📚 Solving Linear Equations for KS3 Cambridge Maths | 剑桥初中数学：解一元一次方程
Linear equations are one of the most important algebra skills in the Cambridge KS3 Mathematics curriculum. They appear in everything from number puzzles to real-life problems involving distance, money and measurement. The examples below reflect the type of algebra question found on page 267, Question 1 of the Cambridge KS3 Mathematics practice materials. This article reviews how to solve linear equations step by step using the balance method.
一元一次方程是剑桥 KS3 数学课程中最重要的代数技能之一。从数字谜题到涉及距离、金钱和测量的现实问题，它们无处不在。以下例子反映了剑桥 KS3 数学练习资料第 267 页第 1 题中常见的代数题型。本文将复习如何使用天平法逐步解一元一次方程。
1. What is a Linear Equation? | 什么是一元一次方程？
An equation is a statement that two expressions are equal. Equations are different from expressions because an equation contains an equals sign and can be solved to find the value of an unknown.
方程是两个表达式相等的陈述。方程与代数式不同，因为方程含有等号，并且可以通过求解找到未知数的值。
to the power of 1. For example, 2x + 3 = 11 is linear because the highest power of x is 1.
一元一次方程中未知数的指数只能为 1。例如，2x + 3 = 11 是一元一次方程，因为 x 的最高次数是 1。
2x + 3 = 11
In this equation, x is the unknown. The left-hand side is 2x + 3 and the right-hand side is 11. The solution is the value of x that makes both sides equal.
在这个方程中，x 是未知数。左边是 2x + 3，右边是 11。解就是使两边相等的 x 的值。
2. The Balance Method | 天平法
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep the scales balanced.
把方程想成一台天平。你对一边做的任何操作，必须对另一边也做同样的操作，才能保持天平平衡。
This means you can add, subtract, multiply or divide both sides by the same non-zero number without changing the solution. This is called the balance method.
这意味着你可以在方程两边同时加上、减去、乘以或除以同一个非零数，而不会改变方程的解。这就是天平法。
- Add the same number to both sides.
- Subtract the same number from both sides.
- Multiply both sides by the same number.
- Divide both sides by the same non-zero number.
两边同时加上同一个数；两边同时减去同一个数；两边同时乘以同一个数；两边同时除以同一个非零数。
Using the balance method keeps the equation true while you work towards isolating the unknown.
使用天平法可以在你逐步分离未知数的过程中保持方程成立。
3. Solving One-Step Equations | 解一步方程
o isolate the unknown. For example, x + 5 = 12 is solved by subtracting 5 from both sides.
一步方程只需要一步逆运算就能求出未知数。例如，x + 5 = 12 可以通过两边同时减去 5 来求解。
x + 5 = 12
x = 12 − 5
x = 7
Similarly, 3x = 18 is solved by dividing both sides by 3.
类似地，3x = 18 可以通过两边同时除以 3 来求解。
3x = 18
x = 18 ÷ 3
x = 6
For a subtraction equation such as x − 4 = 9, add 4 to both sides to get x = 13.
对于减法方程，如 x − 4 = 9，两边同时加上 4，得到 x = 13。
x − 4 = 9
x = 9 + 4
x = 13
4. Solving Two-Step Equations | 解两步方程
ions. Consider 2x + 3 = 11. First subtract 3 from both sides, then divide both sides by 2.
两步方程需要两次逆运算。考虑 2x + 3 = 11。首先两边同时减去 3，然后两边同时除以 2。
2x + 3 = 11
2x = 11 − 3
2x = 8
x = 8 ÷ 2
x = 4
Always undo the addition or subtraction before the multiplication or division. The order of inverse operations follows the reverse of the order of operations.
始终先处理加减法，再处理乘除法。逆运算的顺序与运算顺序相反。
Another example is 3x − 4 = 8. Add 4 to both sides first, then divide by 3.
另一个例子是 3x − 4 = 8。先两边同时加 4，然后两边同时除以 3。
3x − 4 = 8
3x = 8 + 4
3x = 12
x = 12 ÷ 3
x = 4
5. Equations with Brackets | 含括号的方程
nd them first using the distributive law. For example, 3(x + 4) = 21 becomes 3x + 12 = 21.
当方程含有括号时，首先用分配律展开。例如，3(x + 4) = 21 展开为 3x + 12 = 21。
3(x + 4) = 21
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3x + 12 = 21
3x = 21 − 12
3x = 9
x = 3
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