Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

gebra skill that appears in many exercises, including the questions styled for p136_2.pdf.

本剑桥 KS3 数学复习指南逐步讲解如何解一元一次方程。一元一次方程是代数中的核心技能，出现在许多练习中，包括 p136_2.pdf 类型的题目。

By the end of this guide, you should be able to balance an equation, undo operations in the correct order, and check your final answer confidently.

学完本指南后，你应该能够平衡方程、按正确顺序进行逆运算，并自信地检验最终答案。

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1. What Is a Linear Equation? | 什么是一元一次方程

A linear equation is an equation in which the unknown value, usually written as x, only appears to the power of 1.

一元一次方程是指未知数（通常写成 x）只出现一次方的方程。

For example, 2x + 3 = 11 is linear, because the highest power of x is 1.

例如，2x + 3 = 11 是一次方程，因为 x 的最高次幂是 1。

The general form is ax + b = c, where a, b and c are known numbers and x is the unknown.

一般形式为 ax + b = c，其中 a、b、c 是已知数，x 是未知数。

Example: 2x + 3 = 11

An example of a non-linear equation is x² + 3 = 7, because the unknown x is squared.

一个非一次方程的例子是 x² + 3 = 7，因为未知数 x 被平方了。

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2. The Balancing Method | 等号平衡法

Think of an equation as a balanced scale. Whatever you do to one side, you must do exactly the same to the other side.

把方程想象成一架平衡的天平。你对一边做什么，就必须对另一边做完全相同的操作。

We use inverse operations to isolate x. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.

我们使用逆运算来分离出 x。加法和减法互为逆运算，乘法和除法互为逆运算。

For example, to solve x + 5 = 12, subtract 5 from both sides.

例如，要解 x + 5 = 12，等式两边同时减去 5。

x + 5 = 12   →   x = 12 - 5   →   x = 7

Keeping the balance is the most important rule in equation solving.

保持平衡是解方程最重要的规则。

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3. Solving One-Step Equations | 解一步方程

One-step equations need only one inverse operation to find the value of x.

一步方程只需要一次逆运算就能求出 x 的值。

If a number is subtracted from x, add that number to both sides.

如果从 x 中减去一个数，就在等式两边同时加上这个数。

x - 7 = 15   →   x = 15 + 7   →   x = 22

If x is multiplied by a number, divide both sides by that number.

如果 x 乘以一个数，就在等式两边同时除以这个数。

5x = 35   →   x = 35 ÷ 5   →   x = 7

Always do the same operation to both sides so the equation stays balanced.

始终对等式两边进行相同的运算，使方程保持平衡。

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4. Solving Two-Step Equations | 解两步方程

Two-step equations contain two operations, so we undo them in the reverse order.

两步方程包含两个运算，因此我们要按逆向顺序撤销这些运算。

ultiply by 2, then add 3. To solve, reverse the order: subtract 3 first, then divide by 2.

对于 2x + 3 = 11，运算是：先乘以 2，再加 3。解题时逆转顺序：先减 3，再除以 2。

2x + 3 = 11   →   2x = 11 - 3   →   2x = 8   →   x = 4

Do not divide by 2 before subtracting 3, because that changes the equation incorrectly.

不要在减 3 之前除以 2，因为那样会错误地改变方程。

Undo addition or subtraction before multiplication or division in a two-step equation.

在两步方程中，先撤销加法或减法，再撤销乘法或除法。

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5. Equations with Brackets | 含括号的方程

When an equation has brackets, expand them first using the distributive law.

当方程含有括号时，首先用分配律展开括号。

For 3(x + 4) = 27, multiply both terms inside the bracket by 3.

对于 3(x + 4) = 27，将括号内的两项都乘以 3。

3(x + 4) = 27   →   3x + 12 = 27   →   3x = 15   →   x = 5

You can also divide both sides by the number outside the bracket first, if it divides evenly.

如果括号外的数能整除，也可以先两边同时除以括号外的数。

3(x + 4) = 27   →   x + 4 = 9   →   x = 5

Both methods are valid, but you should show your working clearly.

两种方法都正确，但你应该清晰展示解题过程。

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6. Unknown on Both Sides | 未知数在等号两边

Sometimes x appears on both sides of the equals sign. Collect the x terms on one side first.

有时 x 同时出现在等号两边

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