📚 Solving Linear Equations: From Basics to Problem Solving | 解线性方程：从基础到问题解决
Linear equations are the foundation of algebra and appear in every area of mathematics. In KS3, you build the skills to solve equations step by step, moving from simple one‑step problems to ones that involve multiple operations, brackets and even fractions. This article explains each concept clearly, with examples and practice ideas, so you can become confident in solving any linear equation.
线性方程是代数的基础，几乎出现在数学的每个领域。在 KS3 阶段，你会逐步掌握解方程的方法，从简单的一步方程到涉及多种运算、括号乃至分数的方程。本文将逐一解释每个概念，配合范例和练习思路，让你对解任何一元一次方程都充满信心。
1. What Is a Linear Equation? | 什么是线性方程？
2(x + 4) = 18. The aim is to find the value of the variable that makes the equation true.
线性方程就是两个表达式相等的一个陈述，且其中的变量（通常写作 x）的指数为 1。例如 x + 5 = 12、3x - 2 = 10 和 2(x + 4) = 18。我们的目标是求出使等式成立的变量值。
Every linear equation has a left‑hand side (LHS) and a right‑hand side (RHS), separated by an equals sign. Think of the equals sign as a balance: whatever you do to one side, you must do to the other to keep the equation balanced.
每个线性方程都有一个左边（LHS）和一个右边（RHS），由等号分隔。可以把等号想象成天平：你对一边做的任何操作，都必须对另一边做同样的操作，才能保持方程平衡。
2. Solving by Inspection | 通过观察求解
For very simple equations, you can find the solution by looking at them and asking, “What number makes this true?” For example, in x + 3 = 10, you can see that x must be 7 because 7 + 3 = 10.
对于非常简单的方程，你可以通过观察并问自己“什么数值能让等式成立？”来求解。例如，在 x + 3 = 10 中，很容易看出 x 必须是 7，因为 7 + 3 = 10。
Similarly, for 5x = 20, you ask “5 times what equals 20?” utions are small whole numbers, but it becomes impractical for more complicated equations.
同样地，对于 5x = 20，思考“5 乘几等于 20？”答案是 4，所以 x = 4。当解是比较小的整数时，观察法非常有效，但遇到较复杂的方程时就不好用了。
3. Inverse Operations: Addition and Subtraction | 使用逆运算：加法与减法
sides. For instance, x + 9 = 15 → subtract 9 from both sides: x + 9 - 9 = 15 - 9 → x = 6.
为了系统地解方程，我们使用逆运算。如果变量加上了一个数，我们就从两边减去这个数。例如，x + 9 = 15 → 两边同时减去 9：x + 9 - 9 = 15 - 9 → x = 6。
4 = 11 + 4 → y = 15. Always show the same step on both sides to keep the balance correct.
如果是变量减去了一个数，我们就给两边同时加上这个数：y - 4 = 11 → 加 4：y - 4 + 4 = 11 + 4 → y = 15。始终要写明两边都进行相同的运算，以保持天平平衡。
4. Inverse Operations: Multiplication and Division | 使用逆运算：乘法与除法
e both sides by that number. For 7x = 56, divide both sides by 7: 7x ÷ 7 = 56 ÷ 7 → x = 8.
当变量乘以一个数时，我们就用那个数去除两边。对于 7x = 56，两边同时除以 7：7x ÷ 7 = 56 ÷ 7 → x = 8。
, multiplication and division are opposite operations, just like addition and subtraction.
当变量除以一个数时，我们就在两边乘上这个数。对于 x/5 = 4，两边同时乘以 5：(x/5) × 5 = 4 × 5 → x = 20。请记住，乘和除是一对逆运算，就像加和减一样。
The table below summarises the four basic inverse operations.
下面的表格总结了四种基本的逆运算。
| Operation | Inverse Operation | Example |
|---|---|---|
| Addition (+) | Subtraction (−) | x + 6 = 10 → x = 4 |
| Subtraction (−) | Addition (+) | x − 3 = 9 → x = 12 |
| Multiplication (×) | Division (÷) | 4x = 28 → x = 7 |
| Division (÷) | Multiplication (×) | x/7 = 3 → x = 21 |
5. Solving Two‑Step Equations | 解两步方程
A two‑ n: subtract 5 from both sides → 2x = 8. Then undo the multiplication: divide by 2 → x = 4.
两步方程就是变量上进行了两种运算。比如 2x + 5 = 13。首先，消去加法：两边同时减去 5 → 2x = 8。然后，消去乘法：两边除以 2 → x = 4。
ys perform the opposite operations in the reverse order they were applied to the variable.
另一种常见的形式是 3x − 7 = 14。两边加 7 → 3x = 21，然后除以 3 → x = 7。一定要按照与运算顺序相反的顺序来使用逆运算。
d 5 → result 13. The reverse flow is: start from 13, subtract 5, then divide by 2 → x = 4.
一个实用的检验方法是画出运算流程图：从 x 开始，乘 2，再加 5 → 结果为 13。逆向流程则是：从 13 开始，减 5，再除以 2 → x = 4。
6. Equations with Brackets | 带括号的方程
6) = 20. Method 1: expand → 2x + 12 = 20, then subtract 12 → 2x = 8, divide by 2 → x = 4.
方程中出现括号时，可以先展开括号，或者如果括号乘了一个系数，也可以直接除以该系数。以 2(x + 6) = 20 为例。方法一：展开 → 2x + 12 = 20，然后减 12 → 2x = 8，除以 2 → x = 4。
ethods give the same answer. Whichever you use, be careful to apply the operation to the entire side of the equation.
方法二：先给两边同时除以 2 → x + 6 = 10，然后减 6 → x = 4。两种方法结果一样。无论采用哪种，都要确保对整个一边进行运算。
add 1 → 2x = 8, x = 4. Practice both approaches so you can choose the more efficient one.
对于 3(2x − 1) = 21，可以先除以 3 → 2x − 1 = 7，再加 1 → 2x = 8，x = 4。两种方法都多加练习，这样你就可以选择更高效的一种。
7. Equations with Fractions | 带分数的方程
both sides by it. For x/3 + 2 = 5, subtract 2 first → x/3 = 3, then multiply by 3 → x = 9.
分数经常出现在线性方程中。关键是通过两边乘以分母来消去分母。对于 x/3 + 2 = 5，先减 2 → x/3 = 3，再乘以 3 → x = 9。
15. Another way is to multiply both sides by 3 first → 2x = 30, then divide by 2 → x = 15.
如果方程是 (2/3)x = 10，你可以两边同时乘以倒数 3/2：x = 10 × 3/2 = 15。另一种方法是先两边乘 3 → 2x = 30，再除以 2 → x = 15。
3 = 5. The LCM of 2 and 3 is 6. Multiply every term by 6 → 3x + 2x = 30 → 5x = 30 → x = 6.
当分母不同时，通常可以把每一项都乘以最小公倍数（LCM）。例如 x/2 + x/3 = 5。2 和 3 的 LCM 是 6。每一项乘 6 → 3x + 2x = 30 → 5x = 30 → x = 6。
8. Equations with Unknowns on Both Sides | 未知数在等式两边的方程
quations like 5x + 2 = 3x + 10, the variable appears on both sides. Start by getting all x‑ n one side. Subtract 3x from both sides → 2x + 2 = 10. Then subtract 2 → 2x = 8, so x = 4.
在 5x + 2 = 3x + 10 这类方程中，变量出现在两边。首先，把所有含 x 的项移到一边。两边减去 3x → 2x + 2 = 10。再减 2 → 2x = 8，得 x = 4。
You can also collect the x‑ x = 4. The result is the same; choose whichever side makes the coefficient of x positive.
也可以把 x 项集中到右边。两边减 5x → 2 = −2x + 10。减 10 → −8 = −2x，再除以 −2 → x = 4。结果一样；选择能使 x 系数为正的那一边会让计算更顺手。
gn changes. For example, 4x − 7 = x + 5 → subtract x: 3x − 7 = 5 → add 7: 3x = 12 → x = 4.
务必记住：把一项移到等号另一边时，它的符号会改变。例如，4x − 7 = x + 5 → 减 x：3x − 7 = 5 → 加 7：3x = 12 → x = 4。
9. Solving Word Problems with Equations | 用方程解文字题
Many real‑world situations can be modelled by linear equations. Start by reading the problem carefully and identifying the unknown. Represent it with a variable, then translate the words into an equation.
许多实际问题都可以用线性方程来建模。先仔细读题，确定未知量。用一个变量表示它，然后把文字转化为方程。
Example: “Three more than twice a number is 19.” number is 2x. Three more than that is 2x + 3. So 2x + 3 = 19. Subtract 3 → 2x = 16, x = 8.
例题：“一个数的两倍再加 3 等于 19。”设这个数为 x。两倍即为 2x。再加 3 就是 2x + 3。因此 2x + 3 = 19。减 3 → 2x = 16，x = 8。
Example: “The perimeter of a rectangle is 36 cm. Its length is twice its width. Find the dimensions.” then length = 2w. Perimeter = 2(l + w) = 2(2w + w) = 6w. Set 6w = 36 → w = 6, length = 12.
例题：“一个长方形的周长是 36 cm，长是宽的两倍。求长和宽。”设宽为 w，则长 = 2w。周长 = 2(长 + 宽) = 2(2w + w) = 6w。令 6w = 36 → w = 6，长 = 12。
Always define your variable clearly and write down the equation before you solve it. After finding the answer, read the question again to make sure you have given the information asked for.
务必先明确定义变量，并列出方程再求解。求出答案后，再次读题，确认你回答的就是题目所问。
10. Checking and Interpreting Solutions | 检验并解释解
on is correct. For 4x − 9 = 11, x = 5: LHS = 4×5 − 9 = 20 − 9 = 11, RHS = 11, so it works.
解出任何方程后，把你的答案代回原方程进行检验。如果两边得到相同的值，解就是正确的。对 4x − 9 = 11，x = 5：左边 = 4×5 − 9 = 20 − 9 = 11，右边 = 11，成立。
In word problems, also check whether the answer makes sense in the context. Negative lengths, or fractional numbers of people, indicate that either you have made a mistake or the equation does not fit the situation.
在文字题中，还要检查答案在题境中是否合理。负数长度或不是整数的人数，都意味着要么你计算有误，要么所列方程不符合实际情形。
A powerful method is to use inverse operations in reverse order mentally: start from the right‑hand value and perform the opposite steps. This echoes the flow‑diagram approach and helps you spot errors.
一个很好的方法是心中按逆序执行逆运算：从右边的值开始，逆向一步一步操作。这与流程图法一脉相承，有助于发现错误。
ecking will reveal whether the equation is true for all x or for none. At KS3, you will mostly work with equations that have one unique solution.
当方程无解或为恒等式（如 2x + 4 = 2(x + 2)）时，检验将显示方程是对所有 x 成立还是无解。在 KS3 阶段，你主要处理的还是有一个唯一解的方程。
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