📚 Solving Linear Equations at KS3 | KS3 解一元一次方程
Linear equations are one of the most important building blocks in Cambridge Lower Secondary Mathematics. A linear equation is an equation in which the unknown, usually written as x, appears only to the power of 1. In this revision guide, you will learn how to set up, solve and check linear equations using the balance method.
一元一次方程是剑桥初中数学中最重要的基础内容之一。一元一次方程是指未知数（通常用 x 表示）只出现一次方的一次方程。在本复习指南中，你将学习如何建立、求解并检验一元一次方程，重点掌握天平法。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is a statement that two expressions are equal. For example, x + 3 = 10 is a linear equation because the unknown x has an exponent of 1. The solution of the equation is the value of x that makes the statement true.
方程是表示两个表达式相等的语句。例如，x + 3 = 10 是一个一元一次方程，因为未知数 x 的指数为 1。方程的解就是使等式成立的 x 的值。
The equals sign is like a balance point: the left-hand side must always have the same value as the right-hand side.
等号就像一个平衡点：左边必须始终与右边保持相同的值。
2. The Balance Method | 天平法
To solve an equation, you can picture a set of scales in balance. Whatever you do to one side of the equation, you must do exactly the same to the other side. This keeps the scales balanced and gives an equivalent equation.
解方程时，可以想象一台保持平衡的天平。无论对方程的一边做什么运算，另一边也必须做完全相同的运算。这样才能保持天平平衡，并得到同解方程。
Common operations include adding, subtracting, multiplying or dividing both sides by the same non-zero number.
常见的操作包括对两边同时加、减、乘或除以同一个非零数。
x + 5 = 12 → x + 5 − 5 = 12 − 5 → x = 7
3. Solving One-Step Equations | 解一步方程
For equations such as y − 4 = 9, use the inverse operation. Since 4 is subtracted from y, add 4 to both sides: y − 4 + 4 = 9 + 4, so y = 13.
对于 y − 4 = 9 这类方程，使用逆运算。因为 y 减去了 4，所以两边同时加 4：y − 4 + 4 = 9 + 4，得到 y = 13。
For equations such as 7a = 35, divide both sides by 7 to isolate a: a = 5.
对于 7a = 35 这类方程，两边同时除以 7 来分离出 a：a = 5。
Useful inverse operations: addition is undone by subtraction, multiplication by division, squaring by square rooting.
常用的逆运算：加法用减法抵消，乘法用除法抵消，平方用开平方抵消。
y − 4 = 9 → y = 13
4. Solving Two-Step Equations | 解两步方程
A two-step equation has two operations acting on the unknown. For example, in 2x + 3 = 11, x is first multiplied by 2, then 3 is added. Reverse the order of operations: undo addition or subtraction first, then multiplication or division.
两步方程中有两个运算作用于未知数。例如，在 2x + 3 = 11 中，x 先乘以 2，再加上 3。求解时按与运算顺序相反的顺序逆推：先抵消加减，再抵消乘除。
Worked example: 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
例题：2x + 3 = 11。两边先减 3：2x = 8。然后两边除以 2：x = 4。
2x + 3 = 11 → 2x = 8 → x = 4
5. Equations with Brackets | 含括号的方程
If an equation contains brackets, first expand them using the distributive law. For example, 3(x + 2) = 21 becomes 3x + 6 = 21.
如果方程中含有括号，先用分配律展开括号。例如，3(x + 2) = 21 可化为 3x + 6 = 21。
Then solve as a two-step equation: subtract 6 from both sides to get 3x = 15, and divide by 3 to get x = 5.
然后按两步方程求解：两边减 6 得到 3x = 15，再除以 3 得到 x = 5。
Take care with negative signs: 2(x − 4) = 10 expands to 2x − 8 = 10, not 2x − 4 = 10.
注意负号的处理：2(x − 4) = 10 展开后为 2x − 8 = 10，而不是 2x − 4 = 10。
3(x + 2) = 21 → 3x + 6 = 21 → x = 5
6. Unknowns on Both Sides | 未知数在两侧
When equations have unknowns on both sides, collect the unknown terms on one side first. For example, 5x − 2 = 3x + 6. Subtract 3x from both sides: 2x − 2 = 6.
当方程两边都有未知数时，先把含未知数的项移到一边。例如，5x − 2 = 3x + 6。两边同时减 3x：2x − 2 = 6。
Then add 2 to both sides to get 2x = 8, and divide by 2: x = 4.
然后两边加 2 得到 2x = 8，再除以 2：x = 4。
Always keep the equation balanced; moving a term across the equals sign is really adding or subtracting the same term on both sides.
始终保持方程平衡；把一项从等号一边移到另一边，实际上是在两边同时加上或减去相同的项。
5x − 2 = 3x + 6 → 2x − 2 = 6 → x = 4
7. Equations Involving Fractions | 含分数的方程
To solve equations such as x ÷ 4 = 5, multiply both sides by 4 to undo division: x = 20.
对于 x ÷ 4 = 5 这类方程，两边同时乘以 4 以抵消除法：x = 20。
For two-step cases like (2x) ÷ 3 = 8, multiply both sides by 3 first: 2x = 24, then divide by 2: x = 12.
对于 (2x) ÷ 3 = 8 这类两步方程，先两边乘以 3：2x = 24，再除以 2：x = 12。
If there are several terms, multiply every term by the lowest common denominator to clear fractions.
如果有多项含有分数，通常将每一项乘以最小公分母来消去分母。
x ÷ 4 = 5 → x = 20
8. Checking Your Solution | 检验解
Always substitute your solution back into the original equation. For example, if x = 4 in 2x + 3 = 11, the left side is 2(4) + 3 = 8 + 3 = 11, which equals the right side.
一定要把解代回原方程检验。例如，若 2x + 3 = 11 的解为 x = 4，则左边为 2(4) + 3 = 8 + 3 = 11，与右边相等。
This is especially useful in word problems, where the context can tell you whether a negative or fractional answer is sensible.
这在应用题中尤其有用，因为实际背景可以判断负数或分数解是否合理。
2(4) + 3
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