Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are one of the most important topics in the Cambridge KS3 mathematics curriculum. They appear in algebra, problem solving, sequences and even in geometry when calculating unknown angles or lengths. This guide takes you through the key methods step by step, with examples and common errors to avoid.

一元一次方程是剑桥 KS3 数学课程中最重要的主题之一。它们出现在代数、问题解决、数列甚至几何中，例如计算未知角度或长度。本指南将逐步讲解关键方法，并给出例题和需要避免的常见错误。

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1. What is a Linear Equation? | 什么是线性方程？

A linear equation is an equation where the unknown, often written as x, only appears to the first power. There are no terms like x², x³, √x or 1/x. The general form is ax + b = c, where a, b and c are constants and a is not zero.

线性方程是指未知数（通常写成 x）只出现一次方的方程。方程中没有 x²、x³、√x 或 1/x 这样的项。一般形式为 ax + b = c，其中 a、b、c 是常数，且 a 不等于零。

For example, 2x + 5 = 13, x/4 − 1 = 2 and 3(x − 2) = 9 are all linear equations. The word ‘linear’ tells us that if the equation is drawn as a graph, it forms a straight line.

例如，2x + 5 = 13、x/4 − 1 = 2 和 3(x − 2) = 9 都是线性方程。’线性’这个词告诉我们，如果把方程画成图像，它会形成一条直线。

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2. The Balance Method | 等式平衡法

An equation is like a balance scale. The left-hand side and right-hand side have the same value. To keep the balance, whatever you do to one side, you must do exactly the same to the other side.

方程就像一个天平。左边和右边的值相等。为了保持平衡，你对一边做的任何运算，必须对另一边做完全相同的运算。

This is the golden rule of algebra: keep the equation balanced. If you add 3 to one side, add 3 to the other side. If you divide one side by 2, divide the other side by 2 as well.

这是代数的黄金法则：保持方程平衡。如果你在一边加 3，另一边也要加 3。如果你用 2 除一边，另一边也必须用 2 除。

Using the balance method helps you see why solving an equation works. It is not about moving numbers randomly across the equals sign; it is about applying the same operation to both sides.

使用平衡法可以帮助你理解为什么解方程是有效的。它不是把数字随意移过等号，而是对等式两边同时施加相同的运算。

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3. Inverse Operations | 逆运算

To isolate x, we use inverse operations. Addition and subtraction are inverse operations, just like multiplication and division. Applying the inverse operation ‘undoes’ the original operation.

为了把 x 单独留在一边，我们使用逆运算。加法和减法是互逆运算，乘法和除法也是。使用逆运算可以’撤销’原来的运算。

For example, if x has 5 added to it, we subtract 5 from both sides to undo the addition. If x is multiplied by 4, we divide both sides by 4 to undo the multiplication.

例如，如果 x 被加了 5，我们就从两边同时减去 5 来撤销加法。如果 x 被乘以 4，我们就把两边同时除以 4 来撤销乘法。

x + 5 = 12 → x = 12 − 5 → x = 7

4x = 20 → x = 20 ÷ 4 → x = 5

Always apply the inverse operation to the whole side, not just to one term. This prevents many sign and fraction mistakes.

始终要把逆运算应用于整个一边，而不仅仅是某一项。这样可以避免许多符号和分数错误。

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4. Solving One-Step Equations | 解一步方程

A one-step equation needs only one inverse operation. For example, x + 7 = 15 is solved by subtracting 7 from both sides: x = 15 − 7, so x = 8.

一步方程只需要一次逆运算。例如，x + 7 = 15 的解法是从两边同时减去 7：x = 15 − 7，所以 x = 8。

For x − 3 = 10, we add 3 to both sides: x = 10 + 3, so x = 13. For x/6 = 4, we multiply both sides by 6: x = 4 × 6, so x = 24.

对于 x − 3 = 10，我们给两边同时加 3：x = 10 + 3，所以 x = 13。对于 x/6 = 4，我们把两边同时乘以 6：x = 4 × 6，所以 x = 24。

For 5x = 35, we divide both sides by 5: x = 35 ÷ 5, so x = 7. These examples cover the four basic operations: addition, subtraction, multiplication and division.

对于 5x = 35，我们把两边同时除以 5：x = 35 ÷ 5，所以 x = 7。这些例子涵盖了四种基本运算：加、减、乘、除。

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5. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. Usually we undo addition or subtraction first, then undo multiplication or division. This order follows the reverse of the order of operations.

两步方程涉及两种运算。通常我们先撤销加法或减法，再撤销乘法或除法。这个顺序与运算顺序相反。

For example, solve 3x + 2 = 17. First subtract 2 from both sides: 3x = 15. Then divide both sides by 3: x =

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