Solving Linear Equations Step by Step for KS3 | 解一元一次方程分步教学（KS3）

📚 Solving Linear Equations Step by Step for KS3 | 解一元一次方程分步教学（KS3）

Linear equations are one of the most important building blocks in KS3 Cambridge Mathematics. Being able to solve them confidently helps you with algebra, graphs, and real-life problem solving.

线性方程是剑桥 KS3 数学中最重要的基石之一。能够熟练解方程将帮助你更好地学习代数、图像和现实生活中的问题解决。

This article provides a complete revision guide, from the balance method to equations with brackets and fractions, with fully worked examples and common mistakes to avoid.

本文提供一份完整的复习指南，从天平法到含括号和分数的方程，配有详细的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 in which the highest power of the unknown variable is 1.

线性方程是指未知数的最高次数为 1 的方程。

For example, 3x + 5 = 17, 2y − 4 = 10 and a/3 + 2 = 5 are all linear equations.

例如，3x + 5 = 17、2y − 4 = 10 和 a/3 + 2 = 5 都是线性方程。

The goal is to find the value of the unknown that makes both sides equal.

解题目标是找到使方程两边相等的未知数的值。

General form: ax + b = c

Here a, b and c are constants, and x is the unknown variable.

其中 a、b、c 是常数，x 是未知数。

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2. The Balance Method | 天平法

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

把方程想象成一架平衡的天平。你对一边做什么，另一边也必须做同样的操作。

This is called the balance method and it keeps the equation true.

这就是天平法，它能保持方程仍然成立。

To solve an equation, use inverse operations to isolate the variable step by step.

解方程时，使用逆运算逐步将未知数单独留在等号一边。

x + 3 = 10 → x + 3 − 3 = 10 − 3 → x = 7

Adding or subtracting the same amount from both sides does not change the solution.

两边同时加上或减去同一个数，不会改变方程的解。

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3. Solving One-Step Equations: Addition and Subtraction | 解一步方程：加法与减法

If the equation has addition, subtract the same number from both sides.

如果方程中有加法，就在两边同时减去同一个数。

If the equation has subtraction, add the same number to both sides.

如果方程中有减法，就在两边同时加上同一个数。

Example 1: Solve x + 6 = 14.

例 1：解方程 x + 6 = 14。

x + 6 − 6 = 14 − 6 → x = 8

Example 2: Solve y − 5 = 12.

例 2：解方程 y − 5 = 12。

y − 5 + 5 = 12 + 5 → y = 17

Always use the inverse operation to undo addition or subtraction.

始终使用逆运算来消除加法或减法。

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4. Solving One-Step Equations: Multiplication and Division | 解一步方程：乘法与除法

When the variable is multiplied by a number, divide both sides by that number.

当未知数乘以一个数时，两边同时除以这个数。

When the variable is divided by a number, multiply both sides by that number.

当未知数除以一个数时，两边同时乘以这个数。

Example 3: Solve 4x = 28.

例 3：解方程 4x = 28。

4x ÷ 4 = 28 ÷ 4 → x = 7

Example 4: Solve x ÷ 3 = 9.

例 4：解方程 x ÷ 3 = 9。

x ÷ 3 × 3 = 9 × 3 → x = 27

This method works because multiplication and division are inverse operations.

这个方法有效，因为乘法和除法互为逆运算。

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5. Solving Two-Step Equations | 解两步方程

A two-step equation has two operations attached to the unknown. Reverse them in the correct order.

两步方程中未知数带两个运算。需要按照正确顺序逆推求解。

Always undo addition or subtraction first, then undo multiplication or division.

总是先消除加法或减法，再消除乘法或除法。

Example 5: Solve 5x + 3 = 18.

例 5：解方程 5x + 3 = 18。

5x + 3 − 3 = 18 − 3 → 5x = 15

5x ÷ 5 = 15 ÷ 5 → x = 3

Example 6: Solve x/4 − 2 = 6.

例 6：解方程 x/4 − 2 = 6。

x/4 − 2 + 2 = 6 + 2 → x/4 = 8

x/4 × 4 = 8 × 4 → x = 32

Working step by step helps prevent sign and operation mistakes.

分步计算可以帮助避免符号和运算错误。

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6. Equations with Brackets | 含括号的方程

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

当方程含有括号时，先用分配律展开括号。

The distributive law says a(b + c) = ab + ac.

分配律表示 a(b + c) = ab + ac。

Example 7: Solve 3(x + 2) = 21.

例 7：解方程 3(x + 2) = 21。

3 × x + 3 × 2 = 21 → 3x + 6 = 21

3x = 21 − 6 → 3x = 15 → x = 5

Always multiply every term inside the bracket by the number outside.

总是用括号外的数乘以括号内的每一项。

After expanding, continue with the balance method as usual.

展开后，像往常一样继续使用天平法求解。

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7. Equations with Variables on Both Sides | 两边含未知数的方程

If the unknown appears on both sides of the equation, collect variables on one side first.

如果未知数出现在等号两边，先将其合并到等号一边。

Use addition or subtraction to move the smaller variable term to the other side.

用加法或减法将较小的未知数项移到另一边。

Example 8: Solve 5x = 2x + 12.

例 8：解方程 5x = 2x + 12。

5x − 2x = 2x + 12 − 2x → 3x = 12

x = 12 ÷ 3 → x = 4

Example 9: Solve 7x − 5 = 3x + 11.

例 9：解方程 7x − 5 = 3x + 11。

7x − 3x − 5 = 3x − 3x + 11 → 4x − 5 = 11

4x = 11 + 5 → 4x = 16 → x = 4

This technique simplifies the equation into a familiar one-step or two-step form.

这个方法可以将方程化简为熟悉的一步或两步方程形式。

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8. Equations with Fractions | 含分数的方程

Fractions in equations can be cleared by multiplying every term by the common denominator.

方程中的分数可以通过将每一项乘以公分母来消除。

Alternatively, treat the fraction as a division and undo it step by step.

也可以将分数看作除法，并分步进行逆运算。

Example 10: Solve x/5 + 2 = 6.

例 10：解方程 x/5 + 2 = 6。

x/5 + 2 − 2 = 6 − 2 → x/5 = 4

x/5 × 5 = 4 × 5 → x = 20

Example 11: Solve (2x)/3 = 8.

例 11：解方程 (2x)/3 = 8。

2x ÷ 3 × 3 = 8 × 3 → 2x = 24 → x = 12

If there are several fractions, multiply both sides by the least common denominator first.

如果有多个分数，先两边乘以最小公分母。

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9. Using Equations to Solve Word Problems | 用方程解应用题

Many real-life problems can be solved by forming a linear equation. Start by defining the unknown with a letter.

许多实际问题都可以通过建立线性方程来解决。首先用字母设未知数。

Translate the words into mathematical operations and build an equation.

将文字转化为数学运算，并建立方程。

Example 12: The perimeter of a rectangle is 26 cm. The length is 3 cm more than the width. Find the width.

例 12：一个长方形的周长是 26 cm。长比宽多 3 cm。求宽。

Let the width be w cm. Then the length is w + 3 cm.

设宽为 w cm，则长为 w + 3 cm。

Perimeter = 2(length + width)

2(w + 3 + w) = 26 → 2(2w + 3) = 26

4w + 6 = 26 → 4w = 20 → w = 5

So the width is 5 cm and the length is 8 cm.

所以宽为 5 cm，长为 8 cm。

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10. Checking Your Solution | 检验你的答案

After finding a solution, substitute it back into the original equation to check.

求出解后，将其代回原方程进行检验。

If both sides are equal, your answer is correct.

如果两边相等，说明答案正确。

Example 13: For the equation 5x + 3 = 18, we found x = 3.

例 13：对于方程 5x + 3 = 18，我们解得 x = 3。

Left side = 5 × 3 + 3 = 15 + 3 = 18 = Right side

This check proves the solution is valid.

这个检验证明解是正确的。

Always write the check as a separate line to avoid careless errors.

总是将检验过程单独写一行，以避免粗心错误。

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11. Common Mistakes | 常见错误

One common mistake is forgetting to perform the same operation on both sides of the equation.

一个常见错误是忘记对方程两边同时进行相同的运算。

Another mistake is expanding brackets incorrectly, such as 3(x + 2) = 3x + 2.

另一个错误是括号展开不正确，例如 3(x + 2) = 3x + 2。

reful with negative signs when moving terms, especially in equations like 4x − 5 = 2x + 3.

在移项时要小心负号，尤其是在 4x − 5 = 2x + 3 这类方程中。

Always rewrite each step neatly; this makes it easier to spot errors.

始终整齐地写出每一步，这样更容易发现错误。

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12. Practice Questions and Summary | 练习题与总结

Try these practice questions to test your understanding.

尝试下列练习题来检验你的理解。

• Solve x + 9 = 15
• Solve 6x = 42
• Solve 4x − 7 = 25
• Solve 2(x + 5) = 24
• Solve 3x + 2 = x + 10

解答过程应从逆运算开始，步骤清晰，最后代回检验。

The working should start with inverse operations, show clear steps, and finish with substitution checking.

Mastering linear equations gives you a strong foundation for higher-level algebra and graph work in KS3 and beyond.

掌握线性方程将为你 KS3 及更高阶段的代数和图像学习打下坚实基础。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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