Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations form the backbone of algebra at the KS3 level. Being able to solve them confidently unlocks a wide range of problem-solving skills, from handling simple balance problems to tackling complex real-world scenarios.

一元一次方程是 KS3 阶段代数的基础。能够熟练地解方程将开启一系列解题技能，从处理简单的平衡问题到解决复杂的实际场景。

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1. What is a Linear Equation? | 什么是一元一次方程？

A linear equation is an algebraic statement where the variable (usually represented by a letter like x or y) is only raised to the power of 1. This means there are no squares, cubes, or higher powers. The equation must also not have the variable in the denominator of a fraction. The highest degree of the variable is 1, so the graph of such an equation is a straight line (hence ‘linear’).

一元一次方程是一种代数陈述，其中变量（通常用 x 或 y 这样的字母表示）的指数只为 1。这意味着没有平方、立方或更高次幂。方程中的未知数也不能出现在分母中。变量的最高次数为 1，因此这种方程的图像是一条直线（这就是“线性”一词的来源）。

Examples of linear equations: 2x + 3 = 7, 4 – y = 1, 3a/2 = 6.

一元一次方程的例子：2x + 3 = 7、4 – y = 1、3a/2 = 6。

Non-examples: x² + 2 = 6 (the exponent is 2), 5/x = 10 (unknown in the denominator).

非例子：x² + 2 = 6（指数为 2），5/x = 10（未知数在分母）。

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2. The Balancing Method | 天平法

When solving a linear equation, we think of each side of the equals sign as a balance scale. Whatever operation we perform on one side, we must perform the same operation on the other side to keep the balance. This principle is summed up as ‘do the same to both sides’.

解一元一次方程时，我们可以将等号左右两边想象成一个天平。我们对一边进行任何运算，就必须对另一边进行同样的运算，以保持平衡。这个原则可以概括为“等式两边同操相同运算”。

For example, to solve x + 5 = 12, we subtract 5 from both sides: x + 5 – 5 = 12 – 5, so x = 7.

例如，解 x + 5 = 12，两边同时减去 5：x + 5 – 5 = 12 – 5，得到 x = 7。

This method utilises inverse operations: addition and subtraction are inverses, multiplication and division are inverses. Applying the inverse operation isolates the variable.

这一方法利用了逆运算：加法与减法互为逆运算，乘法与除法互为逆运算。运用逆运算可以分离出变量。

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3. Solving One-Step Equations (Addition & Subtraction) | 解一步方程（加法和减法）

One-step equations involve only a single inverse step to isolate the variable. If the equation has a number added to the variable, use subtraction; if a number is subtracted, use addition.

一步方程只需一步逆运算就能分离出变量。如果方程中未知数加了一个数，就用减法；如果未知数减了一个数，就用加法。

Example 1: y + 8 = 15. Subtract 8 from both sides: y = 15 – 8 = 7.

例 1： y + 8 = 15。两边减去 8：y = 15 – 8 = 7。

Example 2: z – 6 = 9. Add 6 to both sides: z = 9 + 6 = 15.

例 2： z – 6 = 9。两边加上 6：z = 9 + 6 = 15。

Rule: To undo an addition, subtract; to undo a subtraction, add.

法则：要消除加法，用减法；要消除减法，用加法。

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4. Solving One-Step Equations (Multiplication & Division) | 解一步方程（乘法和除法）

When the variable is multiplied by a coefficient or divided, we use the inverse operation. For multiplication, we divide both sides; for division, we multiply both sides.

当变量乘以一个系数或被除时，我们使用逆运算。乘法用除以两边解决；除法用乘以两边解决。

Example 1: 5x = 35. Divide both sides by 5: x = 35 / 5 = 7.

例 1： 5x = 35。两边除以 5：x = 35 / 5 = 7。

Example 2: a / 4 = 6. Multiply both sides by 4: a = 6 × 4 = 24.

例 2： a / 4 = 6。两边乘以 4：a = 6 × 4 = 24。

Example 3: (2/3)y = 8. Multiply both sides by the reciprocal 3/2: y = 8 × 3/2 = 12.

例 3： (2/3)y = 8。两边乘以倒数 3/2：y = 8 × 3/2 = 12。

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5. Solving Two-Step Equations | 解两步方程

A two-step equation requires two inverse operations in the correct order. Usually we deal with the constant term (addition/subtraction) first, then the coefficient (multiplication/division). This order is often remembered using the phrase ‘reverse PEMDAS’ or SADMEP.

两步方程需要按正确顺序进行两步逆运算。通常我们先处理常数项（加减），再处理系数（乘除）。这个顺序可以用“逆向 PEMDAS”或“SADMEP”来记忆。

Example: 2x + 3 = 11. Step 1: Subtract 3 from both sides → 2x = 8. Step 2: Divide both sides by 2 → x = 4.

例子： 2x + 3 = 11。第一步：两边减去 3 → 2x = 8。第二步：两边除以 2 → x = 4。

Another example: x/5 – 2 = 1. Add 2 to both sides: x/5 = 3. Multiply by 5: x = 15.

另一个例子： x/5 – 2 = 1。两边加 2：x/5 = 3。乘以 5：x = 15。

If the variable appears in a subtracted term, like 10 – 2x = 4, you might either add 2x to both sides or isolate the term containing x by subtracting 10. Both ways work if you maintain balance.

如果变量出现于减数中，比如 10 – 2x = 4，你可以两边同时加 2x，或者两边减去 10 来隔离含 x 的项。只要保持平衡，两种方法都可行。

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6. Equations with Parentheses (Distributive Law) | 带括号的方程（分配律）

When an equation contains parentheses, the first step is to expand them using the distributive law: a(b + c) = ab + ac. This transforms the equation

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