Linear Equations: Balancing and Solving Techniques | 线性方程：等量平衡与求解技巧

📚 Linear Equations: Balancing and Solving Techniques | 线性方程：等量平衡与求解技巧

Linear equations are one of the most important building blocks in the Cambridge KS3 mathematics curriculum. They appear when an unknown number, usually written as x, is connected to known numbers by addition, subtraction, multiplication or division. Being able to solve these equations confidently is essential for algebra, graphs, sequences and problem solving. This article explains the balancing method, covers one-step and two-step equations, and then moves to brackets, fractions and word problems.

线性方程是剑桥 KS3 数学课程中最重要的基础模块之一。当一个未知数（通常用 x 表示）通过加、减、乘、除与已知数联系在一起时，就会出现线性方程。能够自信地解这些方程，对学习代数、图像、数列和问题解决都至关重要。本文将讲解等量平衡法，覆盖一步方程和两步方程，然后进一步讨论括号、分数系数和应用题。

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

f a linear equation in one variable is ax + b = c, where a, b and c are numbers and a ≠ 0.

线性方程是指未知数只以一次幂出现的方程。也就是说，方程中不会出现 x²、x³ 或 1/x 这样的项，也不会出现两个未知数相乘的项。一元线性方程的一般形式是 ax + b = c，其中 a、b、c 是数，且 a ≠ 0。

inear because the variable has a power greater than 1 or is multiplied by another variable.

例如，x + 5 = 12、4x − 7 = 21 和 x ÷ 3 + 2 = 6 都是线性方程。像 x² + 3 = 12 或 xy = 10 这样的方程不是线性方程，因为未知数的次数大于 1，或者出现了两个未知数相乘。

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2. The Balancing Method | 等量平衡法

Think of an equation as a balance scale. The left side and the right side must have exactly the same value. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. Otherwise, the balance is broken and the equation is no longer true.

把方程想象成一台天平。左边和右边必须保持完全相等的值。如果你对一边加上、减去、乘以或除以一个数，就必须对另一边做完全相同的运算。否则，平衡就会被打破，方程也就不再成立了。

This rule is often called the golden rule of algebra: whatever you do to one side, you must do to the other. The goal is always to isolate the variable x on one side of the equation.

这条规则常被称为代数的黄金法则：你对一边做了什么，就必须对另一边做同样的操作。目标始终是让未知数 x 单独出现在方程的一边。

When solving an equation, work backwards through the operations that have been applied to x. For example, if x has been multiplied by 2 and then 3 has been added, undo the addition of 3 first, and then undo the multiplication by 2.

解方程时，要逆向拆解已经作用在 x 上的运算。例如，如果 x 先被乘以 2，然后加上 3，那么应先撤销加 3，再撤销乘以 2。

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3. Solving One-Step Equations | 解一步方程

A one-step equation only needs one inverse operation to find the value of x. The four basic types are addition, subtraction, multiplication and division.

一步方程只需要一步逆运算就能求出 x 的值。四种基本类型分别是加法、减法、乘法和除法。

f the equation is x + 7 = 15, subtract 7 from both sides. This gives x = 15 − 7, so x = 8.

如果方程是 x + 7 = 15，两边同时减去 7。得到 x = 15 − 7，因此 x = 8。

x + 7 = 15 ⇒ x = 15 − 7 = 8

If the equation is x − 9 = 3, add 9 to both sides. This gives x = 3 + 9, so x = 12.

如果方程是 x − 9 = 3，两边同时加上 9。得到 x = 3 + 9，因此 x = 12。

x − 9 = 3 ⇒ x = 3 + 9 = 12

If the equation is 5x = 30, divide both sides by 5. This gives x = 30 ÷ 5, so x = 6.

如果方程是 5x = 30，两边同时除以 5。得到 x = 30 ÷ 5，因此 x = 6。

5x = 30 ⇒ x = 30 ÷ 5 = 6

If the equation is x ÷ 4 = 9, multiply both sides by 4. This gives x = 9 × 4, so x = 36.

如果方程是 x ÷ 4 = 9，两边同时乘以 4。得到 x = 9 × 4，因此 x = 36。

x ÷ 4 = 9 ⇒ x = 9 × 4 = 36

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4. Solving Two-Step Equations | 解两步方程

A two-step equation contains two operations applied to x. The key is to use inverse operations in the reverse order. In most cases, you undo addition or subtraction first, and then undo multiplication or division.

两步方程包含两个作用在 x 上的运算。关键是按照相反的顺序使用逆运算。在大多数情况下，先撤销加法或减法，然后撤销乘法或除法。

ions on x are: multiply by 2, then add 3. To solve it, first subtract 3 from both sides, then divide by 2.

对于方程 2x + 3 = 11，作用在 x 上的运算是：先乘以 2，然后加上 3。要解这个方程，先两边减去 3，然后两边除以 2。

2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 8 ÷ 2 = 4

is multiplied by 3 and then 4 is subtracted. Add 4 to both sides first, then divide by 3.

另一个例子是 3x − 4 = 17。这里 x 先乘以 3，然后减去 4。先两边加上 4，然后两边除以 3。

3x − 4 = 17 ⇒ 3x = 21 ⇒ x = 21 ÷ 3 = 7

erations on x are: divide by 2, then add 5. Subtract 5 from both sides first, then multiply by 2.

对于 x ÷ 2 + 5 = 9 这样的方程，作用在 x 上的运算是：先除以 2，然后加上 5。先两边减去 5，然后两边乘以 2。

x ÷ 2 + 5 = 9 ⇒ x ÷ 2 = 4 ⇒ x = 4 × 2 = 8

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5. Equations with Brackets | 含括号的方程

wo ways. One way is to expand the bracket first: multiply each term inside the bracket by the number outside. The other way is to divide both sides by the outside number first, if it divides evenly.

当方程含有括号，例如 3(x + 2) = 18 时，可以用两种方法求解。第一种方法是先展开括号：把括号内的每一项都乘以外面的数。第二种方法是先两边除以外面的数，如果它刚好能整除的话。

3x + 6 = 18. Then subtract 6 from both sides to get 3x = 12, and divide by 3 to get x = 4.

使用展开法，3(x + 2) = 18 变成 3x + 6 = 18。然后两边减去 6，得到 3x = 12，再除以 3，得到 x = 4。

3(x + 2) = 18 ⇒ 3x + 6 = 18 ⇒ 3x = 12 ⇒ x = 4

+ 2 = 6. Then subtract 2 from both sides to get x = 4. Both methods give the same answer.

使用先除法的方法，先两边同时除以 3：x + 2 = 6。然后两边减去 2，得到 x = 4。两种方法得到相同的答案。

3(x + 2) = 18 ⇒ x + 2 = 18 ÷ 3 = 6 ⇒ x = 4

every term inside the bracket. For example, 2(x − 4) + 6 = 14 expands to 2x − 8 + 6 = 14.

对于更复杂的括号，一定要仔细展开，并记住乘以括号内的每一项。例如，2(x − 4) + 6 = 14 展开后为 2x − 8 + 6 = 14。

2x − 8 + 6 = 14 ⇒ 2x − 2 = 14 ⇒ 2x = 16 ⇒ x = 8

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6. Variables on Both Sides | 两边含未知数的方程

. The first step is to collect all the x terms on one side and all the constant terms on the other side.

有些线性方程两边都含有未知数 x，例如 5x + 2 = 3x + 10。第一步是把所有含 x 的项移到一边，把所有常数项移到另一边。

2x + 2 = 10. Then subtract 2 from both sides to get 2x = 8, and divide by 2 to get x = 4.

要解 5x + 2 = 3x + 10，两边同时减去 3x，使含 x 的项都在左边。得到 2x + 2 = 10。然后两边减去 2，得到 2x = 8，再除以 2，得到 x = 4。

5x + 2 = 3x + 10 ⇒ 2x + 2 = 10 ⇒ 2x = 8 ⇒ x = 4

s 7 = 3x + 1. Then subtract 1 from both sides to get 6 = 3x, and divide by 3 to get x = 2.

在 7 − 2x = x + 1 这样的方程中，可以两边同时加上 2x，把所有含 x 的项移到右边。得到 7 = 3x + 1。然后两边减去 1，得到 6 = 3x，再除以 3，得到 x = 2。

7 − 2x = x + 1 ⇒ 7 = 3x + 1 ⇒ 6 = 3x ⇒ x = 2

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7. Fractional Coefficients | 分数系数

solve it, first undo the addition, and then undo the division. Subtract 4 from both sides, then multiply both sides by 3.

方程中可能出现变量除以一个数的情况，例如 x ÷ 3 + 4 = 7。要解这个方程，先撤销加法，再撤销除法。两边减去 4，然后两边乘以 3。

x ÷ 3 + 4 = 7 ⇒ x ÷ 3 = 3 ⇒ x = 3 × 3 = 9

3 = 6, then multiply both sides by 3 to get 2x = 18, and finally divide by 2 to get x = 9.

如果变量的系数是分数，例如 (2x) ÷ 3 − 1 = 5，把除以 3 看作一个运算。两边加上 1，得到 (2x) ÷ 3 = 6，然后两边乘以 3，得到 2x = 18，最后除以 2，得到 x = 9。

(2x) ÷ 3 − 1 = 5 ⇒ (2x) ÷ 3 = 6 ⇒ 2x = 18 ⇒ x = 9

6, remember that 1/2 x means x ÷ 2. To isolate x, multiply both sides by 2, giving x = 12.

当方程含有 1/2 x = 6 这样的简单分数时，记住 1/2 x 表示 x ÷ 2。要分离出 x，两边同时乘以 2，得到 x = 12。

x ÷ 2 = 6 ⇒ x = 6 × 2 = 12

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8. Checking Your Solution | 检验方程的解

After solving an equation, you should always check that your value of x works. Substitute the value back into the original equation and simplify both sides. If both sides are equal, the solution is correct.

解完方程后，你应该总是检验 x 的值是否成立。把 x 的值代回原方程，并化简两边。如果两边相等，解就是正确的。

left side: 2 × 4 + 3 = 8 + 3 = 11. The right side is also 11, so the solution is correct.

例如，如果你解 2x + 3 = 11 得到 x = 4，把 4 代入左边：2 × 4 + 3 = 8 + 3 = 11。右边也是 11，所以解是正确的。

Checking is especially useful in tests because it can catch sign errors, arithmetic mistakes and incorrect operation order.

检验在考试中特别有用，因为它可以发现符号错误、计算错误和运算顺序错误。

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9. Word Problems Leading to Linear Equations | 应用题转化为线性方程

Many KS3 problems ask you to form an equation from a written description. The key is to identify the unknown, give it a letter, and translate the words into algebraic operations.

许多 KS3 题目要求你根据文字描述建立方程。关键是要找出未知数，用一个字母表示，然后把文字转化为代数运算。

Example 1: I think of a number, multiply it by 3, add 7, and the result is 22.

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