = 3x + 1. Subtract 1: 6 = 3x. Divide by 3: x = 2.

另一个例子：解 7 − 2x = x + 1。两边加 2x：7 = 3x + 1。减去 1：6 = 3x。除以 3：x = 2。

8. Equations with Fractions | 含分数的方程

To solve equations with fractions, first clear the fraction by multiplying both sides by the denominator. If there are multiple fractions, multiply by the least common multiple of all denominators.

要解含分数的方程，首先通过两边乘以分母来清除分数。如果有多个分数，则乘以所有分母的最小公倍数。

For example, solve x/5 + 7 = 12. Subtract 7 from both sides to get x/5 = 5. Then multiply both sides by 5 to get x = 25.

例如，解 x/5 + 7 = 12。两边减去 7，得到 x/5 = 5。然后两边乘以 5，得到 x = 25。

x/5 + 7 = 12 → x/5 = 5 → x = 5 × 5 → x = 25

Another example: solve 2x/3 = 8. Multiply both sides by 3 to get 2x = 24. Divide by 2 to find x = 12.

另一个例子：解 2x/3 = 8。两边乘以 3，得到 2x = 24。除以 2，得到 x = 12。

9. Checking Your Solution | 检验你的解

Always substitute your solution back into the original equation to check that the left-hand side equals the right-hand side. This step helps you catch small arithmetic mistakes.

始终将你的解代回原方程，检查左边是否等于右边。这一步可以帮助你发现小的算术错误。

For example, if you solved 2x + 4 = 10 and found x = 3, check by calculating 2 × 3 + 4 = 6 + 4 = 10. Since both sides equal 10, the solution is correct.

例如，如果你解 2x + 4 = 10 得到 x = 3，可以通过计算 2 × 3 + 4 = 6 + 4 = 10 来检验。因为两边都等于 10，所以解是正确的。

If the two sides are not equal, go back and check each step. Most errors come from sign changes or incorrect division.

如果两边不相等，请返回检查每一步。大多数错误来自符号变化或不正确的除法。

10. Word Problems | 应用题

Word problems ask you to translate everyday language into an algebraic equation. Start by letting x represent the unknown number, then build the equation step by step.

应用题要求你将日常语言转化为代数方程。首先设 x 代表未知数，然后逐步构建方程。

For example: “Three times the sum of a number and 5 is 27. Find the number.” Let x be the number. The sum of x and 5 is x + 5. Three times this sum is 3(x + 5). The equation is 3(x + 5) = 27. Expand: 3x + 15 = 27. Subtract 15: 3x = 12. Divide by 3: x = 4.

例如：“一个数与 5 的和的 3 倍等于 27。求这个数。”设这个数为 x。x 与 5 的和是 x + 5。这个和的 3 倍是 3(x + 5)。方程为 3(x + 5) = 27。展开：3x + 15 = 27。减去 15：3x = 12。除以 3：x = 4。

Common keywords include sum (+), difference (−), product (×), and quotient (÷). Watch for phrases like “more than” and “less than” because they often reverse the order of terms.

常见关键词包括 sum（和，+）、difference（差，−）、product（积，×）和 quotient（商，÷）。注意“more than”和“less than”这类短语，因为它们常常会颠倒项的顺序。

11. Common Mistakes | 常见错误

Students often make the same mistakes when solving linear equations. Recognising them can help you avoid losing marks.

学生在解一元一次方程时经常犯同样的错误。认识这些错误可以帮助你避免丢分。

Mistake 1: only changing one side of the equation. Always apply the same operation to both sides. Mistake 2: mishandling negative signs, especially when subtracting a negative term. Mistake 3: dividing by the wrong number or forgetting to divide both terms. Mistake 4: not checking the final answer.

错误一：只改变方程的一边。始终对两边进行相同的运算。错误二：处理负号不当，尤其是在减去负项时。错误三：除以错误的数或忘记将两项都除以。错误四：没有检验最终答案。

For example, in 3x − 5 = 10, the correct first step is to add 5 to both sides, giving 3x = 15, not 3x = 5.

例如，在 3x − 5 = 10 中，正确的第一步是两边加 5，得到 3x = 15，而不是 3x = 5。

12. Practice Summary | 练习总结

To master linear equations, practise regularly and follow a clear method: simplify each side, move variable terms to one side, move constant terms to the other side, isolate the variable, and check your answer.

要掌握一元一次方程，需要定期练习并遵循清晰的方法：化简每一边，将变量项移到一边，将常数项移到另一边，分离出变量，并检验答案。

Use the questions in p60_2.pdf to test yourself. After solving each one, verify your solution by substitution. The more you practise, the faster and more accurate you will become in algebraic manipulation.

使用 p60_2.pdf 中的题目进行自测。解完每一题后，通过代入来验证你的解。练习得越多，你的代数运算速度就越快、准确度就越高。

Once you are confident with these skills, you will be well prepared for more advanced topics such as simultaneous equations, inequalities, and quadratic equations.

一旦你对这些技能充满信心，你就为更高级的主题做好了充分准备，例如联立方程、不等式和二次方程。

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Mastering Linear Equations | 掌握一元一次方程