📚 Solving Linear Equations Step by Step | 一元一次方程分步解法
Linear equations are the foundation of algebra at KS3. A linear equation has an unknown, usually written x, and no powers higher than 1. Being able to solve these equations confidently is essential for Cambridge Checkpoint and later IGCSE work. This article explains each method step by step with examples and practice.
线性方程是 KS3 代数的基础。线性方程含有一个未知数，通常写成 x，且未知数的次数不超过 1。熟练掌握解这类方程对 Cambridge Checkpoint 以及后续 IGCSE 学习至关重要。本文通过例题和练习逐步讲解每种方法。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is a statement that two expressions are equal, such as 2x + 3 = 11. The letter x is called the variable or unknown. In KS3, equations usually have one variable and the highest power is 1.
线性方程是表示两个表达式相等的式子，例如 2x + 3 = 11。字母 x 称为变量或未知数。在 KS3 阶段，方程通常只有一个变量，并且最高次数为 1。
The solution of an equation is the value of x that makes the statement true. For 2x + 3 = 11, the solution is x = 4 because 2 × 4 + 3 = 11.
方程的解是使等式成立的 x 的值。对于 2x + 3 = 11，解是 x = 4，因为 2 × 4 + 3 = 11。
2. The Balancing Method | 天平法
Think of an equation as a balanced set of scales. Whatever you do to one side, you must do to the other side to keep the balance. This is the golden rule of solving equations.
把方程想象成一架平衡的天平。无论你对等式一边做什么，都必须对另一边做同样的操作以保持平衡。这是解方程的金法则。
If you add, subtract, multiply or divide one side by a number, do exactly the same to the other side. This keeps the equation true and moves you closer to x alone.
如果你对等式一边加、减、乘或除以一个数，必须对另一边做完全相同的操作。这样等式仍然成立，并帮助你逐步将 x 单独留在一边。
3. One-Step Equations | 一步方程
In a one-step equation, only one operation links the unknown to the numbers. For example, x + 5 = 12. To solve, subtract 5 from both sides.
在一步方程中，未知数与数字之间只有一种运算。例如 x + 5 = 12。求解时，两边同时减去 5。
x + 5 − 5 = 12 − 5 ⇒ x = 7
Check by substituting back: 7 + 5 = 12, so it is correct.
通过代回检验：7 + 5 = 12，所以答案正确。
Another example is 3x = 21. Here x is multiplied by 3, so divide both sides by 3.
另一个例子是 3x = 21。这里 x 乘以 3，所以两边同时除以 3。
3x ÷ 3 = 21 ÷ 3 ⇒ x = 7
Division is the inverse operation of multiplication, just as subtraction is the inverse of addition.
除法是乘法的逆运算，正如减法是加法的逆运算。
4. Two-Step Equations | 两步方程
Two-step equations require two inverse operations. For example, 2x + 3 = 11. Start by undoing the addition, then undo the multiplication.
两步方程需要两步逆运算。例如 2x + 3 = 11。先消去加法，再消去乘法。
Step 1: subtract 3 from both sides. Step 2: divide both sides by 2.
第一步：两边同时减去 3。第二步：两边同时除以 2。
2x + 3 − 3 = 11 − 3 ⇒ 2x = 8
2x ÷ 2 = 8 ÷ 2 ⇒ x = 4
Always work in the opposite order to the order of operations. Undo addition or subtraction first, then multiplication or division.
始终按照与运算顺序相反的顺序求解。先消去加或减，再消去乘或除。
5. Equations with Brackets | 含括号的方程
If an equation has brackets, expand them first using the distributive law. For example, 3(x + 2) = 18.
如果方程含有括号，先用分配律展开。例如 3(x + 2) = 18。
Multiply each term inside the bracket by the number outside: 3 × x + 3 × 2, which gives 3x + 6 = 18.
将括号内的每一项都乘以外面的数：3 × x + 3 × 2，得到 3x + 6 = 18。
Then solve the two-step equation: subtract 6 from both sides, then divide by 3.
然后解两步方程：两边减去 6，再除以 3。
3x + 6 − 6 = 18 − 6 ⇒ 3x = 12
3x ÷ 3 = 12 ÷ 3 ⇒ x = 4
Another example is 5(2x − 1) = 35. Expand to 10x − 5 = 35, then add 5 and divide by 10.
另一个例子是 5(2x − 1) = 35。展开为 10x − 5 = 35，然后加 5 再除以 10。
6. Unknown on Both Sides | 方程两边都有未知数
Some equations have x on both sides, such as 5x + 2 = 3x + 10. The goal is to collect all x terms on one side and numbers on the other.
有些方程两边都有 x，例如 5x + 2 = 3x + 10。目标是将所有含 x 的项移到一边，数字移到另一边。
Subtract the smaller x term from both sides. Here subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 10, so 2x + 2 = 10.
从两边减去较小的 x 项。这里两边减去 3x：5x − 3x + 2 = 3x − 3x + 10，得到 2x + 2 = 10。
Then solve: subtract 2 from both sides, then divide by 2. The solution is x = 4.
然后求解：两边减 2，再除以 2。解为 x = 4。
2x + 2 − 2 = 10 − 2 ⇒ 2x = 8 ⇒ x = 4
You may also need to simplify by combining like terms on each side before collecting variables.
你可能还需要在移项之前先合并每一边的同类项。
7. Equations with Fractions | 含分数的方程
When fractions appear, multiply every term by the lowest common denominator to clear them. For example, x/2 + 3 = 7.
当方程中出现分数时，将每一项乘以最小公分母以消去分母。例如 x/2 + 3 = 7。
Multiply all terms by 2: 2 × (x/2) + 2 × 3 = 2 × 7, giving x + 6 = 14. Then x = 8.
所有项都乘以 2：2 × (x/2) + 2 × 3 = 2 × 7，得到 x + 6 = 14。因此 x = 8。
2 × (x/2) + 2 × 3 = 2 × 7 ⇒ x + 6 = 14 ⇒ x = 8
For equations like (2x − 1)/3 = 5, multiply both sides by 3 first, then solve the resulting two-step equation.
对于像 (2x − 1)/3 = 5 的方程，先两边乘以 3，然后解得到的两步方程。
8. Word Problems and Forming Equations | 应用题与列方程
Many KS3 questions ask you to form an equation from a real-life situation. Identify the unknown, give it a letter, and translate the words into algebra.
许多 KS3 题目要求根据实际情境建立方程。确定未知数，用字母表示，然后将文字转化为代数式。
Example: I think of a number, multiply it by 4, add 7, and the result is 31. Let the number be n. The equation is 4n + 7 = 31.
例题：我想一个数，将它乘以 4，再加 7，结果是 31。设这个数为 n。方程为 4n + 7 = 31。
Solve: 4n = 24, so n = 6. The original number is 6.
求解：4n = 24，所以 n = 6。原来的数是 6。
Another example: The perimeter of a rectangle is 26 cm. Its length is (x + 3) cm and width is x cm. Form and solve an equation.
另一个例子：一个长方形的周长是 26 cm。它的长是 (x + 3) cm，宽是 x cm。列方程并求解。
2(x + 3) + 2x = 26 ⇒ 2x + 6 + 2x = 26 ⇒ 4x + 6 = 26 ⇒ x = 5
So the width is 5 cm and the length is 8 cm.
因此宽为 5 cm，长为 8 cm。
9. Common Mistakes to Avoid | 常见错误与规避
Mistake 1: Forgetting to do the same operation to both sides. For example, writing x + 5 = 12, then x = 7 only after subtracting 5 from the left side but not the right.
错误 1：忘记对等式两边做相同的运算。例如 x + 5 = 12，只在左边减 5 而不在右边减
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