📚 Solving Linear Equations | 解一元一次方程
Linear equations are a central topic in KS3 Cambridge Mathematics. They appear in algebra, problem solving, and even in geometry and data questions. This revision guide explains the key methods step by step, with clear examples and common exam tips.
线性方程是剑桥 KS3 数学的核心主题。它们出现在代数、问题解决、甚至几何和数据问题中。本复习指南逐步解释关键方法，并提供清晰的例题和常见考试技巧。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an equation where the unknown variable, usually written as x, only appears to the power of 1. This means there are no terms like x², x³, or √x.
线性方程是指未知数（通常写作 x）只以一次方出现的方程。这意味着没有 x²、x³ 或 √x 这样的项。
s. The goal is to find the value of x that makes the left-hand side equal to the right-hand side.
线性方程的一般形式为 ax + b = c，其中 a、b、c 是数字。目标是求出使左边等于右边的 x 的值。
3x + 5 = 20
tion because x is only raised to the power of 1. The solution is the number that balances the equation.
例如，3x + 5 = 20 是一个线性方程，因为 x 只以一次方出现。解就是使方程平衡的那个数。
2. Key Vocabulary | 核心词汇
Before solving equations, you need to understand the language used in questions and mark schemes. The most important terms are variable, coefficient, constant, equality, and solution.
在解方程之前，你需要理解题目和评分标准中使用的语言。最重要的术语是变量、系数、常数、等式和解。
- Variable – the unknown letter, often x, y, or n.
- Coefficient – the number multiplying the variable, e.g. 3 in 3x.
- Constant – a number on its own, e.g. 5 in 3x + 5.
- Equation – a statement that two expressions are equal, shown by an equals sign.
- Solution – the value of the variable that makes the equation true.
中文术语如下：
- 变量 – 未知字母，通常是 x、y 或 n。
- 系数 – 乘以变量的数字，例如 3x 中的 3。
- 常数 – 单独的数字，例如 3x + 5 中的 5。
- 方程 – 用等号表示两个表达式相等的陈述。
- 解 – 使方程成立的变量的值。
Knowing these words will help you follow instructions in exams, especially when a question asks for the coefficient or the constant term.
了解这些词汇将帮助你在考试中理解题意，尤其是当题目要求找出系数或常数项时。
3. The Balancing Method | 平衡法
An equation is like a balance scale. Whatever operation you perform on one side, you must perform exactly the same operation on the other side. This is the golden rule of solving equations.
方程就像一架天平。无论你对一边进行什么运算，都必须对另一边进行完全相同的运算。这是解方程的黄金法则。
If you add 4 to the left-hand side, you must add 4 to the right-hand side. If you divide the left-hand side by 2, you must divide the right-hand side by 2. This keeps the equation balanced while you isolate the variable.
如果你在左边加 4，就必须在右边加 4。如果你将左边除以 2，就必须将右边除以 2。这样可以在分离变量时保持方程平衡。
x + 7 = 15 → x + 7 − 7 = 15 − 7
The balancing method works because subtraction undoes addition, and division undoes multiplication. You always use inverse operations to move numbers away from the variable.
平衡法之所以有效，是因为减法可以抵消加法，除法可以抵消乘法。你始终使用逆运算将数字从变量旁边移开。
4. Solving One-Step Equations | 解一步方程
One-step equations require only one inverse operation to find the solution. For addition equations, subtract the same number from both sides.
一步方程只需一次逆运算即可求出解。对于加法方程，两边同时减去同一个数。
Solve x + 9 = 21. Subtract 9 from both sides: x + 9 − 9 = 21 − 9, so x = 12.
解方程 x + 9 = 21。两边同时减去 9：x + 9 − 9 = 21 − 9，所以 x = 12。
x + 9 = 21 → x = 12
ame number to both sides. For example, x − 5 = 8 becomes x − 5 + 5 = 8 + 5, giving x = 13.
对于减法方程，两边同时加上同一个数。例如，x − 5 = 8 变为 x − 5 + 5 = 8 + 5，得到 x = 13。
divide both sides by the coefficient. Solve 5x = 35 by writing 5x ÷ 5 = 35 ÷ 5, so x = 7.
对于乘法方程，两边同时除以系数。解 5x = 35，写出 5x ÷ 5 = 35 ÷ 5，所以 x = 7。
5. Solving Two-Step Equations | 解两步方程
must undo the addition or subtraction first, and then undo the multiplication or division.
两步方程包含两种运算，例如 2x + 3 = 11。你必须先处理加法或减法，再处理乘法或除法。
irst: 2x + 3 − 3 = 11 − 3, which simplifies to 2x = 8. Then divide both sides by 2: x = 4.
对于 2x + 3 = 11，先两边同时减去 3：2x + 3 − 3 = 11 − 3，化简为 2x = 8。然后两边同时除以 2：x = 4。
2x + 3 = 11 → 2x = 8 → x = 4
Always reverse the order of operations from BIDMAS. In the expression 2x + 3, multiplication happens before addition. To solve it, you undo addition first, then division.
始终按照 BIDMAS 的逆序进行运算。在表达式 2x + 3 中，乘法先于加法。解方程时，先逆加法，再逆除法。
6. Equations with Brackets | 含括号的方程
When an equation has brackets, expand them first using the distributive law. This means multiplying the term outside the bracket by each term inside the bracket.
当方程含有括号时，先使用分配律展开括号。这意味着将括号外的项乘以括号内的每一项。
to get 3x + 6 = 18. Then subtract 6 from both sides: 3x = 12. Finally divide by 3: x = 4.
解方程 3(x + 2) = 18。展开左边得到 3x + 6 = 18。然后两边减 6：3x = 12。最后除以 3：x = 4。
3(x + 2) = 18 → 3x + 6 = 18 → x = 4
or example, −2(x − 5) = 10 expands as −2x + 10 = 10 because −2 multiplied by −5 gives +10.
注意含有负号的括号。例如，−2(x − 5) = 10 展开为 −2x + 10 = 10，因为 −2 乘以 −5 等于 +10。
After expanding, the equation becomes a two-step equation, so use the method from the previous section. Expanding brackets correctly is often the most important first step.
展开后，方程就变成两步方程，因此使用上一节的方法。正确展开括号通常是最重要的第一步。
7. Equations with Variables on Both Sides | 两边含未知数的方程
nknown appears on both sides. The first step is to collect the variable terms on one side by subtracting the smaller x-term from both sides.
对于像 5x + 2 = 3x + 10 这样的方程，未知数出现在两边。第一步是通过从两边减去较小的 x 项，将变量项合并到一边。
10, which gives 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8. Divide by 2: x = 4.
两边同时减去 3x：5x − 3x + 2 = 3x − 3x + 10，得到 2x + 2 = 10。然后两边减 2：2x = 8。除以 2：x = 4。
5x + 2 = 3x + 10 → 2x = 8 → x = 4
cting 5x from both sides of the same equation gives −2x + 2 = 10 − 2x, which is less tidy.
你也可以先减较大的 x 项，但这可能会使变量变成负数。例如，对同一个方程两边同时减去 5x 会得到 −2x + 2 = 10 − 2x，这样不太简洁。
Aim to keep the variable positive by moving the smaller x-term first. This reduces mistakes with negative signs and makes the equation easier to read.
尽量先移动较小的 x 项，使变量保持为正数。这可以减少负号错误，并使方程更容易理解。
8. Equations Involving Fractions | 含分数的方程
If an equation contains fractions, you can clear them by multiplying every term by the lowest common denominator. This often makes the equation much easier to solve.
如果方程含有分数，你可以通过将每一项乘以最小公分母来消去分数。这通常会使方程更容易求解。
3 + 1 = 4. First subtract 1 from both sides: x/3 = 3. Then multiply both sides by 3: x = 9
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