Solving Linear Equations: From Balance Model to Word Problems | 解一元一次方程：从天平模型到应用题

📚 Solving Linear Equations: From Balance Model to Word Problems | 解一元一次方程：从天平模型到应用题

Linear equations are one of the most important tools in Cambridge KS3 Mathematics. They appear in algebra questions, number puzzles, geometry problems and real-life word problems. This article explains how to solve them step by step, from simple one-step equations to equations with brackets and variables on both sides.

一元一次方程是剑桥 KS3 数学中最重要的工具之一。它们出现在代数题、数字谜题、几何题和现实生活应用题中。本文逐步讲解如何解这类方程，从简单的一步方程到含括号和两边都有变量的方程。

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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 unknown, usually written as x, is only raised to the power 1. Examples include x + 5 = 12, 3x − 4 = 14 and 2(x + 3) = 18. The word ‘linear’ tells us that the graph of the equation would be a straight line.

线性方程是指未知数（通常写作 x）的指数仅为 1 的方程。例如 x + 5 = 12、3x − 4 = 14 和 2(x + 3) = 18。’线性’ 这个词告诉我们，该方程的图像会是一条直线。

• An equation has an equals sign; an expression does not.
• 方程有等号；表达式没有等号。
• We solve an equation by finding the value of x that makes the statement true.
• 解方程就是找出使等式成立的 x 值。
• Always think of the equals sign as a balance point.
• 始终把等号想象成一个平衡点。

Equation: 2x + 7 = 19

方程：2x + 7 = 19

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2. The Balance Model | 天平模型

The balance model is the key idea behind solving linear equations. If you imagine both sides of the equation sitting on a balance scale, then whatever you do to one side, you must also do to the other side. This keeps the scale balanced and keeps the equation true.

天平模型是解一元一次方程背后的核心思想。想象方程的两边分别放在天平的两端，那么你对一边做的任何操作，也必须对另一边做同样的操作。这样才能保持天平平衡，也保持方程成立。

• Add the same number to both sides.
• 两边同时加上同一个数。
• Subtract the same number from both sides.
• 两边同时减去同一个数。
• Multiply both sides by the same non-zero number.
• 两边同时乘以同一个非零数。
• Divide both sides by the same non-zero number.
• 两边同时除以同一个非零数。

Example: x + 6 = 15

示例：x + 6 = 15

Subtract 6 from both sides: x + 6 − 6 = 15 − 6, so x = 9.

两边同时减去 6：x + 6 − 6 = 15 − 6，所以 x = 9。

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3. Solving Two-Step Equations | 解两步方程

A two-step equation has two operations attached to the unknown. For example, in 4x + 3 = 27, the x is first multiplied by 4 and then 3 is added. To solve it, we reverse the operations in the opposite order: first subtract 3, then divide by 4.

两步方程在未知数上有两种运算。例如 4x + 3 = 27，x 先乘以 4，然后加上 3。解这类方程时，我们要按相反顺序逆运算：先减去 3，再除以 4。

4x + 3 = 27

Step 1: Subtract 3 from both sides.

第一步：两边同时减去 3。

4x + 3 − 3 = 27 − 3

4x = 24

Step 2: Divide both sides by 4.

第二步：两边同时除以 4。

4x ÷ 4 = 24 ÷ 4

x = 6

Always check by substituting x = 6 back into the original equation: 4(6) + 3 = 24 + 3 = 27. It works.

一定要把 x = 6 代回原方程检验：4(6) + 3 = 24 + 3 = 27。成立。

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4. Dealing with Negative Coefficients | 处理负系数

Sometimes the unknown has a negative sign, as in 10 − 2x = 4. One method is to add 2x to both sides first so that the x-term becomes positive. Another method is to isolate −2x and then divide by −2. Both methods give the same answer.

有时未知数带负号，例如 10 − 2x = 4。一种方法是先在两边同时加上 2x，使 x 项变为正。另一种方法是先把 −2x 单独放在一边，然后除以 −2。两种方法得到相同答案。

Method 1: Make the x-term positive.

方法一：使 x 项为正。

10 − 2x = 4

10 − 2x + 2x = 4 + 2x

10 = 4 + 2x

10 − 4 = 2x

6 = 2x

x = 3

Be careful with signs when dividing by a negative number. For example, −5x = 20 means x = 20 ÷ (−5), so x = −4.

除以负数时要特别小心符号。例如 −5x = 20 表示 x = 20 ÷ (−5)，所以 x = −4。

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5. Equations with Brackets | 含括号的方程

When an equation contains brackets, the first step is usually to expand them using the distributive law. For example, 3(x + 4) means 3 times x plus 3 times 4, which is 3x + 12. Then solve the resulting two-step equation.

当方程中含有括号时，第一步通常是使用分配律展开括号。例如 3(x + 4) 表示 3 乘 x 加 3 乘 4，即 3x + 12。然后解得到的两步方程。

3(x + 4) = 27

Expand: 3x + 12 = 27

展开：3x + 12 = 27

Subtract 12: 3x = 15

减去 12：3x = 15

Divide by 3: x = 5

除以 3：x = 5

• If the bracket has a minus sign outside, such as 7 − 2(x + 1), expand carefully: −2(x + 1) = −2x − 2.
• 如果括号外是负号，如 7 − 2(x + 1)，要小心展开：−2(x + 1) = −2x − 2。
• Always multiply every term inside the bracket by the number outside.
• 始终把括号内的每一项都乘以外面的数。

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6. Equations with Variables on Both Sides | 两边含变量的方程

Some equations have x-terms on both sides, such as 5x + 6 = 3x + 14. The strategy is to collect all x-terms on one side and all number terms on the other side. Usually we choose the side with the larger x-coefficient to keep x positive.

有些方程两边都有 x 项，例如 5x + 6 = 3x + 14。解题策略是把所有 x 项移到一边，把所有数字项移到另一边。通常选择 x 系数较大的一边，使 x 保持为正。

5x + 6 = 3x + 14

Subtract 3x from both sides: 2x + 6 = 14

两边同时减去 3x：2x + 6 = 14

Subtract 6 from both sides: 2x = 8

两边同时减去 6：2x = 8

Divide by 2: x = 4

除以 2：x = 4

Check: Left side = 5(4) + 6 = 26; Right side = 3(4) + 14 = 26. Both sides match.

检验：左边 = 5(4) + 6 = 26；右边 = 3(4) + 14 = 26。两边相等。

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7. Forming Equations from Word Problems | 根据应用题列方程

Word problems often describe a situation in ordinary English. The trick is to translate the words into algebra. Words like ‘sum’, ‘more than’ and ‘total’ suggest addition; ‘difference’, ‘less than’ and ‘minus’ suggest subtraction; ‘product’ and ‘times’ suggest multiplication; ‘quotient’ and ‘shared equally’ suggest division.

应用题经常用普通英语描述一个情景。关键是要把文字翻译成代数语言。’sum’、’more than’、’total’ 表示加法；’difference’、’less than’、’minus’ 表示减法；’product’、’times’ 表示乘法；’quotient’、’shared equally’ 表示除法。

English word | 中文关键词	Operation | 运算
sum, total, more than | 和、总共、比……多	Addition | 加
difference, less than, minus | 差、比……少、减	Subtraction | 减
product, times, double | 积、倍、翻倍	Multiplication | 乘
quotient, divided by, half | 商、除以、一半	Division | 除

Example: ‘Three more than twice a number is 19.’ Let the number be x. Twice the number is 2x. Three more than that is 2x + 3. So the equation is 2x + 3 = 19. Solving gives x = 8.

示例：’一个数的两倍再加三等于 19。’ 设这个数为 x。两倍就是 2x。再加三就是 2x + 3。因此方程是 2x + 3 = 19。解得 x = 8。

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8. Worked Example: A Typical Page 168-Style Problem | 典型例题：第168页题型演练

A common Cambridge KS3 exercise asks students to solve an equation and then use it in a short word problem. Consider this typical two-part question: (a) Solve 4(x − 2) = 5x − 14. (b) The perimeter of a rectangle is given by the expressions 4(x − 2) and 5x − 14. Find the length and width when x is the solution from part (a).

剑桥 KS3 常见练习题要求学生先解方程，再把它用于简短应用题。请看这个典型两小问：(a) 解方程 4(x − 2) = 5x − 14。(b) 一个矩形的两条邻边分别用 4(x − 2) 和 5x − 14 表示。当 x 为第 (a) 题的解时，求长和宽。

Part (a): Expand the left side.

第 (a) 题：展开左边。

4x − 8 = 5x − 14

Subtract 4x from both sides: −8 = x − 14

两边同时减去 4x：−8 = x − 14

Add 14 to both sides: 6 = x

两边同时加上 14：6 = x

So x = 6.

所以 x = 6。

Part (b): Substitute x = 6 into each expression.

第 (b) 题：把 x = 6 代入每个表达式。

4(x − 2) = 4(6 − 2) = 4(4) = 16

5x − 14 = 5(6) − 14 = 30 − 14 = 16

The two adjacent sides are both 16 units, so the rectangle is actually a square with side length 16 units.

两条邻边都为 16 个单位，所以这个矩形实际上是一个边长为 16 个单位的正方形。

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

Many students lose marks not because they do not understand equations, but because of small mistakes in signs, order of operations or checking. This section lists the most common errors and gives simple ways to avoid them.

许多学生丢分并不是因为不懂方程，而是因为符号、运算顺序或检查中的小错误。本节列出最常见的错误，并给出简单的避免方法。

• Mistake: Adding when you should subtract. Fix: Ask ‘What operation is attached to x?’, then do the opposite.
• 错误：该减时却加了。对策：问自己 ‘x 上带有哪种运算？’，然后做逆运算。
• Mistake: Forgetting to multiply every term inside a bracket. Fix: Rewrite the bracket as two separate products before simplifying.
• 错误：忘记乘以括号内的每一项。对策：先把括号写成两个独立的乘积，再化简。
• Mistake: Losing the negative sign when dividing. Fix: Write the division step clearly and keep the sign with the number.
• 错误：除法时丢失负号。对策：清楚写出除法步骤，并把符号与数字放在一起。
• Mistake: Not checking the final answer. Fix: Substitute your x-value back into the original equation.
• 错误：没有检验最终答案。对策：把你的 x 值代回原方程。

A neat layout is very important. Write one step per line and line up the equals signs. This makes it much easier to see if the balance has been kept.

整洁的书写格式非常重要。每行只写一步，并把等号对齐。这样更容易看清是否保持了平衡。

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10. Quick Practice Checklist | 快速练习清单

Use this checklist when you practise KS3 linear equations. It will help you build speed and accuracy for Cambridge tests and Checkpoint-style assessments.

练习 KS3 一元一次方程时，请使用这份清单。它可以帮助你在剑桥测试和 Checkpoint 风格评估中提高速度和准确性。

• Identify the type: one-step, two-step, brackets, or variables on both sides.
• 判断类型：一步方程、两步方程、含括号方程，还是两边含变量方程。
• Expand brackets first if needed.
• 如有括号，先展开。
• Collect x-terms on one side and numbers on the other.
• 把所有 x 项移到一边，数字项移到另一边。
• Reverse the order of operations: undo addition or subtraction before multiplication or division.
• 逆运算顺序：先消去加或减，再消去乘或除。
• Check your answer by substitution.
• 通过代入检验答案。
• Write a clear conclusion: x = …
• 写出清晰结论：x = …

With regular practice, solving linear equations becomes automatic. Start with simple one-step equations and gradually move to harder problems involving brackets, negative coefficients and word problems.

通过定期练习，解一元一次方程会变得自动化。从简单的一步方程开始，逐步过渡到含括号、负系数和应用题的较难问题。

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