Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the foundation of algebra, allowing us to find unknown values by applying a series of logical operations. Understanding how to solve them is an essential skill for KS3 Cambridge Mathematics, as it builds the bridge between arithmetic and advanced mathematical reasoning.

线性方程是代数的基础，它们让我们能够通过一系列逻辑运算求出未知数的值。掌握如何解方程是剑桥KS3数学的基本技能，因为它架起了算术与进阶数学推理之间的桥梁。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation is an algebraic statement in which the variable (often represented by a letter, such as x) is raised to the power of one. This means the highest exponent of the variable is 1. Graphically, a linear equation always produces a straight line.

线性方程是一个代数等式，其中变量（通常用字母表示，如 x）的指数为 1。也就是说变量的最高次幂为 1。在图像上，线性方程总是产生一条直线。

For example, 2x + 3 = 7 is a linear equation because x appears only as x¹. In contrast, x² + 4 = 13 is not linear, as the power of x is 2.

例如，2x + 3 = 7 是一个线性方程，因为 x 仅以 x¹ 的形式出现。相反，x² + 4 = 13 不是线性方程，因为 x 的指数为 2。

In the Cambridge KS3 syllabus, you will work exclusively with equations where the unknown appears to the first power. The key principle is to isolate the variable on one side of the equation while keeping the equation balanced.

在剑桥KS3大纲中，你只会接触到未知数为一次幂的方程。核心原理是在保持等式平衡的前提下，将变量单独移到等式的一边。

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2. The Balancing Method | 平衡法

Every equation has two sides separated by an equals sign. The balance method states that whatever operation you perform on one side, you must perform exactly the same operation on the other side. This maintains equality.

每个方程都有由等号分隔的两边。平衡法规定，你在方程一边进行的任何运算，必须在另一边进行完全相同的运算，这样才能保持相等。

Think of it as a set of scales: if you add a weight to the left pan, you must add the same weight to the right pan to keep the scales level. Similarly, if you subtract, multiply, or divide, you must apply it to both sides.

可以把它想象成一台天平：如果你在左边的托盘上添加一个砝码，就必须在右边的托盘上也添加同样的砝码，才能使天平保持水平。同样地，减、乘、除操作也都必须在两边同时进行。

When solving an equation, your goal is always to end with the variable on its own on one side, such as x = 5. To achieve this, use inverse operations: addition undoes subtraction, multiplication undoes division, and vice versa.

解方程时，你的目标始终是将变量单独留在等式的一边，例如 x = 5。要做到这一点，就要使用逆运算：加法抵消减法，乘法抵消除法，反过来也一样。

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3. Solving Simple One-step Equations | 解简单的一步方程

One-step equations require only a single operation to isolate the variable. For example, consider x + 4 = 10. The inverse of adding 4 is subtracting 4, so subtract 4 from both sides:

一步方程只需要一次运算就能将变量单独分离出来。例如，考虑方程 x + 4 = 10。加 4 的逆运算是减 4，因此两边同时减去 4：

x + 4 − 4 = 10 − 4, so x = 6.

If the equation is x − 7 = 5, add 7 to both sides because addition undoes subtraction:

如果方程是 x − 7 = 5，那么两边同时加上 7，因为加法可以抵消减法：

x − 7 + 7 = 5 + 7, so x = 12.

When a variable is multiplied, such as 3x = 12, divide both sides by 3 to undo the multiplication:

当变量被乘时，例如 3x = 12，两边同时除以 3 即可抵消乘法：

3x ÷ 3 = 12 ÷ 3, so x = 4.

If the variable is divided, such as x/4 = 2, multiply both sides by 4. In the Cambridge Checkpoint style, write the step clearly:

如果变量被除，例如 x/4 = 2，那么两边同时乘以 4。在剑桥 Checkpoint 的解题风格中，要清楚地写出步骤：

x/4 × 4 = 2 × 4, so x = 8.

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4. Solving Two-step Equations | 解两步方程

Two-step equations involve two operations. A typical example is 2x + 3 = 11. The first step is to eliminate the constant term that is not attached to x. Here, subtract 3 from both sides:

两步方程包含两种运算。典型例子是 2x + 3 = 11。第一步是消去没有附着在 x 上的常数项。这里，两边先减去 3：

2x + 3 − 3 = 11 − 3 → 2x = 8.

The second step is to divide by the coefficient of x, which is 2:

第二步是除以 x 的系数，即 2：

2x ÷ 2 = 8 ÷ 2 → x = 4.

Always reverse the order of operations when solving: deal with addition or subtraction first, then multiplication or division. This ensures a systematic approach.

解方程时始终要逆向运算顺序：先处理加减，再处理乘除。这能保证解题步骤的系统性。

Another form is when the variable appears after a division, such as x/5 − 2 = 3. Start by adding 2 to both sides:

另一种形式是变量出现在除法之后，例如 x/5 − 2 = 3。先从两边同时加 2 开始：

x/5 − 2 + 2 = 3 + 2 → x/5 = 5.

Then multiply both sides by 5:

然后两边同时乘以 5：

x = 5 × 5 = 25.

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5. Equations with Brackets | 带括号的方程

When an equation contains brackets, such as 3(x + 2) = 15, the first priority is to expand the brackets. Multiply each term inside the bracket by the term outside:

当方程中包含括号时，例如 3(x + 2) = 15，首先要展开括号。用括号外的数乘以括号内的每一项：

3(x + 2) → 3 × x + 3 × 2 = 3x + 6.

Now the equation becomes 3x + 6 = 15, which is a two-step equation. Subtract 6 from both sides, then divide by 3:

现在方程变成了 3x + 6 = 15，这是一个两步方程。两边减去 6，然后除以 3：

3x = 9 → x = 3.

In some cases, you might have a negative sign before the bracket, like 5 − 2(x − 1) = 3. Be careful: the negative sign belongs to the term outside, so expand as −2(x − 1) = −2x + 2. The equation becomes:

有时括号前会有一个负号，例如 5 − 2(x − 1) = 3。要小心：这个负号和括号外的数是一个整体，所以展开为 −2(x − 1) = −2x + 2。方程变为：

5 − 2x + 2 = 3 → 7 − 2x = 3.

Then solve by subtracting 7 from both sides and dividing by −2. This attention to sign rules is crucial for Cambridge KS3 assessments.

然后两边减去 7，再除以 −2 求解。这种对符号规则的关注在剑桥KS3评估中非常重要。

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6. Equations with Variables on Both Sides | 含有两边变量的方程

When the unknown appears on both sides of the equation, like 5x + 2 = 3x + 10, the strategy is to collect all variable terms on one side and all constants on the other.

当未知数出现在方程的两边时，例如 5x + 2 = 3x + 10，解题策略是把所有含有变量的项移到一边，所有常数移到另一边。

Start by subtracting the smaller variable term from both sides. Here, subtract 3x from each side:

先从两边减去较小的变量项开始。这里，两边减去 3x：

5x − 3x + 2 = 3x − 3x + 10 → 2x + 2 = 10.

Now it is a simple two-step equation: subtract 2, then divide by 2, to obtain x = 4.

现在它变成了一个简单的两步方程：减去 2，再除以 2，得到 x = 4。

If the variable term with the smaller coefficient is on the right, you can still operate the same way, or you could add the variable term to both sides to avoid negative coefficients. Both methods are acceptable.

如果系数较小的变量项在右边，同样可以这样操作，或者也可以将变量项移到左边以避免出现负系数。两种方法都可以接受。

Example: 8 − 3x = 2x + 3. Add 3x to both sides to bring variables together:

例如：8 − 3x = 2x + 3。两边同时加上 3x 以合并变量项：

8 = 5x + 3, then subtract 3 → 5 = 5x, so x = 1.

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7. Forming Equations from Word Problems | 从应用题建立方程

Many KS3 Cambridge questions require you to translate a written situation into an equation. Look for key phrases: ‘sum’ means addition, ‘difference’ means subtraction, ‘product’ means multiplication, and ‘quotient’ or ‘shared equally’ indicates division.

许多剑桥KS3题目要求你将文字情境转化为方程。请注意关键词：“和”意味着加法，“差”意味着减法，“积”意味着乘法，“商”或“平均分”表示除法。

For instance: ‘I think of a number, double it, and add 7; the result is 23.’ Let the unknown number be n. Then 2n + 7 = 23.

例如：“我心里想了一个数，把它乘以 2，再加上 7，结果是 23。”设这个未知数为 n，那么方程为 2n + 7 = 23。

Solve it: subtract 7 from both sides → 2n = 16, then divide by 2 → n = 8. Always check your answer by substituting back into the word problem.

解方程：两边减 7 → 2n = 16，再除以 2 → n = 8。一定要将答案代回原来的文字情境中进行检验。

Age problems are also common: ‘Sam is 4 years older than Tom. The sum of their ages is 22. Find Tom’s age.’ Let Tom’s age be t, then Sam’s age is t + 4. The equation: t + (t + 4) = 22 → 2t + 4 = 22 → 2t = 18 → t = 9. Tom is 9, Sam is 13.

年龄问题也很常见：“山姆比汤姆大 4 岁。他们的年龄之和是 22 岁。求汤姆的年龄。”设汤姆的年龄为 t，则山姆的年龄为 t + 4。方程为：t + (t + 4) = 22 → 2t + 4 = 22 → 2t = 18 → t = 9。汤姆 9 岁，山姆 13 岁。

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8. Checking Your Solution | 验证你的解

Substitution is the most reliable way to verify your answer. Take the value you found for the variable and plug it into the original equation, simplifying each side separately. If both sides are equal, your solution is correct.

代入法是最可靠的验证方法。把你求出的变量值代入原方程，分别化简方程的左右两边。如果两边相等，你的解就是正确的。

For the equation 3(x − 2) = x + 4, suppose you found x = 5. Check left side: 3(5 − 2) = 3 × 3 = 9. Right side: 5 + 4 = 9. Both equal 9, so x = 5 is correct.

对于方程 3(x − 2) = x + 4，假设你求出 x = 5。检验左边：3(5 − 2) = 3 × 3 = 9。右边：5 + 4 = 9。两边都等于 9，所以 x = 5 正确。

In Cambridge Checkpoint exams, checking not only confirms accuracy but can also help you spot arithmetic errors before moving to the next question.

在剑桥 Checkpoint 考试中，验算不仅能确认答案的准确性，还能在继续做下一题之前发现计算错误。

If the two sides give different numbers, retrace your steps carefully. Look for sign errors, mishandling of brackets, or simple arithmetic mistakes.

如果两边得出的数字不同，就要仔细回溯你的解题步骤。检查符号错误、括号处理不当或简单的算术错误。

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9. Common Mistakes and How to Avoid Them | 常见错误及如何避免

Many students forget to keep the equation balanced. For example, dividing only part of one side or adding a number to only one term breaks the equality.

许多学生会忘记保持等式平衡。例如，只除了一边的某一部分，或只给某一项加上一个数，都会破坏等式的平衡。

Another frequent mistake is mishandling negative signs when expanding brackets. Remember that −1(x − 3) expands to −x + 3, not −x − 3.

另一个常见错误是展开括号时处理不好负号。牢记 −1(x − 3) 展开后是 −x + 3，而不是 −x − 3。

Errors also occur when solving equations with fractions. To eliminate a fraction like x/3, multiply the whole term by 3, not just the variable part. Write (x/3) × 3 = x, but if there are other terms, multiply them too.

解含有分数的方程时也容易出错。为了消去像 x/3 这样的分数，要将整个项乘以 3，而不只是变量部分。写为 (x/3) × 3 = x，但如果还有其他项，也必须同时乘以 3。

A useful strategy is to use clear steps and write each new line directly below the previous one, showing your operation on both sides. This makes it easier to track your reasoning.

一个有用的策略是步骤要清晰，并且把每新的一行直接写在上一行的下方，同时展示你对两边所做的运算。这样更容易追踪你的推理过程。

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10. Practice Examples with Step-by-step Solutions | 带分步解答的练习题

Let’s work through a range of examples to solidify the techniques. Try each one before reading the solution.

我们来做一些不同种类的例题，以巩固所学技巧。在看答案之前，先自己尝试解答每一题。

Equation / 方程	Solution Steps / 解题步骤
4x − 7 = 9	Add 7 to both sides: 4x = 16. Divide by 4: x = 4.
2(3x + 1) = 20	Expand: 6x + 2 = 20. Subtract 2: 6x = 18. Divide by 6: x = 3.
5x + 3 = 2x + 12	Subtract 2x: 3x + 3 = 12. Subtract 3: 3x = 9. Divide by 3: x = 3.
x/4 + 5 = 7	Subtract 5: x/4 = 2. Multiply by 4: x = 8.
3(x − 2) = 2(x + 1)	Expand: 3x − 6 = 2x + 2. Subtract 2x: x − 6 = 2. Add 6: x = 8.

Repeat these examples and invent your own to build fluency. Regular practice is the key to mastering linear equations for KS3 Cambridge Mathematics.

反复练习这些例题，并自己编题练习，以提高熟练度。定期练习是在剑桥KS3数学中掌握线性方程的关键。

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11. Real-life Applications of Linear Equations | 线性方程的实际应用

Linear equations are not just abstract schoolwork; they model real-world situations like calculating costs, distances, and conversions. For example, a mobile phone plan charges a fixed monthly fee plus a rate per minute. The total cost can be expressed as a linear equation, and you can solve for unknown usage.

线性方程并不只是抽象的学校作业，它们能模拟现实世界中的各种情境，比如计算费用、距离和换算。例如，某手机套餐收取固定月租和每分钟通话费。总费用可以表示为一个线性方程，你可以通过它求出未知的通话时长。

In science experiments, relationships such as speed = distance / time often lead to linear equations when two of the three quantities are known. Solving equations thus becomes a tool for inquiry.

在科学实验中，速度 = 距离 / 时间 这样的关系，当三个量中有两个已知时，常常会引出线性方程。因此，解方程就成为探究问题的一种工具。

Being able to form and solve equations empowers you to analyse situations quantitatively and make informed decisions, whether you are budgeting your pocket money or planning a journey.

能够建立并求解方程，会让你有能力定量地分析各种情况并做出明智的决定，无论是规划零花钱的预算，还是安排一次出行。

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12. Summary and Key Points | 总结与要点

To recap, solving linear equations involves isolating the unknown using inverse operations while keeping the equation balanced. Always remove brackets first, then collect like terms, and deal with addition/subtraction before multiplication/division.

概括来说，解线性方程就是要在保持方程式平衡的前提下，运用逆运算将未知数分离出来。始终要先去掉括号，再合并同类项，并且在乘除之前先处理加减。

Check your final answer by substituting it back into the original problem. Practice regularly, pay attention to negative signs, and write every step clearly. These habits will serve you well in Cambridge KS3 Mathematics and beyond.

通过将最终答案代回原题来检验。要经常练习，注意负号，并且把每一步都清楚地写出来。这些习惯将在剑桥KS3数学以及更远的未来让你受益匪浅。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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