📚 Solving Linear Equations and Inequalities | 解一元一次方程与不等式
Linear equations are one of the most important building blocks in Cambridge KS3 mathematics. They appear in many real-life problems, from splitting a bill to calculating unknown measurements. In this article, you will learn how to solve simple and more advanced linear equations, as well as how to work with inequalities and represent them on a number line. Clear steps and careful checking will help you avoid common errors and gain confidence in algebra.
线性方程是剑桥 KS3 数学中最重要的基础模块之一。它们在许多现实生活问题中都会出现，从分摊账单到计算未知长度。本文将带你学习如何解简单和更复杂的一元一次方程，以及如何处理不等式并在数轴上表示它们。清晰的步骤和认真检查将帮助你避免常见错误，并增强代数信心。
1. What is a Linear Equation? | 什么是线性方程？
mple, 3x + 2 = 14 is a linear equation because x appears only to the first power. The word 'linear' comes from the fact that its graph is always a straight line. Your goal is to find the value of x that makes the equation true.
线性方程是指未知数（通常写作 x）的最高次数不超过 1 的方程。例如，3x + 2 = 14 就是线性方程，因为 x 只出现一次方。“线性”一词来源于它的图像总是一条直线。你的目标是找出使方程成立的 x 的值。
3x + 2 = 14
2. The Balancing Method | 等式平衡法
To solve an equation, you must keep both sides balanced. Think of an equation as a set of scales. Whatever you do to one side, you must also do to the other side. This means you can add, subtract, multiply or divide both sides by the same non-zero number without changing the solution.
解方程时，你必须保持两边平衡。可以把方程想象成一架天平。你对一边做什么运算，另一边也必须做同样的运算。这意味着你可以在两边同时加、减、乘或除以同一个非零数，而不会改变方程的解。
For example, to solve x + 5 = 12, subtract 5 from both sides to get x = 7.
例如，解 x + 5 = 12 时，两边同时减去 5，得到 x = 7。
3. Solving Two-Step Equations | 解两步方程
Many KS3 equations require two steps. First undo the addition or subtraction, then undo the multiplication or division. Always follow the reverse order of operations. Start by removing the number added or subtracted, then deal with the coefficient of x.
许多 KS3 方程需要两步来解。首先消去加法或减法，然后再消去乘法或除法。始终按照运算顺序的逆序来进行。先去掉加减的常数项，再处理 x 前面的系数。
1: subtract 2 from both sides to get 3x = 12. Step 2: divide both sides by 3 to get x = 4.
例如，解 3x + 2 = 14。第一步：两边减 2，得 3x = 12。第二步：两边除以 3，得 x = 4。
4. Expanding Brackets First | 先展开括号
ves 2x + 6. After expanding, solve the equation using the two-step method described above.
如果方程含有括号，应先用分配律展开。例如，2(x + 3) 表示 2 × x + 2 × 3，得到 2x + 6。展开后再用前面介绍的两步法来解方程。
mes 2x + 6 = 16. Subtract 6 from both sides to get 2x = 10, then divide by 2 to get x = 5.
例如，2(x + 3) = 16 变为 2x + 6 = 16。两边减 6 得 2x = 10，再除以 2 得 x = 5。
5. Variables on Both Sides | 两边含未知数
When x appears on both sides of the equation, collect all x terms on one side by adding or subtracting. Then collect the number terms on the other side. This often takes three or four clear steps, so write each step neatly to avoid mistakes.
当 x 同时出现在方程两边时，通过加法或减法把含 x 的项集中到一边，再把数字项集中到另一边。这通常需要三到四个清晰的步骤，因此请把每一步写整齐，以免出错。
es to get 3x + 1 = 10. Subtract 1 from both sides to get 3x = 9. Divide by 3 to get x = 3.
例如，解 5x + 1 = 2x + 10。两边减 2x 得 3x + 1 = 10。两边减 1 得 3x = 9。除以 3 得 x = 3。
6. Inequality Symbols | 不等式符号
r equal to. Solving inequalities is similar to solving equations, but you must remember one special rule when multiplying or dividing by a negative number.
不等式通过符号来比较两个表达式：< 表示小于，> 表示大于，≤ 表示小于等于，≥ 表示大于等于。解不等式与解方程类似，但必须记住在乘以或除以负数时有一条特殊规则。
| Symbol | 符号 | Meaning | 含义 |
| < | less than | 小于 |
| > | greater than | 大于 |
| ≤ | less than or equal to | 小于等于 |
| ≥ | greater than or equal to | 大于等于 |
7. Solving Linear Inequalities | 解线性不等式
To solve an inequality, use the same balancing method as for equations. However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, −2x < 6 becomes x > −3 after dividing both sides by −2.
解不等式时，使用与解方程相同的平衡法。但是，如果你在两边乘以或除以一个负数，必须反转不等号。例如，−2x < 6 两边同时除以 −2 后变成 x > −3。
3 to get x ≤ 3. Since we divided by a positive number, the inequality sign stays the same.
另一个例子：解 3x + 1 ≤ 10。两边减 1 得 3x ≤ 9。两边除以 3 得 x ≤ 3。因为除以的是正数，所以不等号方向不变。
8. Number Line Representations | 数轴表示
≥ because the endpoint is included. Draw an arrow in the direction of all possible values.
不等式可以在数轴上表示。严格不等式（< 或 >）用空心圆表示，因为端点不包含在内。≤ 或 ≥ 用实心圆表示，因为端点包含在内。用箭头表示所有可能取值的方向。
o the right. x ≤ −1 is shown with a closed circle at −1 and an arrow pointing to the left.
例如，x > 2 在数轴上用 2 处的空心圆和指向右边的箭头表示。x ≤ −1 用 −1 处的实心圆和指向左边的箭头表示。
9. Common Mistakes | 常见错误
One common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Another is performing an operation on only one side of the equation. Always write each step clearly and check your answer by substituting it back into the original equation or inequality.
一个常见错误是在乘或除以负数时忘记反转不等号。另一个错误是只在方程的一边进行运算。请始终清晰写出每一步，并通过将答案代入原方程或不等式来检查。
For example, if you solve −4x > 12, you must divide by −4 and reverse the sign to get x < −3. Forgetting the reversal would give x > −3, which is wrong.
例如，解 −4x > 12 时，必须除以 −4 并反转符号，得到 x < −3。如果忘记反转，就会得到 x > −3，这是错误的。
10. Checking Your Answer by Substitution | 代入检验答案
check: 5(3) + 1 = 16 and 2(3) + 10 = 16. Both sides are equal, so the solution is correct.
解完方程后，将答案代回原方程进行检验。例如，如果你认为 x = 3 是 5x + 1 = 2x + 10 的解，检验：5(3) + 1 = 16，2(3) + 10 = 16。两边相等，所以解是正确的。
rue. If x ≤ 3, try x = 1 in 3x + 1 ≤ 10: 3(1) + 1 = 4, which is less than 10, so it works.
对于不等式，从解集中选一个值代入，检查不等式是否成立。如果 x ≤ 3，试试 x = 1 代入 3x + 1 ≤ 10：3(1) + 1 = 4，确实小于 10，所以成立。
11. Word Problems | 应用题
Word problems often require you to write and solve a linear equation. Read the problem carefully, define the unknown as x, and translate the words into an equation using the clues given. Then solve using the balancing method and check that your answer makes sense in the real-life context.
应用题通常要求你列出并解一个线性方程。仔细读题，把未知数设为 x，并根据题目线索把文字转化为方程。然后用平衡法求解，并检查答案在实际情况中是否合理。
For example, 'Three times a number plus 5 equals 20.' x. The equation is 3x + 5 = 20. Subtract 5 to get 3x = 15, then divide by 3 to get x = 5.
例如，“一个数的三倍加 5 等于 20。”设这个数为 x。方程为 3x + 5 = 20。减 5 得 3x = 15，再除以 3 得 x = 5。
12. Exam-Style Tips | 考试技巧
In Cambridge KS3 exams, marks are given for method, not just the final answer. Show every step clearly. If a question asks you to represent an inequality on a number line, label the scale and use the correct open or closed circle. Always double-check signs when negatives are involved.
在剑桥 KS3 考试中，分数不仅给最终答案，也给解题过程。要清晰展示每一步。如果题目要求在数轴上表示不等式，请标注刻度，并使用正确的空心或实心圆。涉及负数时务必仔细检查符号。
Keep your working neat and logical. If you get stuck, go back to the balancing method and think about what operation can undo the current step. Practice with mixed equations and inequalities will make your algebra much stronger.
保持书写整洁、逻辑清晰。如果卡住了，回到平衡法，想一想哪一步运算可以消去当前项。混合练习方程和不等式会让你的代数能力大大提高。
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