📚 Solving Linear Equations: From One-Step to Multi-Step | 解一元一次方程：从一步到多步
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in algebra, problem solving, and even in real-life situations such as working out costs, distances, or unknown amounts. In this revision guide, you will learn how to solve linear equations step by step, from simple one-step examples to more challenging equations with brackets and fractions. The key idea is to keep the equation balanced while you isolate the unknown variable.
一元一次方程是 KS3 数学中最重要的基础内容之一。它们出现在代数、问题解决，甚至现实生活中的情境中，例如计算成本、距离或未知数量。在本复习指南中，你将逐步学习如何解一元一次方程，从简单的一步方程到含有括号和分数的更有挑战性的方程。核心思路是：在分离未知变量的过程中始终保持等式平衡。
1. What Is a Linear Equation? | 什么是一元一次方程？
Linear equations form a straight line when drawn on a graph, which is why they are called 'linear'.
一元一次方程是指未知变量（通常写作 x）的指数仅为 1 的方程。这意味着在一元一次方程中你永远不会看到 x²、x³ 或更高次幂。例如，2x + 3 = 11 是一个一元一次方程，但 x² + 3 = 11 不是。一元一次方程在图上画出来是一条直线，这就是它们被称为 "线性" 的原因。
2. The Balance Method | 天平法
The balance method is the most reliable way to solve linear equations. Think of an equation as a set of scales: the left-hand side must always equal the right-hand side. Whatever you do to one side, you must do to the other side. This keeps the equation balanced while you gradually isolate the variable. For example, if you add 4 to the left, you must add 4 to the right.
天平法是解一元一次方程最可靠的方法。把方程想象成一架天平：左边必须始终等于右边。你对一边做的任何操作，都必须对另一边做同样的操作。这样在你逐步分离变量时，方程始终保持平衡。例如，如果你在左边加 4，就必须在右边也加 4。
3. One-Step Equations | 一步方程
subtract 7 from both sides to get x = 8. If 3x = 21, divide both sides by 3 to get x = 7.
一步方程只需要一次运算就能求出 x 的值。你使用逆运算来撤销对 x 施加的运算。例如，如果 x + 7 = 15，两边同时减去 7，得到 x = 8。如果 3x = 21，两边同时除以 3，得到 x = 7。
x + 7 = 15
x = 15 − 7
x = 8
4. Two-Step Equations | 两步方程
subtract 5 from both sides first to get 2x = 12, then divide both sides by 2 to get x = 6.
两步方程涉及两种运算，例如乘法和加法。总是先撤销加法或减法，再撤销乘法或除法。例如，在 2x + 5 = 17 中，先两边同时减去 5，得到 2x = 12，然后两边同时除以 2，得到 x = 6。
2x + 5 = 17
2x = 17 − 5
2x = 12
x = 12 ÷ 2
x = 6
5. Equations with Brackets | 带括号的方程
o-step equation: subtract 12 from both sides to get 3x = 15, and divide by 3 to get x = 5.
当方程含有括号时，你需要先用分配律展开括号。将括号外的项乘以括号内的每一项。例如，3(x + 4) = 27 变为 3x + 12 = 27。然后按两步方程来解：两边同时减去 12，得到 3x = 15，再除以 3，得到 x = 5。
3(x + 4) = 27
3x + 12 = 27
3x = 15
x = 5
6. Unknown on Both Sides | 未知数在两边
3x + 2 = 14. Then subtract 2 from both sides to get 3x = 12, and divide by 3 to get x = 4.
如果未知数出现在方程的两边，你的第一步是把所有含 x 的项移到一边。选择系数较大的一边以避免出现负数。例如，在 5x + 2 = 2x + 14 中，两边同时减去 2x，得到 3x + 2 = 14。然后两边同时减去 2，得到 3x = 12，再除以 3，得到 x = 4。
5x + 2 = 2x + 14
5x − 2x + 2 = 14
3x + 2 = 14
3x = 12
x = 4
7. Equations with Fractions | 分数方程
both sides to get x = 9. Always check that your answer makes the original fraction valid.
含有分数的方程看起来可能很难，但你可以通过将每一项乘以最小公分母来消去分数。例如，在 x/3 + 2 = 5 中，将每一项乘以 3，得到 x + 6 = 15。然后两边同时减去 6，得到 x = 9。一定要检查你的答案是否能使命题中的分数有意义。
x/3 + 2 = 5
3 × (x/3) + 3 × 2 = 3 × 5
x + 6 = 15
x = 15 − 6
x = 9
8. Checking Your Solution | 检验你的解
ches the right-hand side. If it does not match, you have made a mistake and should go back over your steps.
解出任何方程后，一定要把你的答案代回原方程，检查它是否成立。例如，如果你解出 4x − 3 = 13 并得到 x = 4，通过计算 4 × 4 − 3 来检验。结果是 16 − 3 = 13，与右边相等。如果不相等，说明你犯了错误，应该回头检查你的步骤。
4(4) − 3 = 16 − 3 = 13 ✓
9. Common Mistakes to Avoid | 常见错误
Many students lose marks by making avoidable mistakes. The most common ones include forgetting to apply an operation to both sides, combining unlike terms such as x and x², and misusing negative signs. Always write each step clearly and check your work. Another common error is dividing by the wrong number at the final step, so pay special attention to the coefficient of x.
许多学生因为可避免的错误而失分。最常见的错误包括忘记对两边同时进行运算，将 x 和 x² 这类不同类项合并，以及误用负号。一定要把每一步写清楚并检查你的作业。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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