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 and appear in almost every area of mathematics. A linear equation is an equation where the unknown variable is raised only to the power of one, such as x + 3 = 7 or 2x − 5 = 9. In KS3 Mathematics, students learn to solve these equations by performing the same operation on both sides, keeping the equation balanced. Mastering this skill builds strong problem-solving habits and prepares you for more advanced topics like graphs, inequalities, and simultaneous equations.

一元一次方程是代数的基础，几乎在数学的每个领域都会出现。线性方程是指未知数的次数只有一次（即一次幂）的方程，例如 x + 3 = 7 或 2x − 5 = 9。在 KS3 数学中，学生通过对方程两边同时进行相同运算来求解，从而保持等式平衡。掌握这一技巧能培养良好的解题习惯，并为进一步学习图像、不等式和联立方程等高级主题做好准备。

1. Understanding Equations | 认识方程

An equation is a mathematical statement that says two expressions are equal, linked by an equals sign ‘=’. Think of it as a balanced scale: whatever you do to one side, you must do to the other to keep it balanced. The goal when solving is to isolate the variable on one side and find its value.

方程是表明两个表达式相等的数学陈述，用等号“=”连接。可以把它想象成一架平衡的天平：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。解题的目标是将变量单独放在一边，求出它的值。

2. The Balancing Method | 平衡法

The balancing method involves using inverse operations to ‘undo’ what is being done to the variable. For example, if a number is added to x, subtract that number from both sides. If x is multiplied by a number, divide both sides by that number. Always perform the same operation on both sides to maintain equality.

平衡法涉及使用逆运算来“撤销”对变量所做的运算。例如，如果给 x 加了一个数，那么两边同时减去这个数。如果 x 被一个数乘，那么两边同时除以这个数。始终在等号两边进行相同运算以维持相等关系。

3. Solving x + a = b | 解 x + a = b 型方程

When you see an equation such as x + 5 = 12, ask yourself: what number plus 5 gives 12? Using the balancing method, subtract 5 from both sides: x + 5 − 5 = 12 − 5, which simplifies to x = 7. The solution is x = 7.

当你看到像 x + 5 = 12 这样的方程时，问问自己：什么数加上 5 等于 12？用平衡法，两边同时减去 5：x + 5 − 5 = 12 − 5，化简得 x = 7。解为 x = 7。

4. Solving x − a = b | 解 x − a = b 型方程

For an equation like x − 4 = 9, the inverse operation of subtraction is addition. Add 4 to both sides: x − 4 + 4 = 9 + 4, giving x = 13. Always check your answer by substituting it back: 13 − 4 = 9, which is true.

对于像 x − 4 = 9 这样的方程，减法的逆运算是加法。两边同时加 4：x − 4 + 4 = 9 + 4，得到 x = 13。一定要把答案代回原方程检验：13 − 4 = 9，成立。

5. Solving ax = b | 解 ax = b 型方程

When x is multiplied by a coefficient, such as in 3x = 15, divide both sides by the coefficient to isolate x. Dividing both sides by 3 gives x = 15 ÷ 3, so x = 5. Remember that 3x means 3 × x, and the inverse of multiplication is division.

当 x 被一个系数乘时，比如 3x = 15，两边同时除以该系数就能把 x 单独解出来。两边除以 3 得到 x = 15 ÷ 3，所以 x = 5。记住 3x 表示 3 × x，而乘法的逆运算是除法。

6. Solving x ÷ a = b | 解 x ÷ a = b 型方程

If the equation is x/4 = 7, this means x divided by 4 equals 7. The inverse operation is multiplication, so multiply both sides by 4: (x/4) × 4 = 7 × 4, resulting in x = 28. Check: 28 ÷ 4 = 7, correct.

如果方程是 x/4 = 7，这意味着 x 除以 4 等于 7。逆运算是乘法，所以两边同时乘以 4：(x/4) × 4 = 7 × 4，得到 x = 28。检验：28 ÷ 4 = 7，正确。

7. Solving Two-Step Equations | 解两步方程

A two-step equation contains two operations, like 2x + 3 = 11. First, undo the addition or subtraction: subtract 3 from both sides to get 2x = 8. Then undo the multiplication: divide both sides by 2 to get x = 4. Always reverse the order of operations – start with addition/subtraction before multiplication/division.

两步方程包含两种运算，例如 2x + 3 = 11。首先，撤销加减法：两边同时减去 3，得到 2x = 8。然后撤销乘法：两边同时除以 2，得到 x = 4。一定要逆着运算顺序来——先处理加减法，再处理乘除法。

8. Equations with Brackets | 带括号的方程

When an equation contains brackets, such as 3(x + 4) = 21, you can either expand the brackets first or divide both sides by the coefficient. For example, expand: 3x + 12 = 21, then subtract 12: 3x = 9, then divide by 3: x = 3. Alternatively, divide both sides by 3 right away: x + 4 = 7, then subtract 4 to get x = 3. Both methods are valid.

当方程中含有括号时，比如 3(x + 4) = 21，可以先把括号展开，也可以两边先除以系数。例如，展开：3x + 12 = 21，然后减去 12：3x = 9，再除以 3：x = 3。另一种方法，直接两边除以 3：x + 4 = 7，再减去 4 得到 x = 3。两种方法都有效。

9. Equations with Variables on Both Sides | 变量在等号两边的方程

For equations like 5x − 2 = 3x + 6, the strategy is to collect all variable terms on one side and constant terms on the other. Subtract 3x from both sides to get 2x − 2 = 6. Then add 2 to both sides: 2x = 8. Finally divide by 2 to get x = 4. Aim to have a positive coefficient for x when moving terms.

对于像 5x − 2 = 3x + 6 这样的方程，策略是把所有含变量的项移到一边，常数项移到另一边。两边同时减去 3x，得到 2x − 2 = 6。然后两边加 2：2x = 8。最后除以 2 得 x = 4。移项时尽量让 x 的系数为正。

10. Checking Your Solution | 检验你的解

Always substitute your answer back into the original equation to verify it. For instance, if you solved 4x − 5 = 15 and got x = 5, check by replacing x with 5: 4(5) − 5 = 20 − 5 = 15, which matches the right side. This habit catches arithmetic errors and ensures your solution is correct.

永远要把答案代回原方程进行检验。举个例子，如果你解 4x − 5 = 15 得到 x = 5，就用 5 替换 x 来检验：4(5) − 5 = 20 − 5 = 15，与右边一致。这个习惯能发现计算错误，确保解正确。

11. Common Mistakes | 常见错误

Many KS3 students forget to perform the same operation on both sides, or they apply operations in the wrong order. Others mishandle negative signs, for example in −x = 4, they might think x = 4 instead of multiplying both sides by −1 to get x = −4. Take care with brackets and always write each step clearly to avoid missing details.

许多 KS3 学生会忘记在两边进行相同运算，或者弄错运算顺序。另一些同学处理不好负号，比如在 −x = 4 中，可能会错误地认为 x = 4，而正确做法是两边同乘 −1 得到 x = −4。留意括号，并清楚地写出每一步，以免遗漏细节。

12. Practice and Real-Life Applications | 练习与实际应用

Linear equations are not just for textbooks; they appear in everyday problems like calculating costs, distances, and converting measurements. Regular practice with increasing difficulty—starting from one-step equations to those with brackets and variables on both sides—builds fluency. Use flashcards, online quizzes, and past papers to sharpen your skills. The more you practise, the more intuitive balancing equations will become.

一元一次方程不仅出现在课本上，在日常生活中也会遇到，比如计算费用、距离以及单位换算。通过逐步增加难度的练习——从一步方程到含括号和两边都有变量的方程——能提升熟练度。利用抽认卡、在线测验和历年真题来磨炼技巧。练习越多，解方程时的平衡感就会越直观。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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