Solving Linear Equations and Inequalities for Cambridge KS3 | 剑桥 KS3 线性方程与不等式解法

📚 Solving Linear Equations and Inequalities for Cambridge KS3 | 剑桥 KS3 线性方程与不等式解法

Linear equations and inequalities form a central part of the Cambridge KS3 Mathematics curriculum. They give learners the tools to represent unknown quantities, solve real-world problems, and build the algebra skills needed for IGCSE. This article explains key methods step by step, from simple one-step equations to inequalities, with exam-style guidance.

线性方程和不等式是剑桥 KS3 数学课程的核心内容。它们为学习者提供了表示未知量、解决实际问题的工具，并培养 IGCSE 阶段所需的代数能力。本文将逐步讲解关键方法，从简单的一步方程到不等式，并配套考试风格的指导。

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

A linear equation is an algebraic statement where two expressions are equal, and the unknown variable has an exponent of 1. For example, x + 5 = 9 is linear, while x² + 3 = 7 is not linear because x has a power of 2. Linear equations can always be rearranged into the form ax + b = c, where a, b and c are numbers.

线性方程是两个表达式相等并且未知变量的指数为 1 的代数等式。例如，x + 5 = 9 是线性方程，而 x² + 3 = 7 不是线性方程，因为 x 的指数是 2。线性方程总能整理成 ax + b = c 的形式，其中 a、b 和 c 是数字。

x + 5 = 9

In this equation the unknown is x. Solving the equation means finding the value of x that makes the statement true.

在这个方程中，未知数是 x。解方程意味着找出使等式成立的 x 的值。

2. Balancing Method: The Core Rule | 平衡法：核心规则

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other. This preserves equality. If you add, subtract, multiply or divide one side by a number, you must perform the same operation on the other side.

把方程想象成一台天平。无论你对等式的一边做什么，都必须对另一边做同样的操作。这样才能保持相等。如果你对等式的一边加、减、乘或除以一个数，你必须在另一边也做同样的运算。

If x + 5 = 9, then x + 5 − 5 = 9 − 5

This balance rule is the foundation of all equation solving. It helps you isolate the variable without changing the meaning of the equation.

这个平衡规则是所有方程求解的基础。它帮助你在不改变方程含义的情况下分离出变量。

3. Solving Equations with Addition and Subtraction | 用加减法解方程

To solve an equation like x + 7 = 12, subtract 7 from both sides. This removes the +7 on the left and leaves x alone. To solve x − 4 = 10, add 4 to both sides.

要解 x + 7 = 12 这样的方程，需要在两边同时减去 7。这样会消去左边的 +7，使 x 单独留下。要解 x − 4 = 10，则在两边同时加上 4。

x + 7 − 7 = 12 − 7 → x = 5

x − 4 + 4 = 10 + 4 → x = 14

Always check your answer by substituting it back into the original equation. If x = 5, then 5 + 7 = 12, which is correct.

始终把答案代回原方程进行检验。如果 x = 5，那么 5 + 7 = 12，这是正确的。

4. Solving Equations with Multiplication and Division | 用乘除法解方程

When a variable is multiplied by a number, divide both sides by that number. For example, if 4x = 20, divide both sides by 4. When the variable is divided by a number, multiply both sides by that number.

当变量乘以一个数时，两边同时除以这个数。例如，如果 4x = 20，两边同时除以 4。当变量除以一个数时，两边同时乘以这个数。

4x ÷ 4 = 20 ÷ 4 → x = 5

x ÷ 3 × 3 = 6 × 3 → x = 18

Remember that division and multiplication are inverse operations. Using the inverse operation on both sides is the quickest way to isolate the unknown.

记住，除法和乘法是互逆运算。在两边同时使用逆运算是分离未知数最快的方法。

5. Two-Step Equations | 两步方程

Some equations require two operations. For example, 2x + 3 = 11 has both multiplication and addition. First undo the addition by subtracting 3 from both sides. Then undo the multiplication by dividing both sides by 2.

有些方程需要两步运算。例如，2x + 3 = 11 同时包含乘法和加法。首先通过两边同时减去 3 来消除加法，然后通过两边同时除以 2 来消除乘法。

2x + 3 − 3 = 11 − 3 → 2x = 8

2x ÷ 2 = 8 ÷ 2 → x = 4

You can check this by substituting x = 4: 2 × 4 + 3 = 8 + 3 = 11, so the solution is correct.

你可以通过代入 x = 4 来检验：2 × 4 + 3 = 8 + 3 = 11，所以解是正确的。

6. Equations with Brackets | 含括号的方程

If an equation contains brackets, you can either expand them first or divide both sides by the coefficient outside the bracket. For example, in 3(x + 2) = 15, dividing both sides by 3 is often faster than expanding.

如果方程含有括号，你可以先展开括号，或者将两边同时除以括号外的系数。例如，在 3(x + 2) = 15 中，两边同时除以 3 通常比展开更快。

3(x + 2) ÷ 3 = 15 ÷ 3 → x + 2 = 5

x + 2 − 2 = 5 − 2 → x = 3

Sometimes expanding first is useful, especially when the bracket has more than one term. For example, 2(x − 4) = 10 becomes 2x − 8 = 10 after expanding.

有时先展开括号也很有用，特别是当括号内有不止一个项时。例如，2(x − 4) = 10 展开后变成 2x − 8 = 10。

7. Equations with Unknowns on Both Sides | 未知数在等号两边的方程

If an equation has the unknown on both sides, collect all x terms on one side and all number terms on the other. For example, 5x + 2 = 3x + 10. Subtract 3x from both sides to move the x terms together.

如果方程中的未知数在等号两边，需要将所有含 x 的项移到一边，将所有数字项移到另一边。例如，5x + 2 = 3x + 10。两边同时减去 3x，把含 x 的项集中在一起。

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

2x + 2 − 2 = 10 − 2 → 2x = 8 → x = 4

Always be careful with signs when moving terms. A positive term becomes negative if it moves to the other side, and vice versa.

移项时要特别注意符号。一个正项移到另一边就变成负项，反之亦然。

8. Introduction to Inequalities | 不等式入门

Inequalities compare two expressions using symbols such as <, >, ≤ and ≥. Unlike equations, they often have many possible solutions. For example, x > 3 means all numbers greater than 3, not just one value.

不等式使用 <、>、≤ 和 ≥ 等符号来比较两个表达式。与方程不同，不等式通常有许多可能的解。例如，x > 3 表示所有大于 3 的数，而不仅仅是一个值。

x > 3

The symbol ≤ means ‘less than or equal to’, and ≥ means ‘greater than or equal to’. These are used when the boundary value is included in the solution.

符号 ≤ 表示 ‘小于或等于’，≥ 表示 ‘大于或等于’。当边界值包含在解中时，就使用这些符号。

9. Solving Simple Inequalities | 解简单不等式

Solving inequalities works in the same way as solving equations. However, there is one key rule: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

解不等式的方法与解方程相同。但是有一个关键规则：如果两边同时乘以或除以一个负数，必须反转不等号的方向。

−2x < 8

Divide both sides by −2 and reverse the sign:

两边同时除以 −2，并反转不等号：

−2x ÷ (−2) > 8 ÷ (−2) → x > −4

This rule is important because negative numbers change the order of values. For instance, −5 is less than −4, so reversing the sign keeps the statement true.

这条规则很重要，因为负数会改变数值的大小顺序。例如，−5 小于 −4，所以反转不等号才能保持不等式成立。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

A common error is to perform an operation on only one side of the equation. This breaks the balance rule and leads to a wrong answer. Always apply the same operation to both sides.

一个常见错误是只对等式的一侧进行运算。这会破坏平衡规则并导致错误答案。必须始终对两边同时进行相同的运算。

Another common mistake is forgetting to reverse the inequality sign when dividing by a negative number. In exam questions, highlight the negative divisor to remind yourself.

另一个常见错误是在除以负数时忘记反转不等号。在考试题目中，可以高亮标记负数除数来提醒自己。

Finally, always check your solution by substituting it back into the original equation or inequality. This takes only a few seconds and can help you spot simple arithmetic errors before you move on.

最后，始终将解代回原方程或不等式进行检验。这只需要几秒钟，却可以帮助你在继续作答前发现简单的算术错误。

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Solving Linear Equations and Inequalities for Cambridge KS3 | 剑桥 KS3 线性方程与不等式解法