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 foundations of Key Stage 3 mathematics. They appear in nearly every Cambridge Lower Secondary checkpoint paper, and the skills you build here will be used again in algebra, graphs and problem solving. This article explains how to solve linear equations step by step, from one-step problems to equations with brackets and unknowns on both sides.

一元一次方程是 KS3 数学最重要的基础之一。它们几乎出现在每一张剑桥初中 checkpoint 试卷中，你在这里建立的技能将在代数、图像和问题解决中反复使用。本文将逐步讲解如何解一元一次方程，从一步方程到含括号和未知数在等号两侧的方程。

1. What Is an Equation? | 什么是方程？

An equation is a mathematical statement that says two expressions are equal. It always contains an equals sign (=). For example, x + 4 = 9 is an equation because it says that when 4 is added to x, the result is 9.

方程是一个数学陈述，说明两个表达式相等。它总是包含等号（=）。例如，x + 4 = 9 是一个方程，因为它表示 x 加上 4 后等于 9。

An expression such as 2x + 3 has no equals sign, so it is not an equation. The letter x is called the variable or unknown, and solving the equation means finding the value of x that makes the statement true.

像 2x + 3 这样的表达式没有等号，因此它不是方程。字母 x 被称为变量或未知数，解方程意味着求出使等式成立的 x 的值。

x + 4 = 9 means x = 5

In this case, x = 5 is the solution because 5 + 4 = 9. Understanding this idea is the first step towards solving more complicated equations.

在这种情况下，x = 5 是解，因为 5 + 4 = 9。理解这个思想是解更复杂方程的第一步。

2. The Balance Method | 天平法

Think of an equation as a balance scale. The left side and the right side must have exactly the same value. Whatever you do to one side, you must also do to the other side to keep the equation balanced.

把方程想象成一个天平。左边和右边的数值必须完全相同。你对一边做的任何操作，也必须对另一边做同样的操作，以保持方程平衡。

This rule is the golden rule of algebra: you can add, subtract, multiply or divide, but you must apply the same operation to both sides. This idea allows us to isolate x by undoing operations in reverse order.

这是代数的黄金法则：你可以加、减、乘或除，但必须对等号两边施加相同的运算。这一思想使我们能够通过逆序撤销运算来分离出 x。

If a = b, then a + c = b + c

You should always write each step clearly below the previous one. This makes your working easy to follow and helps you spot mistakes quickly.

你应该始终把每一步清楚地写在上一步的下方。这使你的解题过程易于理解，也能帮助你快速发现错误。

3. Solving One-Step Equations | 解一步方程

A one-step equation needs only one inverse operation to solve. For example, solve x + 5 = 12. To undo ‘+5’, subtract 5 from both sides.

一步方程只需一次逆运算即可求解。例如，解 x + 5 = 12。为了撤销 ‘+5’，两边同时减去 5。

x + 5 − 5 = 12 − 5 ⇒ x = 7

If the equation is x − 3 = 8, add 3 to both sides: x = 11. If it is 4x = 20, divide both sides by 4: x = 5. If it is x ÷ 6 = 2, multiply both sides by 6: x = 12.

如果方程是 x − 3 = 8，两边同时加 3：x = 11。如果是 4x = 20，两边同时除以 4：x = 5。如果是 x ÷ 6 = 2，两边同时乘以 6：x = 12。

The four basic inverse operations are addition and subtraction, multiplication and division. Always choose the opposite operation to move a number away from x.

四种基本逆运算是加与减、乘与除。始终选择相反的运算把数字从 x 旁边移走。

4. Solving Two-Step Equations | 解两步方程

Two-step equations require two inverse operations. Consider 2x + 3 = 11. First undo the ‘+3’, then undo the ‘×2’.

两步方程需要两次逆运算。考虑 2x + 3 = 11。首先撤销 ‘+3’，然后撤销 ‘×2’。

2x + 3 − 3 = 11 − 3 ⇒ 2x = 8
2x ÷ 2 = 8 ÷ 2 ⇒ x = 4

Notice that we operate on the whole side, not just part of it. Writing each step on a new line helps you avoid errors and shows clear working.

注意，我们对整个一边进行运算，而不仅仅是其中一部分。每一步写在新的行上可以帮助你避免错误，并呈现清晰的解题过程。

Always check: when x = 4, 2(4) + 3 = 8 + 3 =

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