Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the building blocks of algebra. In KS3 Cambridge Mathematics, students learn to find unknown values by balancing operations on both sides of an equals sign. Mastering this topic is essential for tackling more complex problems, from geometry to data handling.

一元一次方程是代数的基石。在KS3剑桥数学课程中，学生通过等号两边进行平衡运算来求解未知数。掌握这一内容对于处理从几何到数据处理的更复杂问题至关重要。

1. What is an Equation? | 什么是方程？

An equation is a mathematical sentence stating that two expressions have the same value, connected by the equals sign ‘=’. For example, 3x + 4 = 19 is an equation because it claims that the expression 3x+4 is equal to 19.

方程是表明两个表达式具有相同值的数学语句，用等号 ‘=’ 连接。例如 3x + 4 = 19 就是一个方程，它声称表达式 3x+4 等于 19。

The letter x (or any variable) represents an unknown number we need to find. The left-hand side and right-hand side must stay in balance, just like a seesaw.

字母 x（或任意变量）表示我们需要找出的未知数。左边和右边必须像跷跷板一样保持平衡。

2. The Balance Method | 天平法

Think of an equation as a perfectly balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep the scale balanced. This is the golden rule for solving equations.

把方程想像成一个完全平衡的天平。你对一边做的任何操作，必须对另一边做完全相同的操作，以保持天平平衡。这是解方程的黄金法则。

If you add, subtract, multiply or divide by a number on one side, you must perform the identical operation on the other side. This method allows you to isolate the variable step by step.

如果你在一边加、减、乘或除以一个数，你必须在另一边执行完全相同的运算。这种方法让你逐步把变量单独解出来。

3. Solving One-step Equations | 解一步方程

In a one-step equation, there is only one operation separating you from the solution. For example, x + 7 = 15. To undo the addition of 7, subtract 7 from both sides: x + 7 − 7 = 15 − 7, so x = 8.

在一步方程中，你与答案之间只隔着一个运算。例如 x + 7 = 15。要撤销加7，两边同时减7：x + 7 − 7 = 15 − 7，得出 x = 8。

x + 7 = 15 → x = 15 − 7 → x = 8

Similarly, if you have x − 3 = 10, add 3 to both sides: x = 13. For multiplication, 4x = 20, divide both sides by 4: x = 5. For division, x/6 = 2, multiply both sides by 6: x = 12.

类似地，如果你有 x − 3 = 10，两边加3：x = 13。对于乘法，4x = 20，两边除以4：x = 5。对于除法，x/6 = 2，两边乘6：x = 12。

The key is to recognise the inverse operation: addition ↔ subtraction, multiplication ↔ division, squaring ↔ square root, etc.

关键在于识别逆运算：加法 ↔ 减法，乘法 ↔ 除法，平方 ↔ 平方根等。

4. Solving Two-step Equations | 解两步方程

Two-step equations involve two operations, such as 2x + 3 = 11. The aim is to unravel the expression by reversing the order of operations (usually reversing BIDMAS: undo addition/subtraction first, then multiplication/division).

两步方程包含两个运算，例如 2x + 3 = 11。目标是通过颠倒运算顺序（通常颠倒BIDMAS：先撤销加减法，再撤销乘除法）来拆解表达式。

For 2x + 3 = 11, subtract 3 from both sides first: 2x = 8. Then divide both sides by 2: x = 4. Always do the same to both sides.

对于 2x + 3 = 11，先两边减3：2x = 8。然后两边除以2：x = 4。始终对两边做相同的操作。

2x + 3 = 11 → 2x = 8 → x = 4

Another example: x/5 − 2 = 3. Add 2 to both sides: x/5 = 5. Multiply both sides by 5: x = 25.

另一个例子：x/5 − 2 = 3。两边加2：x/5 = 5。两边乘5：x = 25。

5. Equations with Brackets | 带括号的方程

When brackets appear, such as 3(x + 2) = 15, expand them first. Multiply the number outside by each term inside: 3×x + 3×2 = 3x + 6 = 15. Then solve as a two-step equation: 3x = 9, so x = 3.

当出现括号时，例如 3(x + 2) = 15，首先展开它们。用外面的数乘以括号内的每一项：3×x + 3×2 = 3x + 6 = 15。然后按照两步方程来解：3x = 9，所以 x = 3。

Alternatively, you can divide both sides by the factor outside the brackets first if it divides neatly: 3(x + 2) = 15, divide both sides by 3 to get x + 2 = 5, then x = 3. Both approaches are valid.

另一种方法是，如果括号外的因数能整除，可以先两边除以该因数：3(x + 2) = 15，两边除以3得到 x + 2 = 5，然后 x = 3。两种方法都有效。

Be careful with a negative sign outside brackets, e.g. −2(x − 3) = 8. Expand: −2x + 6 = 8, then subtract 6: −2x = 2, divide by −2: x = −1.

注意括号外的负号，例如 −2(x − 3) = 8。展开：−2x + 6 = 8，然后减6：−2x = 2，除以 −2：x = −1。

6. Equations with Fractions | 带分数的方程

Fractions make equations look harder, but the balance method still works. For instance, (2x)/3 = 6. Multiply both sides by 3 to clear the denominator: 2x = 18, then x = 9.

分数让方程看起来更难，但天平法仍然有效。例如 (2x)/3 = 6。两边同乘3以消除分母：2x = 18，然后 x = 9。

If you have an equation like x/4 + 1 = 3, treat it as a two-step equation: subtract 1 from both sides → x/4 = 2, then multiply by 4 → x = 8.

如果你遇到方程 x/4 + 1 = 3，把它当作两步方程：两边减1 → x/4 = 2，然后乘4 → x = 8。

Another common type is (x + 3)/2 = 5. Multiply both sides by 2: x + 3 = 10, then x = 7. You can also think of clearing the fraction first.

另一种常见类型是 (x + 3)/2 = 5。两边乘2：x + 3 = 10，然后 x = 7。你也可以先去除分母。

Always aim to eliminate the fraction early, but ensure you apply the multiplication to every term on both sides.

总是争取尽早去掉分母，但要确保对两边每一项都进行乘法。

7. Equations with Variables on Both Sides | 变量在方程两边

Equations like 5x + 2 = 3x + 10 require collecting like terms. Subtract the smaller x-coefficient term from both sides. Here, subtract 3x: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4.

像 5x + 2 = 3x + 10 这样的方程需要合并同类项。从两边减去系数较小的 x 项。这里，减去 3x：2x + 2 = 10。然后减2：2x = 8，所以 x = 4。

If the variable appears on both sides with negative coefficients, be systematic: e.g. 2x − 5 = 4x + 1. You can subtract 2x from both sides: −5 = 2x + 1. Then subtract 1: −6 = 2x, divide by 2: x = −3.

如果变量在两边都带有负系数，要有条理：例如 2x − 5 = 4x + 1。你可以两边减 2x：−5 = 2x + 1。然后减1：−6 = 2x，除以2：x = −3。

The goal is to get all the variable terms on one side and the constants on the other, using inverse operations.

目标是利用逆运算将所有含变量的项移到一边，常数项移到另一边。

8. Checking Your Solution | 检验你的解

After finding a solution, always substitute it back into the original equation. For x = 4 in 2x + 3 = 11: 2×4 + 3 = 8 + 3 = 11, which matches. If both sides equal the same number, the solution is correct.

求出解之后，一定要把它代回原方程检验。对于 x = 4 代入 2x + 3 = 11：2×4 + 3 = 8 + 3 = 11，吻合。如果两边得出相同的数，解就是正确的。

Checking not only confirms your answer but also helps catch arithmetic slip-ups, such as sign errors or misapplied operations.

检验不仅能确认答案，还能帮你发现算术失误，比如符号错误或运算误用。

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

Many practical problems require writing an equation before solving. Suppose the perimeter of a rectangle is 30 cm, and its length is twice its width. Let w be the width, then length = 2w. The perimeter P = 2(length + width) = 2(2w + w) = 6w. Set 6w = 30, so w = 5 cm, length = 10 cm.

许多实际问题需要先列出方程再求解。假设一个长方形的周长是30 cm，长是宽的两倍。设宽为 w，那么长 = 2w。周长 P = 2(长+宽) = 2(2w + w) = 6w。令 6w = 30，因此 w = 5 cm，长 = 10 cm。

Translating English into algebra is a skill: ‘sum’ means addition, ‘difference’ subtraction, ‘product’ multiplication, ‘quotient’ division. ‘Is’, ‘equals’, ‘the same as’ all indicate the equals sign.

把英语翻译成代数是一项技能：’sum’ 表示加法，’difference’ 减法，’product’ 乘法，’quotient’ 除法。’is’、’equals’、’the same as’ 都表示等号。

10. Common Mistakes to Avoid | 常见错误避免

Forgetting to balance both sides: If you do something to one side, you must do it to the other. 忘记平衡两边：如果你对一边进行操作，必须对另一边进行相同操作。
Mis-handling negative signs: When expanding brackets like −(x − 3), be careful: it becomes −x + 3. 错误处理负号：展开括号如 −(x − 3) 时要小心：变为 −x + 3。
Incorrect inverse order: In x/2 + 3 = 7, some subtract 3 first (correct), others multiply by 2 first (wrong order unless you apply it correctly to the whole left side). 逆运算顺序错误：在 x/2 + 3 = 7 中，一些人先减3（正确），另一些人先乘2（错误，除非你对整个左边正确应用）。
Not checking the solution: A quick substitution would reveal obvious errors. 不检验解：快速代回就能发现明显错误。

11. Practice Tips | 练习技巧

Start with simple one-step equations and gradually increase difficulty. Always write each line of working clearly, showing the operation performed on both sides. Use a new line for each step.

从简单的一步方程开始，逐步增加难度。每步计算都清晰写出，注明在两边执行的操作。每一步换一行。

Draw a line down the middle of your page to visualise the two sides of the equation. This can help students who struggle with layout keep their work organised.

在页面中间画一条竖线，以形象化方程的两边。这能帮助书写布局困难的学生保持整洁。

When you feel confident, try applying equations to real-life situations, like calculating costs, ages, or measures in shapes. This strengthens understanding and shows why algebra matters.

当你感到有信心时，尝试将方程应用于实际情境，比如计算费用、年龄或图形中的度量。这可以加深理解，并展示代数的重要性。

12. Summary | 总结

Solving linear equations relies on one core principle: do the same to both sides. Whether dealing with brackets, fractions, or variables on both sides, the balance method stays consistent. Practise the steps, check your answers, and you will build a solid foundation in algebra.

解一元一次方程依靠一个核心原则：对两边做相同的操作。无论处理括号、分数还是两边都有变量，天平法始终一致。练习步骤，检验答案，你就能在代数中打下坚实的基础。

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Solving Linear Equations | 解一元一次方程