📚 Solving Linear Equations | 解一元一次方程
Linear equations are the foundation of algebra at KS3. A linear equation contains an unknown value, usually written as x, and the highest power of x is 1. In this article, you will learn how to solve linear equations step by step using the balance method, check answers, and apply these skills to word problems. This matches the style of questions in Cambridge Lower Secondary Mathematics, including page 40 exercise 2.
线性方程是 KS3 代数的基础。线性方程含有一个未知数，通常用 x 表示，并且 x 的最高次数是 1。本文将教你用天平法逐步解线性方程、检验答案，并把技能应用到文字题中。这与剑桥初中数学第 40 页练习 2 的题型一致。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is an equation where the unknown appears only to the first power. It can be written in forms such as ax + b = c, ax − b = c, or ax + b = cx + d. The word ‘linear’ means the graph of the equation is a straight line. At KS3, you will mostly solve equations with one unknown.
线性方程是指未知数只出现一次方的方程。它可以写成 ax + b = c、ax − b = c 或 ax + b = cx + d 等形式。’线性’ 的意思是方程的图像是一条直线。在 KS3 阶段，你主要解含一个未知数的方程。
For example, 3x + 2 = 14 and x − 5 = 9 are both linear equations. They contain the variable x and no powers like x² or x³. Solving means finding the value of x that makes the equation true.
例如，3x + 2 = 14 和 x − 5 = 9 都是线性方程。它们包含变量 x，并且没有 x² 或 x³ 这样的高次项。解方程就是求出使等式成立的 x 的值。
2. The Balance Method | 天平法
The balance method treats an equation like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. If you add, subtract, multiply or divide one side by a number, do the same to the other side. This is the golden rule of solving equations.
天平法把方程看作一架平衡的天平。你对一边做的任何操作，都必须对另一边做同样的操作才能保持平衡。如果你对一边加、减、乘或除以一个数，另一边也要做同样的事。这是解方程的黄金法则。
- Keep the equation balanced at every step. | 每一步都保持方程平衡。
- Use inverse operations to undo steps. | 用逆运算撤销步骤。
- Always write the new line directly under the previous one. | 始终把新的一行直接写在上一行下面。
3. Solving by Adding or Subtracting | 用加法或减法解方程
To solve equations such as x − 5 = 9, add 5 to both sides. This gives x − 5 + 5 = 9 + 5, so x = 14. For x + 7 = 20, subtract 7 from both sides: x = 13. Addition and subtraction are inverse operations.
要解 x − 5 = 9 这样的方程，给两边都加 5。得到 x − 5 + 5 = 9 + 5，所以 x = 14。对于 x + 7 = 20，两边都减 7：x = 13。加法和减法互为逆运算。
x − 5 = 9 → x = 14
If the equation has a negative sign, be careful. For x − 8 = 3, add 8 to both sides: x = 11. For 12 = x + 4, subtract 4 from both sides to get x = 8.
如果方程带有负号，要小心。对于 x − 8 = 3，两边都加 8：x = 11。对于 12 = x + 4，两边都减 4 得到 x = 8。
4. Solving by Multiplying or Dividing | 用乘法或除法解方程
If the unknown is multiplied or divided, use the inverse operation. For 4x = 32, divide both sides by 4: x = 8. For x ÷ 3 = 9, multiply both sides by 3: x = 27. Multiplication and division are inverse operations.
如果未知数被乘或除，就用逆运算。对于 4x = 32，两边同除以 4：x = 8。对于 x ÷ 3 = 9，两边同乘以 3：x = 27。乘法和除法互为逆运算。
4x = 32 → x = 8
When the coefficient of x is negative, divide by the negative number. For −5x = 20, divide both sides by −5: x = −4. Write the division line clearly so you do not lose the negative sign.
当 x 的系数为负数时，要除以这个负数。对于 −5x = 20，两边都除以 −5：x = −4。清楚地写出除法步骤，以免丢掉负号。
5. Two-Step Equations | 两步方程
A two-step equation has two operations. For example, 2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4. Always undo addition or subtraction before multiplication or division when following the order of operations backwards.
两步方程包含两种运算。例如 2x + 3 = 11。先两边减 3：2x = 8。再两边除以 2：x = 4。逆向遵循运算顺序时，通常先撤销加法或减法，再撤销乘法或除法。
2x + 3 = 11 → 2x = 8 → x = 4
For 7 + 3x = 19, subtract 7 first to get 3x = 12, then divide by 3: x = 4. The order of undoing matters: reverse the last operation first.
对于 7 + 3x = 19，先减 7 得到 3x = 12，再除以 3：x = 4。撤销运算的顺序很重要：先撤销最后一步运算。
6. Equations with Brackets | 含括号的方程
When an equation has brackets, expand them first. For 3(x + 2) = 21, expand to 3x + 6 = 21. Then subtract 6 from both sides: 3x = 15. Finally divide by 3: x = 5. You can check by substituting x = 5 back: 3(5 + 2) = 3 × 7 = 21.
当方程含有括号时，先展开括号。对于 3(x + 2) = 21，展开得 3x + 6 = 21。然后两边减 6：3x = 15。最后除以 3：x = 5。你可以把 x = 5 代回检验：3(5 + 2) = 3 × 7 = 21。
If there is a negative sign before a bracket, multiply all terms inside by that negative. For 2(x − 4) = 10, expand to 2x − 8 = 10. Then add 8 and divide by 2: x = 9.
如果括号前有负号，把括号内的所有项都乘以这个负号。对于 2(x − 4) = 10，展开得 2x − 8 = 10。然后加 8 再除以 2：x = 9。
7. Variables on Both Sides | 变量在等式两边
If the variable appears on both sides, collect like terms first. For 5x − 2 = 2x + 10, subtract 2x from both sides: 3x − 2 = 10. Then add 2 to both sides: 3x = 12. Divide by 3: x = 4. Always move the smaller variable term to avoid negative coefficients.
如果变量出现在等式两边，先合并同类项。对于 5x − 2 = 2x + 10，两边减 2x：3x − 2 = 10。然后两边加 2：3x = 12。除以 3：x = 4。通常移动较小的变量项以避免负系数。
5x − 2 = 2x + 10 → 3x = 12 → x = 4
For 7x + 5 = 3x − 7, subtract 3x from both sides to get 4x + 5 = −7. Then subtract 5 and divide by 4: x = −3. Working line by line prevents mistakes.
对于 7x + 5 = 3x − 7，两边减 3x 得到 4x + 5 = −7。然后减 5 再除以 4：x = −3。逐行书写可以避免错误。
8. Equations with Fraction Coefficients | 分数系数方程
Some equations include fractional coefficients, such as ⅓x = 4. To clear the fraction, multiply both sides by the denominator: x = 12. For x/5 = 3, multiply both sides by 5: x = 15. You can also write fractions using a slash, but the method is the same.
有些方程含有分数系数，如 ⅓x = 4。为了消去分数，两边同乘以分母：x = 12。对于 x/5 = 3，两边同乘以 5：x = 15。你也可以用斜线写分数，但方法相同。
If you have ½(x + 3) = 4, first multiply both sides by 2 to get x + 3 = 8, then subtract 3: x = 5. Clearing fractions early makes the equation simpler.
如果遇到 ½(x + 3) = 4，先两边乘以 2 得到 x + 3 = 8，再减 3：x = 5。尽早消去分数会使方程更简单。
9. Checking Your Solution | 检验解
Substitute your answer back into the original equation to check it works. For 2(x − 3) = 8, if x = 7, then 2(7 − 3) = 2 × 4 = 8. The left side equals the right side, so the solution is correct. Checking helps you catch arithmetic mistakes.
把你的答案代回原方程检验是否成立。对于 2(x − 3) = 8，如果 x = 7，那么 2(7 − 3) = 2 × 4 = 8。左边等于右边，因此解是正确的。检验可以帮助你发现计算错误。
Write the checking step as: Left side = 2(7 − 3) = 8, Right side = 8. Since both sides are equal, x = 7 is the solution. In exams, a quick check can save marks.
写出检验步骤：左边 = 2(7 − 3) = 8，右边 = 8。因为两边相等，所以 x = 7 是解。在考试中，快速检验可以帮你保住分数。
10. Word Problems | 文字题
Word problems turn real situations into equations. Example: ‘A number plus 6 equals 13.’ Let the number be x, so x + 6 = 13. Solving gives x = 7. Always define the variable first and write an equation that matches the words.
文字题把实际情况转化为方程。例如：’一个数加 6 等于 13。’ 设这个数为 x，那么 x + 6 = 13。解得 x = 7。一定要先定义变量，并写出与文字匹配的方程。
| Word phrase | 文字表述 | Equation | 方程 |
|---|---|
| A number plus 8 is 15. | 一个数加 8 等于 15。 | x + 8 = 15 |
| Five times a number is 35. | 一个数的 5 倍是 35。 | 5x = 35 |
| Three less than a number is 9. | 一个数减 3 等于 9。 | x − 3 = 9 |
For multi-step word problems, break the sentence into parts. ‘Twice a number plus 5 is 19’ becomes 2x + 5 = 19, so x = 7. Underline key phrases such as ‘twice’, ‘plus’ and ‘is’ to help translate.
对于多步文字题，把句子拆成几个部分。’一个数的两倍加 5 等于 19′ 变成 2x + 5 = 19，所以 x = 7。在 ‘两倍’、’加’ 和 ‘等于’ 等关键短语下面划线可以帮助翻译。
11. Common Errors | 常见错误
Students often forget to keep the equation balanced. For example, in 2x + 3 = 11, they might subtract 3 from the left only. Another common error is mishandling negative signs, such as writing x − 4 = 9 as x = 5 instead of x = 13. Always show each step clearly.
学生常常忘记保持方程平衡。例如在 2x + 3 = 11 中，他们可能只在左边减 3。另一个常见错误是负号处理错误，例如把 x − 4 = 9 写成 x = 5，而不是 x = 13。每一步都要写清楚。
- Do the same operation on both sides. | 两边做同样的运算。
- Check your final answer by substitution. | 用代入法检查最终答案。
- When dividing by a negative, keep the sign. | 除以负数时，保留符号。
Another mistake is expanding brackets incorrectly when a negative sign is outside. For 3 − 2(x − 1), students write 3 − 2x − 2 instead of 3 − 2x + 2. Take time with signs and write small intermediate steps.
另一个错误是括号外有负号时展开出错。对于 3 − 2(x − 1)，学生会写成 3 − 2x − 2，而不是 3 − 2x + 2。在符号上多花时间，并写出小的中间步骤。
12. Cambridge KS3 Exam Tips | 剑桥 KS3 考试技巧
In Cambridge KS3 exams, you must show your method. Marks are often awarded for correct steps even if the final answer is wrong. Write each line of working underneath the previous one. If a question says ‘solve’, give the value of x clearly. For word problems, define your variable with a sentence.
在剑桥 KS3 考试中，你必须展示解题方法。即使最终答案错误，正确的步骤也常常能得分。把每一步计算写在上一行的下面。如果题目说 ‘解方程’，清楚地给出 x 的值。对于文字题，用一句话定义你的变量。
Time management is important. Solve the easier equations first and return to harder ones. Always leave a few minutes to check solutions by substitution, especially when negative or fractional answers appear.
时间管理很重要。先解较简单的方程，再回来解较难的。始终留几分钟用代入法检查答案，尤其是出现负数或分数答案时。
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