📚 Mastering Linear Equations | 掌握一元一次方程
Linear equations are one of the most important building blocks in Cambridge KS3 mathematics. They appear when we balance numbers, solve puzzles, and describe real-life relationships. This article explains the key ideas step by step, from one-step equations to equations with variables on both sides, and shows you how to avoid common mistakes.
一元一次方程是剑桥 KS3 数学中最重要的基础模块之一。当我们平衡数字、解答谜题以及描述现实生活中的数量关系时，都会用到方程。本文从一步方程到变量在等号两边的方程，逐步讲解核心概念，并告诉你如何避免常见错误。
1. What Is an Equation? | 什么是方程？
x + 5 = 11. The letter x is called a variable because its value can change or is unknown.
方程是说明两个表达式相等的数学陈述。它总是含有等号（=），例如 x + 5 = 11。字母 x 叫做变量，因为它的值可以改变或暂时未知。
An expression such as 3x + 2 does not have an equals sign, so it is not an equation. Equations ask us to find the value of the unknown that makes the statement true.
像 3x + 2 这样的式子没有等号，因此它不是方程。方程要求我们找出使等式成立的未知数的值。
olution is x = 6 because 6 + 5 = 11. Finding this solution is called solving the equation.
例如，在方程 x + 5 = 11 中，解是 x = 6，因为 6 + 5 = 11。求出这个解的过程叫做解方程。
Equation: x + 5 = 11
Expression: 3x + 2
2. The Balance Method | 天平法
Think of an equation as a balance scale. The left side and the right side must always have the same value. If you add, subtract, multiply, or divide one side by a number, you must do exactly the same to the other side to keep the equation balanced.
把方程想象成一架天平。左边和右边必须始终保持相同的值。如果你对一边加上、减去、乘以或除以一个数，就必须对另一边做完全相同的运算，才能保持方程平衡。
This rule is the foundation of solving every linear equation. Never perform an operation on only one side, or the equation will become false.
这条规则是解每一个一元一次方程的基础。永远不要只对一边进行运算，否则方程就会不成立。
le: To solve x + 3 = 10, subtract 3 from both sides. You get x + 3 − 3 = 10 − 3, so x = 7.
例：要解 x + 3 = 10，两边同时减去 3。得到 x + 3 − 3 = 10 − 3，所以 x = 7。
x + 3 = 10 → x = 7
3. Solving One-Step Equations | 解一步方程
To solve a one-step equation, use the inverse operation. Addition and subtraction are inverse operations; multiplication and division are also inverse operations.
解一步方程时，要使用逆运算。加法和减法互为逆运算；乘法和除法也互为逆运算。
If the equation has addition, subtract the same number from both sides. If it has subtraction, add the same number. If it has multiplication, divide. If it has division, multiply.
如果方程中有加法，就在两边同时减去同一个数。如果有减法，就加上同一个数。如果有乘法，就除以同一个数。如果有除法，就乘以同一个数。
Example 1: Solve x + 7 = 12. Subtract 7 from both sides: x + 7 − 7 = 12 − 7, so x = 5.
例 1：解 x + 7 = 12。两边同时减去 7：x + 7 − 7 = 12 − 7，所以 x = 5。
Example 2: Solve x − 4 = 9. Add 4 to both sides: x − 4 + 4 = 9 + 4, so x = 13.
例 2：解 x − 4 = 9。两边同时加上 4：x − 4 + 4 = 9 + 4，所以 x = 13。
Example 3: Solve 4x = 20. Divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, so x = 5.
例 3：解 4x = 20。两边同时除以 4：4x ÷ 4 = 20 ÷ 4，所以 x = 5。
Example 4: Solve x ÷ 5 = 3. Multiply both sides by 5: x ÷ 5 × 5 = 3 × 5, so x = 15.
例 4：解 x ÷ 5 = 3。两边同时乘以 5：x ÷ 5 × 5 = 3 × 5，所以 x = 15。
x + 7 = 12 → x = 5
4x = 20 → x = 5
4. Solving Two-Step Equations | 解两步方程
A two-step equation has two operations attached to the variable. Solve it by undoing the operations in reverse order: first undo addition or subtraction, then undo multiplication or division.
两步方程中，变量带有两步运算。解题时要按相反顺序逐步消除运算：先消除加法或减法，再消除乘法或除法。
2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
例：解 2x + 3 = 11。先两边减去 3：2x = 8。然后两边除以 2：x = 4。
Another example: Solve 5x − 7 = 23. Add 7 to both sides: 5x = 30. Then divide by 5: x = 6.
另一个例子：解 5x − 7 = 23。两边加上 7：5x = 30。然后除以 5：x = 6。
Always show each step clearly. This helps you spot errors and earns marks in exams, even if the final answer is wrong.
每一步都要写清楚。这有助于发现错误，也能在考试中获得步骤分，即使最终答案错误。
2x + 3 = 11 → 2x = 8 → x = 4
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first. Multiply every term inside the bracket by the number outside.
当方程含有括号时，首先要展开括号。把括号内的每一项都乘以外面的数。
= 18. Expand to get 3x + 6 = 18. Subtract 6 from both sides: 3x = 12. Divide by 3: x = 4.
例：解 3(x + 2) = 18。展开得 3x + 6 = 18。两边减去 6：3x = 12。再除以 3：x = 4。
2x − 1) = 20. Expand to get 8x − 4 = 20. Add 4 to both sides: 8x = 24. Divide by 8: x = 3.
另一个例子：解 4(2x − 1) = 20。展开得 8x − 4 = 20。两边加上 4：8x = 24。再除以 8：x = 3。
If the bracket has a minus sign in front, be careful with signs: −2(x − 3) = −2x + 6.
如果括号前面是负号，要注意符号：−2(x − 3) = −2x + 6。
3(x + 2) = 18 → 3x + 6 = 18 → x = 4
6. Variables on Both Sides | 变量在等号两边
art by collecting the variable terms on one side and the constant terms on the other side.
有些方程等号两边都有变量，例如 5x + 2 = 3x + 10。首先把变量项移到一边，常数项移到另一边。
x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Divide
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