📚 Cambridge IGCSE Mathematics: Solving Linear Equations and Inequalities | 剑桥 IGCSE 数学：解一元一次方程与不等式
e more advanced topics, including simultaneous equations, quadratic equations, and graphical methods. This article explains the key techniques step by step, with clear examples, common errors, and exam-focused advice.
一元一次方程与不等式是剑桥 IGCSE 数学课程中代数的基石。熟练掌握求解 2x + 3 = 11 这类方程以及 3x − 5 ≤ 7 这类不等式，将帮助你攻克更高级的主题，包括联立方程、二次方程和图像法。本文逐步讲解核心技巧，并配有清晰的示例、常见错误和考试导向的建议。
1. What Is a Linear Equation? | 什么是一元一次方程？
in the form ax + b = 0, where a and b are constants, a ≠ 0, and x is the unknown. The word 'linear' For example, 4x + 7 = 19 is linear, but x² + 3 = 12 is not linear because it contains x².
一元一次方程是指可以写成 ax + b = 0 形式的方程，其中 a 和 b 为常数，a ≠ 0，x 为未知数。'线性'这个词表示 x 的最高次数为 1。例如，4x + 7 = 19 是线性的，但 x² + 3 = 12 不是线性方程，因为它含有 x²。
3 + 7 = 19. You can check any solution by substituting it back into the original equation.
一元一次方程的解是使方程成立的 x 值。对于 4x + 7 = 19，解为 x = 3，因为 4 × 3 + 7 = 19。你可以将解代回原方程来检验任何解是否正确。
ax + b = 0 ⇒ x = −b ÷ a
2. The Balancing Method | 天平法：等式平衡法
Think of an equation as a balanced set of scales. Whatever you do to one side, you must do to the other side to keep the balance. This is called the balancing method. If you add, subtract, multiply, or divide one side by a number, you must do the same to the other side.
可以把方程想象成一组平衡的天平。你对一侧所做的任何操作，都必须对另一侧做相同的操作，以保持平衡。这称为等式平衡法。如果你对一侧加、减、乘或除以一个数，你必须对另一侧进行相同的运算。
This idea is the foundation of all equation solving. It ensures that the equality remains true while you isolate the variable. Avoid 'moving' terms across the equals sign without understanding the operation involved.
这一思想是所有方程求解的基础。它确保你在隔离变量时等式仍然成立。避免在不理解运算含义的情况下把项'移'过等号。
Add the same amount to both sides: x − 5 = 2 ⇒ x = 7
两边同加一个数：x − 5 = 2 ⇒ x = 7
Subtract the same amount from both sides: x + 4 = 9 ⇒ x = 5
两边同减一个数：x + 4 = 9 ⇒ x = 5
Multiply both sides by the same amount: x ÷ 3 = 6 ⇒ x = 18
两边同乘一个数：x ÷ 3 = 6 ⇒ x = 18
Divide both sides by the same non-zero amount: 5x = 20 ⇒ x = 4
两边同除以一个非零数：5x = 20 ⇒ x = 4
3. Solving Basic Linear Equations | 解基础一元一次方程
To solve a basic linear equation, collect all variable terms on one side and all constant terms on the other side. Perform inverse operations in the correct order: first undo addition or subtraction, then undo multiplication or division.
解基础一元一次方程时，将所有含变量的项移到一侧，将所有常数项移到另一侧。按正确顺序进行逆运算：先消去加法或减法，再消去乘法或除法。
Example: Solve 2x + 3 = 11.
示例：解方程 2x + 3 = 11。
2x + 3 = 11 → 2x = 8 → x = 4
both sides by 2 to obtain x = 4. Always check: 2 × 4 + 3 = 11, so the solution is correct.
首先两边减去 3，隔离 2x 项，得到 2x = 8。然后两边除以 2，得到 x = 4。始终检验：2 × 4 + 3 = 11，因此解是正确的。
4. Equations with Brackets | 含括号的方程
= ab + ac. After expansion, collect like terms and then solve using the balancing method.
当方程含有括号时，第一步通常是使用分配律展开括号：a(b + c) = ab + ac。展开后，合并同类项，然后使用平衡法求解。
Example: Solve 3(x + 2) = 21.
示例：解方程 3(x + 2) = 21。
3(x + 2) = 21 → 3x + 6 = 21 → 3x = 15 → x = 5
so divide both sides by 3 first: x + 2 = 7, then x = 5. Both methods give the same result.
将 3(x + 2) 展开为 3x + 6。两边减去 6，然后除以 3。你也可以先两边除以 3：x + 2 = 7，然后 x = 5。两种方法得到相同结果。
5. Unknowns on Both Sides | 两边都含未知数的方程
If a linear equation has variable terms on both sides, choose one side to collect all the variable terms. It is often easiest to keep the variable term on the left-hand side, but either side works as long as you use inverse operations correctly.
如果一元一次方程两边都含变量项，选择一侧来集中所有变量项。通常将变量项保留在左侧最方便，但无论哪一侧都可以，只要正确使用逆运算即可。
Example: Solve 5x + 4 = 2x + 13.
示例：解方程 5x + 4 = 2x + 13。
5x + 4 = 2x + 13 → 5x − 2x = 13 − 4 → 3x = 9 → x = 3
stants on the right. Divide by 3 to find x = 3. Check: 5 × 3 + 4 = 19 and 2 × 3 + 13 = 19.
两边减去 2x，将 x 项集中到左侧。然后两边减去 4，将常数集中到右侧。除以 3 得到 x = 3。检验：5 × 3 + 4 = 19，2 × 3 + 13 = 19。
6. Equations with Fractions | 含分数的方程
To solve an equation containing fractions, multiply every term by the lowest common denominator (LCD) of all fractions. This clears the fractions and produces a simpler equation. Be careful to multiply both the variable terms and the constant terms.
解含有分数的方程时，将每一项都乘以所有分数的最小公分母（LCD）。这样可以消去分数，得到一个更简单的方程。注意变量项和常数项都要乘。
Example: Solve x/3 + 1 = 5/6.
示例：解方程 x/3 + 1 = 5/6。
x/3 + 1 = 5/6 → 2x + 6 = 5 → 2x = −1 → x = −½
nd (5/6) × 6 = 5. Then solve 2x + 6 = 5 to get x = −½. Substitution confirms the solution.
3 和 6 的最小公分母是 6。将每一项都乘以 6：(x/3) × 6 = 2x，1 × 6 = 6，(5/6) × 6 = 5。然后解 2x + 6 = 5 得到 x = −½。代入检验可确认解的正确性。
7. Introduction to Inequalities | 不等式入门
y has a range of solutions. For example, x > 3 means every number greater than 3 is a solution, including 3.1, 4, and 100.
不等式使用符号 <、>、≤ 或 ≥ 来比较两个表达式。与方程不同，不等式有一系列解。例如，x > 3 表示大于 3 的每个数都是解，包括 3.1、4 和 100。
| Symbol | Meaning | Example |
| < | less than 小于 | x < 5 |
| > | greater than 大于 | x > −2 |
| ≤ | less than or equal to 小于等于 | x ≤ 8 |
| ≥ | greater than or equal to 大于等于 | x ≥ 0 |
r >), and closed circles for inclusive inequalities (≤ or ≥) when drawing number lines.
在数轴上表示时，严格不等式（< 或 >）用空心圆，包含不等式（≤ 或 ≥）用实心圆。
8. Solving Linear Inequalities | 解一元一次不等式
Solving a linear inequality is very similar to solving a linear equation. You can add, subtract, multiply, or divide both sides by a positive number without changing the direction of the inequality. However, multiplying or dividing both sides by a negative number reverses the inequality sign.
解一元一次不等式与解一元一次方程非常相似。你可以在两边进行加、减、乘或除以一个正数，而不改变不等号的方向。但是，两边同乘或同除以一个负数时，不等号方向要改变。
Example: Solve 4x − 7 ≤ 9.
示例：解不等式 4x − 7 ≤ 9。
4x − 7 ≤ 9 → 4x ≤ 16 → x ≤ 4
Add 7 to both sides, then divide by 4. Since 4 is positive, the inequality sign stays the same. The solution is all real numbers less than or equal to 4.
两边加 7，然后除以 4。由于 4 是正数，不等号方向保持不变。解为所有小于等于 4 的实数。
Example: Solve −2x < 10.
示例：解不等式 −2x < 10。
−2x < 10 → x > −5
Divide both sides by −2. Because you divide by a negative number, the < sign becomes >. So the solution is x > −5. This is the most common error in inequalities, so underline the sign change in your working.
两边除以 −2。因为除以了负数，< 号变为 >。因此解为 x > −5。这是不等式中最常见的错误，因此在解题过程中请标记符号变化。
9. Graphing Inequalities on a Number Line | 在数轴上表示不等式
or inclusive inequalities (≤ or ≥). Shade the region that contains all possible solutions.
不等式可以在数轴上直观地表示。严格不等式（< 或 >）使用空心圆，包含不等式（≤ 或 ≥）使用实心圆。将包含所有可能解的区域涂上阴影。
Example: Represent x ≥ −1 on a number line.
示例：在数轴上表示 x ≥ −1。
Draw a number line, mark −1 with a closed circle, and shade to the right because x is greater than or equal to −1. For x < 2, draw an open circle at 2 and shade to the left.
画一条数轴，在 −1 处标实心圆，并向右涂阴影，因为 x 大于等于 −1。对于 x < 2，在 2 处画空心圆，并向左涂阴影。
Some questions ask you to write the inequality shown on a number line. Check the type of circle and the direction of the shading carefully.
有些题目要求写出数轴上显示的不等式。请仔细检查圆的类型和阴影方向。
10. Compound Inequalities | 复合不等式
than 2 and less than or equal to 5. You can solve compound inequalities by isolating the variable in the middle.
复合不等式将两个不等式合并为一个表达式，例如 2 < x ≤ 5。这表示 x 大于 2 且小于等于 5。你可以通过将变量隔离在中间来解复合不等式。
Example: Solve 3 < 2x + 1 ≤ 9.
示例：解不等式 3 < 2x + 1 ≤ 9。
3 < 2x + 1 ≤ 9 → 2 < 2x ≤ 8 → 1 < x ≤ 4
Subtract 1 from all three parts, then divide every part by 2. The solution is x greater than 1 and less than or equal to 4. On a number line, use an open circle at 1 and a closed circle at 4, with shading between them.
从三部分同时减去 1，然后将每一部分都除以 2。解为 x 大于 1 且小于等于 4。在数轴上，1 处用空心圆，4 处用实心圆，两者之间涂阴影。
11. Common Mistakes and How to Avoid Them | 常见错误与如何避免
Many marks are lost in algebra through avoidable errors. The most frequent mistakes include forgetting to change the inequality sign when dividing by a negative, incorrectly expanding brackets, and performing operations on only one side of the equation.
代数中许多失分来自可避免的错误。最常见的错误包括：除以负数时忘记改变不等号方向、括号展开错误，以及只对等式一侧进行运算。
Always check your solution by substituting it into the original equation or inequality.
始终将解代入原方程或不等式进行检验。
Show every step of working. Jumping steps increases the chance of sign errors.
展示每一步解题过程。跳步会增加符号错误的风险。
When multiplying or dividing an inequality by a negative number, flip the inequality sign immediately.
当不等式两边同乘或同除以负数时，立即改变不等号方向。
For fraction equations, multiply every term by the LCD, not just the fractions.
对于分数方程，将每一项都乘以最小公分母，而不仅仅是分数项。
Careful layout and consistent checking will help you avoid these pitfalls in the exam.
清晰的书写布局和持续的检验能帮助你在考试中避开这些陷阱。
12. Exam-Style Practice and Tips | 考试题型练习与技巧
Cambridge IGCSE questions often test linear equations and inequalities in real-life contexts, such as temperature, money, and measurements. You may need to form an equation from a word problem, solve it, and interpret the answer.
剑桥 IGCSE 考试常在实际生活情境中考查一元一次方程和不等式，例如温度、金钱和测量。你可能需要根据文字题建立方程，求解并解释答案。
Example: A taxi charges a fixed fee of $3 plus $2 per kilometre. If the total cost is $15, find the distance travelled.
示例：一辆出租车收取固定费用 3 美元，每公里加收 2 美元。如果总费用为 15 美元，求行驶距离。
3 + 2d = 15 → 2d = 12 → d = 6 km
n the exam, state the units clearly and check that your answer makes sense in the context.
设 d 为以公里为单位的距离。方程为 3 + 2d = 15。解方程得到 d = 6。在考试中，请清楚写出单位，并检查答案在情境中是否合理。
For inequality word problems, phrases like 'at least' mean ≥, 'at most' mean ≤, 'more than' mean >, and 'less than' mean <. Convert these phrases accurately before solving.
对于不等式文字题，'至少'意为 ≥，'至多'意为 ≤，'多于'意为 >，'少于'意为 <。在解题前准确转换这些表达。
Master these techniques and practise past paper questions to build speed and confidence for your Cambridge IGCSE Mathematics examination.
掌握这些技巧并练习历年真题，为你的剑桥 IGCSE 数学考试提高速度和信心。
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