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. In KS3 Mathematics, you learn to find the value of an unknown variable, usually represented by a letter like x or y, by making both sides of the equation balance. Mastering this skill opens the door to more advanced topics such as graphs, simultaneous equations, and problem solving in science and engineering.

一元一次方程是代数的基础。在 KS3 数学中，你将学习通过让等式两边保持平衡来求出未知量（通常用 x 或 y 表示）的值。掌握这一技能后，你就能顺利走进图像、联立方程以及科学和工程问题求解等更高阶的内容。

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1. Understanding an Equation | 认识方程

An equation is a mathematical statement that two expressions are equal. It always contains an equals sign (=). The left-hand side (LHS) must have exactly the same value as the right-hand side (RHS). Our job is to find the value of the unknown that makes this true.

方程是表示两个表达式相等的一种数学陈述。它总是包含一个等号 (=)。左边 (LHS) 的值必须恰好等于右边 (RHS) 的值。我们的任务就是找出使这个等式成立的未知数的值。

For example, in the equation x + 5 = 12, the unknown is x. We can see that x must be 7 because 7 + 5 = 12. In more complex equations, we apply systematic methods.

例如，在方程 x + 5 = 12 中，未知数为 x。我们可以看出 x 一定是 7，因为 7 + 5 = 12。在更复杂的方程中，我们则需要运用系统的方法来求解。

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2. The Balancing Method | 天平法

Think of an equation as a balance scale. Whatever you do to one side, you must do exactly the same to the other side to keep the scale balanced. This idea is central to solving equations correctly.

把方程想象成一个天平。无论你对一边做了什么操作，都必须对另一边做完全一样的操作，这样才能保持天平平衡。这个想法是正确解方程的核心。

The operations allowed are: adding, subtracting, multiplying, or dividing both sides by the same non‑zero number. You can also swap the two sides (the equation remains true). Always aim to isolate the unknown on one side.

允许的操作包括：加、减、乘或除以同一个非零的数。你也可以交换等式的两边（方程依然成立）。我们的目标永远是把未知数单独移到一边。

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3. One‑Step Equations | 一步方程

One‑step equations require only a single operation to find the unknown. Examples are x + 3 = 10, x – 4 = 2, 3x = 15, and x / 5 = 2.

一步方程只需要进行一次运算就可以求出未知数。例子有 x + 3 = 10、x – 4 = 2、3x = 15 和 x / 5 = 2。

For x + 3 = 10, subtract 3 from both sides: x + 3 – 3 = 10 – 3 → x = 7. For 3x = 15, divide both sides by 3: 3x / 3 = 15 / 3 → x = 5.

对于 x + 3 = 10，两边同时减去 3：x + 3 – 3 = 10 – 3 → x = 7。对于 3x = 15，两边同时除以 3：3x / 3 = 15 / 3 → x = 5。

Always check your answer by substituting the value back into the original equation.

最好总是把求出的值代入原方程进行检验。

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4. Two‑Step Equations | 两步方程

Two‑step equations involve a combination of two operations. A common form is 2x + 5 = 13. First, remove the constant added or subtracted, then deal with the coefficient of x.

两步方程涉及两种运算的组合。常见的形式如 2x + 5 = 13。首先，消去加上或减去的常数项，然后再处理 x 的系数。

Step 1: Subtract 5 from both sides → 2x = 8. Step 2: Divide both sides by 2 → x = 4.

第一步：两边同时减去 5 → 2x = 8。第二步：两边同时除以 2 → x = 4。

Remember the reversed order of operations: undo addition/subtraction before multiplication/division.

记住运算顺序是反过来的：先消去加减，再处理乘除。

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5. Equations with Brackets | 含有括号的方程

When an equation contains brackets, expand them first using the distributive law. For example, 3(x + 2) = 15 becomes 3x + 6 = 15.

当方程含有括号时，首先运用分配律展开括号。比如 3(x + 2) = 15 展开后变为 3x + 6 = 15。

Then solve as a two‑step equation: subtract 6 from both sides → 3x = 9, divide by 3 → x = 3.

然后当作两步方程来解：两边同时减去 6 → 3x = 9，再除以 3 → x = 3。

Alternatively, you can divide both sides by the factor outside the bracket first, but expanding is usually safer for beginners.

你也可以先把两边同时除以括号外的因数，但对初学者来说，先展开往往更稳妥。

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6. Unknowns on Both Sides | 两边都有未知数的方程

Equations like 5x – 3 = 2x + 9 have the unknown x on both sides. The strategy is to collect all x‑terms on one side and all constant terms on the other.

像 5x – 3 = 2x + 9 这样的方程两边都含有未知数 x。对策是把所有含 x 的项集中到一边，常数项集中到另一边。

Subtract 2x from both sides: 5x – 2x – 3 = 9 → 3x – 3 = 9. Then add 3 to both sides: 3x = 12, so x = 4.

两边同时减去 2x：5x – 2x – 3 = 9 → 3x – 3 = 9。然后两边同时加 3：3x = 12，因此 x = 4。

Always aim to have a positive coefficient for x at the end. If the coefficient becomes negative, multiply or divide both sides by –1 at the final step.

始终要设法让最后 x 的系数为正。如果系数变成了负数，在最后一步将两边同时乘以或除以 –1。

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7. Fractional Equations | 含有分数的方程

When an equation contains fractions, multiply every term on both sides by the lowest common denominator (LCD) to clear the fractions. For instance, x/3 + 2 = 4. Multiply through by 3: 3×(x/3) + 3×2 = 3×4 → x + 6 = 12. Then x = 6.

当方程含有分数时，将两边每一项都乘以最小公分母 (LCD) 来去掉分母。例如 x/3 + 2 = 4，两边乘以 3：3×(x/3) + 3×2 = 3×4 → x + 6 = 12，于是 x = 6。

Another example: (2x)/5 = x – 3. Multiply both sides by 5: 2x = 5(x – 3) → 2x = 5x – 15. Rearrange: 2x – 5x = –15 → –3x = –15 → x = 5.

另一个例子：(2x)/5 = x – 3。两边乘以 5：2x = 5(x – 3) → 2x = 5x – 15。移项：2x – 5x = –15 → –3x = –15 → x = 5。

Check that no denominator becomes zero with your solution; if it does, that value is not allowed.

注意检验你的解是否会让某个分母为零；如果是，那个值就是无效解。

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8. Word Problems Leading to Linear Equations | 应用题列出一元一次方程

Many real‑world problems can be solved by setting up a linear equation. Read the problem carefully, identify the unknown quantity, assign a variable, and write an equation based on the given relationships.

许多实际问题都可以通过列出一元一次方程来解决。仔细读题，找出未知量，设定一个变量，根据给出的关系写出一个方程。

Example: “I think of a number, double it, add 7, and the result is 23. What is the number?” Let the number be n → 2n + 7 = 23 → 2n = 16 → n = 8.

例如：“我想了一个数，把它加倍，再加 7，结果是 23。这个数是多少？”设这个数为 n → 2n + 7 = 23 → 2n = 16 → n = 8。

Always translate key words: “sum” means +, “difference” means –, “product” means ×, “quotient” means ÷, “is” becomes =.

要习惯翻译关键词：“和”代表 +，“差”代表 –，“积”代表 ×，“商”代表 ÷，“是”变成 =。

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9. Checking Your Solution | 检验你的解

Substituting the value you found back into the original equation helps catch mistakes. Do this for every equation you solve, especially during exams.

把求出的值代回到原方程中，有助于发现错误。每解完一个方程都要这样做，特别是在考试时。

If your solution is correct, both sides will give exactly the same number. For x = 4 in 2x + 5 = 13: LHS = 2(4) + 5 = 8 + 5 = 13 = RHS. It works.

如果你求出的解是对的，两边会得到完全相同的数。对于 2x + 5 = 13 中的 x = 4：左边 = 2(4) + 5 = 8 + 5 = 13 = 右边。正确。

If the sides do not match, re‑trace your steps. Common errors include arithmetic mistakes, sign errors, or forgetting to apply an operation to every term.

如果两边不相等，就需要回溯解题步骤。常见的错误包括计算错误、符号错误，或者忘记了把运算应用到每一项上。

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

• Forgetting to do the same thing to both sides. Always write down what you are doing to both sides. Example: x + 3 = 7 → subtract 3 from both sides.

  忘记对两边做同样的操作。每次都应写下对两边做了什么。示例：x + 3 = 7 → 两边同时减去 3。

• Misapplying the distributive law. Remember 3(x + 2) = 3x + 6, not 3x + 2.

  错误地使用分配律。记住 3(x + 2) = 3x + 6，而不是 3x + 2。

• Sign errors when moving terms. Moving –2x to the other side becomes +2x. Carefully track signs.

  移项时的符号错误。将 –2x 移到另一边要变成 +2x。请小心处理符号。

• Dividing by zero. Never divide both sides by an expression that could be zero unless you are sure it is non‑zero.

  除以零。永远不要用可能为零的表达式去除方程两边，除非你确定它非零。

• Not simplifying before solving. Combine like terms first whenever possible.

  解之前没有化简。请尽可能先合并同类项。

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11. Equations with Negative Coefficients | 系数为负数的方程

When x has a negative coefficient, such as –x = 5 or –3x = 12, multiply or divide both sides by –1 to make the coefficient positive. –x = 5 → x = –5.

当 x 的系数为负数时，比如 –x = 5 或 –3x = 12，可以在两边同时乘以或除以 –1，让系数变正。–x = 5 → x = –5。

For –3x = 12, divide both sides by –3: –3x / –3 = 12 / –3 → x = –4. This small step prevents sign confusion.

对于 –3x = 12，两边同时除以 –3：–3x / –3 = 12 / –3 → x = –4。这简单的一步能避免符号混淆。

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12. Practice Makes Permanent | 熟能生巧

Linear equations appear again in graphical work, in formulas, and in higher topics. The more you practise, the more automatic the balancing method becomes. Try solving problems of increasing difficulty, from simple one‑step equations to those with brackets, fractions, and variables on both sides.

一元一次方程在图像、公式以及更高的知识点中会反复出现。练习得越多，天平法就越能成为你的自然反应。试着从简单的一步方程做起，逐步挑战含有括号、分数和两边都有变量的题目。

Use online quizzes, past paper questions, and mental maths to improve your speed and accuracy. Remember: every equation, no matter how complicated, can be untangled with the same basic principles.

利用在线测验、往年真题和心算来提升速度和准确率。记住：再复杂的方程，都可以用同样的基本原理一步步理清。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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