📚 Cambridge KS3 Maths: Solving Linear Equations with Perimeter Problems | 剑桥 KS3 数学：用周长问题解一元一次方程
Linear equations are one of the most useful tools in Key Stage 3 mathematics. In this lesson, we focus on a common Cambridge Lower Secondary question style: using the perimeter of a rectangle to form a linear equation. By the end, you will be able to set up, solve, and check equations from word problems.
线性方程是初中数学中最有用的工具之一。本节课我们重点讲解剑桥初中常见的题型：利用矩形的周长建立一元一次方程。学完本课后，你将能够根据文字题建立方程、求解并检验结果。
1. What Are Linear Equations? | 什么是线性方程？
. For example, 2x + 5 = 17 is a linear equation because x appears only to the first power.
线性方程是指未知数的最高次数为 1 的方程。例如 2x + 5 = 17 就是一个线性方程，因为 x 只出现一次方。
e. If we replace x with 6 in 2x + 5 = 17, we get 2 × 6 + 5 = 17, so x = 6 is the solution.
方程的解是使方程成立的未知数的值。如果把 x = 6 代入 2x + 5 = 17，就得到 2 × 6 + 5 = 17，所以 x = 6 是方程的解。
In Cambridge KS3, you will meet linear equations in many contexts, including number puzzles, angles, sequences, and perimeter problems.
在剑桥 KS3 课程中，你会在许多情境中遇到线性方程，包括数字谜题、角度、数列和周长问题。
2. Key Skills: Collecting Like Terms | 关键技能：合并同类项
Before solving equations, you need to be confident with collecting like terms. Like terms have exactly the same variable part. For example, 2x and 3x are like terms, but 2x and 3 are not.
在解方程之前，你需要熟练掌握合并同类项。同类项是指变量部分完全相同的项。例如 2x 和 3x 是同类项，但 2x 和 3 不是同类项。
fficients while keeping the variable unchanged. For example, 2x + 3x = 5x and 4x - x = 3x.
合并同类项时，把系数相加或相减，变量保持不变。例如 2x + 3x = 5x，4x - x = 3x。
When an expression contains brackets, expand them first. For example, 2(x + 1) becomes 2x + 2.
当表达式中含有括号时，要先去括号。例如 2(x + 1) 去括号后得到 2x + 2。
2(x + 1) = 2x + 2
3. Understanding Perimeter Expressions | 理解周长表达式
rectangle with length l and width w, the perimeter P is given by the formula P = 2(l + w).
图形的周长是围绕它外部的总长度。对于长为 l、宽为 w 的矩形，周长 P 由公式 P = 2(l + w) 给出。
P = 2(l + w)
If the length and width are given as algebraic expressions, we can substitute them into the perimeter formula to build an equation.
如果长和宽是用代数式给出的，我们可以把它们代入周长公式，从而建立一个方程。
For example, if a rectangle has length (2x + 3) cm and width (x - 1) cm, the perimeter expression is:
例如，如果一个矩形的长为 (2x + 3) cm，宽为 (x - 1) cm，那么周长表达式为：
P = 2[(2x + 3) + (x - 1)]
4. Setting Up the Equation | 建立方程
Once you have the perimeter expression, read the word problem carefully to find the given perimeter value. In our example, the perimeter is 34 cm.
得到周长表达式后，仔细阅读文字题，找到已知的周长值。在我们的例子中，周长是 34 cm。
Set the algebraic expression equal to the known perimeter. This gives the equation:
让代数表达式等于已知
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