Solving Linear Equations with Brackets and Fractions | 解含括号与分数的线性方程

📚 Solving Linear Equations with Brackets and Fractions | 解含括号与分数的线性方程

Linear equations are the foundation of algebra at KS3. They appear in problem solving, graphs, sequences and many later topics. This article explains how to solve equations that involve brackets and fractions, using clear steps and the balance method.

线性方程是 KS3 代数的基础。它们在解决问题、图像、数列以及许多后续主题中都会出现。本文讲解如何用清晰的步骤和平衡法来解含有括号与分数的方程。

At Cambridge KS3 level, you are expected to move beyond simple one-step equations. You need to handle equations such as 2(x − 3) = 10 and (2x + 1)/3 = 5 confidently. The key is to keep the equation balanced at every step and to check your answer by substitution.

在剑桥 KS3 阶段，你需要从简单的一步方程提升到更复杂的类型。你要能够自信地解诸如 2(x − 3) = 10 和 (2x + 1)/3 = 5 这样的方程。关键是在每一步都保持等式平衡，并通过代入来检验答案。

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

A linear equation is an equation where the unknown, usually written as x, is only raised to the power of 1. This means the graph of the equation forms a straight line. Examples include x + 3 = 7, 2x − 5 = 9 and 3(x + 1) = 12.

线性方程是未知数（通常写作 x）的指数仅为 1 的方程。这意味着它的图像是一条直线。例如 x + 3 = 7、2x − 5 = 9 和 3(x + 1) = 12。

The word ‘linear’ comes from ‘line’. If you plot the solutions of a linear equation in two variables, you get a straight line. In KS3, most linear equations have one unknown and one solution, although some special equations may have no solution or infinitely many solutions.

‘线性’ 一词来自 ‘直线’。如果你绘制含有两个变量的线性方程的解，你会得到一条直线。在 KS3 中，大多数线性方程只有一个未知数和一个解，但有些特殊方程可能无解或有无穷多个解。

2. The Balance Method | 天平法

Think of an equation as a balance scale. The left side and the right side must always have the same value. Whatever you do to one side, you must do exactly the same to the other side. This is called the balance method.

把方程想象成一个天平。左边和右边必须始终保持相等的值。你对一边做的任何操作，必须对另一边做完全相同的操作。这叫做天平法。

For example, if you add 3 to the left side, you must add 3 to the right side. If you divide the left side by 2, you must divide the right side by 2 as well. This keeps the equation balanced and leads to the correct solution.

例如，如果你给左边加 3，就必须给右边也加 3。如果你把左边除以 2，就必须把右边也除以 2。这样能保持等式平衡，从而得到正确的解。

3. Solving Simple One-Step Equations | 解简单一步方程

A one-step equation needs only one inverse operation to solve. If the equation has addition, you subtract. If it has multiplication, you divide. The goal is to isolate x on one side.

一步方程只需要一次逆运算就可以解出。如果方程中有加法，你就做减法。如果有乘法，你就做除法。目标是让 x 单独在一边。

Example 1: Solve x + 5 = 12.

例 1：解方程 x + 5 = 12。

x + 5 − 5 = 12 − 5 → x = 7

Example 2: Solve 4x = 20.

例 2：解方程 4x = 20。

4x ÷ 4 = 20 ÷ 4 → x = 5

Example 3: Solve x ÷ 3 = 6.

例 3：解方程 x ÷ 3 = 6。

x ÷ 3 × 3 = 6 × 3 → x = 18

4. Solving Two-Step Equations | 解两步方程

A two-step equation involves two operations. You must undo them in reverse order. Usually you undo addition or subtraction first, then undo multiplication or division.

两步方程涉及两种运算。你必须按相反的顺序去掉它们。通常先去加法或减法，再去乘法或除法。

Example: Solve 2x + 3 = 11.

例：解方程 2x + 3 = 11。

Step 1: Subtract 3 from both sides.

步骤 1：两边同时减去 3。

2x + 3 − 3 = 11 − 3 → 2x = 8

Step 2: Divide both sides by 2.

步骤 2：两边同时除以 2。

2x ÷ 2 = 8 ÷ 2 → x = 4

Always reverse the order of operations. Since the original equation multiplies by 2 and then adds 3, you first undo the addition and then undo the multiplication.

一定要按运算的相反顺序来解。因为原方程先乘以 2 再加 3，所以你要先去掉加法，再去掉乘法。

5. Expanding Brackets First | 先去括号

When an equation contains brackets, you normally expand them before applying the balance method. Use the distributive law: a(b + c) = ab + ac. This makes the equation easier to solve.

当方程含有括号时，通常先去括号再使用天平法。使用分配律：a(b + c) = ab + ac。这样方程会更容易解。

Example 1: Solve 3(x + 2) = 21.

例 1：解方程 3(x + 2) = 21。

Step 1: Expand the bracket.

步骤 1：展开括号。

3x + 6 = 21

Step 2: Subtract 6 from both sides.

步骤 2：两边同时减去 6。

3x = 15

Step 3: Divide both sides by 3.

步骤 3：两边同时除以 3。

x = 5

Example 2: Solve 2(x − 4) = x + 5.

例 2：解方程 2(x − 4) = x + 5。

Step 1: Expand the left side.

步骤 1：展开左边。

2x − 8 = x + 5

Step 2: Subtract x from both sides.

步骤 2：两边同时减去 x。

x − 8 = 5

Step 3: Add 8 to both sides.

步骤 3：两边同时加 8。

x = 13

6. Equations with Fractions | 含分数的方程

Equations with fractions can look difficult, but they become easier if you work step by step. You can treat a fraction bar as a division. For example, x/3 means x divided by 3.

含有分数的方程看起来可能很难，但只要一步步来就会变得简单。你可以把分数线看作除法。例如 x/3 表示 x 除以 3。

Example 1: Solve x/3 = 4.

例 1：解方程 x/3 = 4。

x ÷ 3 × 3 = 4 × 3 → x = 12

Example 2: Solve (x + 1)/2 = 5.

例 2：解方程 (x + 1)/2 = 5。

Step 1: Multiply both sides by 2.

步骤 1：两边同时乘以 2。

x + 1 = 10

Step 2: Subtract 1 from both sides.

步骤 2：两边同时减去 1。

x = 9

7. Clearing Denominators | 去分母

When an equation has more than one fraction, it is often best to clear the denominators first. Multiply every term on both sides by the lowest common multiple (LCM) of all denominators. This removes the fractions and gives a simpler equation.

当方程含有多个分数时，通常最好先去分母。两边每一项都乘以所有分母的最小公倍数 (LCM)。这样可以消去分数，得到一个更简单的方程。

Example: Solve (x + 1)/2 = (x − 1)/3.

例：解方程 (x + 1)/2 = (x − 1)/3。

The denominators are 2 and 3. The LCM is 6. Multiply every term by 6.

分母是 2 和 3。最小公倍数是 6。把每一项都乘以 6。

6 × (x + 1)/2 = 6 × (x − 1)/3

Simplify each side.

化简两边。

3(x + 1) = 2(x − 1)

Expand the brackets.

展开括号。

3x + 3 = 2x − 2

Subtract 2x from both sides.

两边同时减去 2x。

x + 3 = −2

Subtract 3 from both sides.

两边同时减去 3。

x = −5

8. Checking Your Solution | 检验你的解

Always substitute your answer back into the original equation to check it. If the left side equals the right side, your solution is correct. This is a very important habit in exams.

一定要把你的答案代回原方程进行检验。如果左边等于右边，你的解就是正确的。这是考试中非常重要的习惯。

Example: Check x = 4 for the equation 2x + 3 = 11.

例：检验 x = 4 是否满足方程 2x + 3 = 11。

Left side = 2 × 4 + 3 = 8 + 3 = 11

Right side = 11

Since the left side equals the right side, x = 4 is correct.

因为左边等于右边，所以 x = 4 是正确的。

If the two sides are not equal, you have made a mistake. Go back and check each step carefully.

如果两边不相等，说明你出错了。回去仔细检查每一步。

9. Common Mistakes to Avoid | 常见错误与避免方法

Many students lose marks on linear equations because of small mistakes. Here are the most common ones and how to avoid them.

许多学生在解线性方程时因为小错误而失分。以下是最常见的错误以及如何避免它们。

Forgetting to do the same operation on both sides.

忘记在两边做相同的运算。
Incorrectly expanding brackets, such as writing 3(x + 2) = 3x + 2 instead of 3x + 6.

去括号错误，例如把 3(x + 2) 写成 3x + 2，而不是 3x + 6。
Mishandling negative signs, especially with expressions like 2(x − 4).

处理负号错误，尤其是像 2(x − 4) 这样的表达式。
Forgetting to multiply every term when clearing denominators.

去分母时忘记乘以每一项。
Not checking the final answer by substitution.

没有通过代入检验最终答案。

Take your time with each step. Write one operation per line and keep your working neat. This makes it much easier to spot any errors.

每一步都要慢慢来。每行只写一个运算，并保持书写整洁。这样更容易发现错误。

10. Practice Questions | 练习题

Try these questions to test your understanding. Solve each equation and check your answer by substitution.

尝试以下题目来检验你的理解。解出每个方程，并通过代入检验答案。

Solve x + 7 = 15.

解方程 x + 7 = 15。
Solve 5x − 2 = 18.

解方程 5x − 2 = 18。
Solve 4(x + 3) = 32.

解方程 4(x + 3) = 32。
Solve 2(x − 5) = x + 7.

解方程 2(x − 5) = x + 7。
Solve (x + 2)/4 = 3.

解方程 (x + 2)/4 = 3。
Solve (2x + 1)/3 = (x + 4)/2.

解方程 (2x + 1)/3 = (x + 4)/2。

Work through these carefully, showing every step. If you get stuck, revisit the matching section above and try again.

认真完成这些题目，写出每一步。如果卡住了，回顾前面相应的章节再试一次。

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Solving Linear Equations with Brackets and Fractions | 解含括号与分数的线性方程