📚 Solving Linear Equations for KS3 | 解一元一次方程
Linear equations are a cornerstone of the KS3 Cambridge mathematics curriculum. They appear in algebra, problem-solving, and even in topics like sequences and geometry. This article explains the key methods step by step, with examples and practice questions to build your confidence.
一元一次方程是剑桥 KS3 数学课程的核心内容。它们出现在代数、问题求解，甚至数列和几何等主题中。本文逐步讲解关键方法，并提供例题和练习题，帮助你建立信心。
1. What Is a Linear Equation? | 什么是一元一次方程？
oot. Examples include x + 5 = 9, 3x − 2 = 10, and 2(x + 1) = 8. These equations are called "linear" because their graphs are straight lines if plotted on coordinate axes.
一元一次方程是未知数只出现一次幂的方程。这意味着变量不会被平方、立方或开方。例如 x + 5 = 9、3x − 2 = 10 和 2(x + 1) = 8 都属于这类方程。这类方程被称为 "线性" 方程，因为如果在坐标轴上画图，它们的图像是一条直线。
General form: ax + b = 0, where a ≠ 0
2. The Balance Method | 天平法
Think of an equation as a pair of balanced scales. Whatever you do to one side, you must do to the other to keep the scales balanced. This is the golden rule of solving equations. You can add, subtract, multiply, or divide, as long as you apply the same operation to both sides.
把方程想象成一架平衡的天平。无论你对一边做什么，都必须对另一边做同样的操作，以保持天平平衡。这是解方程的黄金法则。你可以加、减、乘或除，但必须对方程两边同时进行相同的操作。
If a = b, then a + c = b + c, a − c = b − c, a × c = b × c, a ÷ c = b ÷ c (c ≠ 0)
3. Solving Simple One-Step Equations | 解简单一步方程
both sides. For 4x = 20, divide both sides by 4. For x ÷ 3 = 6, multiply both sides by 3.
一步方程只需要一次逆运算来分离未知数。例如，对于 x + 7 = 12，两边同时减 7 得到 x = 5。对于 x − 3 = 9，两边同时加 3。对于 4x = 20，两边同时除以 4。对于 x ÷ 3 = 6，两边同时乘以 3。
x + 7 = 12 → x = 12 − 7 = 5
4x = 20 → x = 20 ÷ 4 = 5
x ÷ 3 = 6 → x = 6 × 3 = 18
4. Solving Two-Step Equations | 解两步方程
is 5x − 7 = 18; first add 7 to both sides to get 5x = 25, then divide by 5 to find x = 5.
两步方程包含两种运算。例如，2x + 3 = 11。首先，消除加法或减法，然后消除乘法或除法。两边同时减 3：2x = 8。然后两边同时除以 2：x = 4。另一个例子是 5x − 7 = 18；首先两边同时加 7 得到 5x = 25，然后两边同时除以 5 得到 x = 5。
2x + 3 = 11
2x = 11 − 3 = 8
x = 8 ÷ 2 = 4
5. Equations with Brackets | 含括号的方程
ide the bracket, such as 2(x − 1) = 8, expand carefully: 2x − 2 = 8, so 2x = 10 and x = 5.
当方程含有括号时，首先使用分配律展开括号，然后按常规方法求解。例如，3(x + 2) = 21。展开得到 3x + 6 = 21，两边同时减 6 得到 3x = 15，然后两边同时除以 3 得到 x = 5。如果括号外是负号，例如 2(x − 1) = 8，要仔细展开：2x − 2 = 8，所以 2x = 10，x = 5。
3(x + 2) = 21
3x + 6 = 21
3x = 15 → x = 5
6. Equations with Unknowns on Both Sides | 未知数在等式两边的方程
hoose to move the variable to the left or the right, as long as you keep the equation balanced.
如果未知数出现在等式两边，就把变量项集中到一边，常数项集中到另一边。例如，5x − 3 = 2x + 9。两边同时减 2x 得到 3x − 3 = 9。然后两边同时加 3：3x = 12。最后两边同时除以 3：x = 4。你可以选择把变量移到左边或右边，只要保持方程平衡即可。
5x − 3 = 2x + 9
5x − 2x = 9 + 3
3x = 12 → x = 4
7. Equations Involving Fractions | 含分数的方程
x = 8. For (2x) ÷ 3 = 6, multiply both sides by 3 to get 2x = 18, then divide by 2: x = 9.
当方程中含有分数时，将每一项乘以分母以消去分数。例如，x ÷ 4 + 1 = 3。首先两边同时减 1：x ÷ 4 = 2。然后两边同时乘以 4：x = 8。或者，先整体乘以 4：x + 4 = 12，然后 x = 8。对于 (2x) ÷ 3 = 6，两边同时乘以 3 得到 2x = 18，然后两边同时除以 2：x = 9。
x ÷ 4 + 1 = 3
x ÷ 4 = 2 → x = 8
(2x) ÷ 3 = 6 → 2x = 18 → x = 9
8. Forming Equations from Word Problems | 从文字题列方程
nown number. Let the unknown be x, then translate the words into an equation. For example: "I think of a number, multiply it by 5, add 7, and the result is 32." Let the number be n. Then 5n + 7 = 32. Solve: 5n = 25, so n = 5. Another example: "When I double a number and subtract 9, the result is 13." This gives 2n − 9 = 13, so 2n = 22 and n = 11.
文字题通常用语言描述一个情景，并要求你找出未知数。设未知数为 x，然后把文字翻译成方程。例如："我想一个数，将它乘以 5，再加上 7，结果是 32。" 设这个数为 n。那么 5n + 7 = 32。求解：5n = 25，所以 n = 5。另一个例子："当我把一个数乘以 2 再减去 9，结果是 13。" 这给出 2n − 9 = 13，所以 2n = 22，n = 11。
5n + 7 = 32 → 5n = 25 → n = 5
2n − 9 = 13 → 2n = 22 → n = 11
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
our answer by substitution. Other errors include moving terms without changing their signs and forgetting to reverse the order of inverse operations.
常见错误包括忘记对等式两边同时进行运算、展开括号时错误处理负号，以及除法错误。例如，在 3(x − 2) =
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图