Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the foundation of algebra at KS3 level. They show the relationship between numbers and a variable, usually represented by a letter such as x. By learning to solve these equations step by step, you develop logical thinking that will be essential for all future mathematics topics. In this article, drawn from the core ideas found on page 124 and beyond, we will explore everything from simple one-step equations to more challenging problems involving brackets and unknowns on both sides.

线性方程是 KS3 阶段代数学习的基础。它们展示数字与变量（通常用 x 这样字母表示）之间的关系。通过一步步学会解这些方程，你可以培养逻辑思维能力，这对将来所有数学主题都至关重要。在这篇文章中，我们将深入探讨从简单的一步方程到更复杂的带有括号和两边均有未知数的题目，这些内容均提炼自第 124 页及后续核心考点。

1. Understanding Variables and Constants | 理解变量与常数

A variable is a symbol used to represent an unknown number. In the expression 3x + 2, ‘x’ is the variable and ‘3’ and ‘2’ are constants. Variables allow us to write general rules and solve problems where some quantity is not yet known. At KS3, you will often see letters like x, y or n used as variables.

变量是用来表示未知数的一个符号。在表达式 3x + 2 中，’x’ 是变量，’3′ 和 ‘2’ 是常数。变量让我们能够写出一般规则，并解决一些数量尚未知晓的问题。在 KS3 阶段，你会经常看到 x、y 或 n 这样的字母被用作变量。

When an expression contains an equals sign, it becomes an equation. The equals sign shows that two mathematical expressions have the same value. For example, 5 + y = 13 is an equation stating that when you add some number y to 5, you get 13.

当一个表达式含有等号时，它就成为了方程。等号表示两个数学表达式具有相同的值。例如，5 + y = 13 就是一个方程，表明当你给 5 加上某个数 y 时，你会得到 13。

• Variable: a letter standing for an unknown value (e.g. x)
• 变量：一个代表未知值的字母（如 x）
• Constant: a fixed number that does not change (e.g. 7)
• 常数：一个固定的、不会变化的数（如 7）
• Equation: a statement that two expressions are equal, containing ‘=’
• 方程：一个说明两个表达式相等且包含 ‘=’ 的陈述

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2. Writing Algebraic Expressions | 书写代数表达式

Before solving equations, you must be comfortable writing algebraic expressions from words. In KS3, you might be asked to write ‘I think of a number, multiply it by 4 and then add 7’ as 4n + 7. Notice that we do not use a multiplication sign like × when writing algebra; we simply write the number next to the variable.

在解方程之前，你必须能够自如地根据文字写出代数表达式。在 KS3 中，你可能会被要求将 “我想一个数，乘 4 然后加 7” 写成 4n + 7。请注意，书写代数时我们不使用 × 这样的乘号；我们直接将数字写在变量旁边。

Division is shown using a fraction line. The phrase ‘a number divided by 5’ becomes x/5 or x ÷ 5. Brackets are used to group parts of an expression. For instance, ‘triple the sum of a number and 6’ is written as 3(y + 6).

除法用分数线表示。”一个数除以 5″ 写成 x/5 或 x ÷ 5。括号用来将表达式的一部分进行分组。例如，”一个数加 6 的三倍” 写作 3(y + 6)。

Words (English)	Expression	中文表述
A number increased by 10	n + 10	一个数增加 10
5 less than a number	x – 5	比一个数少 5
Double a number and subtract 3	2m – 3	一个数的两倍再减3
Half of a number plus 4	t/2 + 4	一个数的一半加4

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3. Simplifying Expressions by Collecting Like Terms | 合并同类项化简表达式

Like terms are terms that contain exactly the same variable raised to the same power. For example, 3a and 7a are like terms, but 3a and 5b are not. When simplifying, we add or subtract the coefficients (the numbers in front of the variables) and keep the variable part unchanged.

同类项是指含有完全相同变量且该变量的指数相同的项。例如，3a 和 7a 是同类项，但 3a 和 5b 不是。化简时，我们只需要将系数（变量前的数字）相加或相减，变量部分保持不变。

Consider the expression 4x + 2y – x + 3y. First group like terms: (4x – x) and (2y + 3y). This simplifies to 3x + 5y. This skill is vital because when we solve equations we often need to simplify each side first.

考虑表达式 4x + 2y – x + 3y。首先把同类项分组：(4x – x) 和 (2y + 3y)。化简得到 3x + 5y。这项技能非常重要，因为在解方程时我们往往需要先化简等号两边。

Remember that constants without variables are also like terms with each other. So 5 + 3k – 2 + k simplifies to 4k + 3.

请记住，没有变量的常数项彼此之间也是同类项。因此 5 + 3k – 2 + k 可化简为 4k + 3。

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4. Solving One-Step Equations | 解一步方程

One-step equations require only a single inverse operation to isolate the variable. If the equation is x + 7 = 15, we subtract 7 from both sides to get x = 8. The golden rule is: whatever you do to one side of the equation, you must do exactly the same to the other side to keep it balanced.

一步方程只需进行一次逆运算即可将变量分离出来。如果方程是 x + 7 = 15，我们将两边同时减去 7，得到 x = 8。黄金法则是：对方程的一边做什么操作，就必须对另一边做完全相同的操作，以保持平衡。

Other examples include solving 4y = 28 by dividing both sides by 4, giving y = 7. For a division equation like p/3 = 9, we multiply both sides by 3 to find p = 27.

其他例子包括解 4y = 28，两边同时除以 4，得到 y = 7。对于像 p/3 = 9 这样的除法方程，我们将两边同时乘以 3，得出 p = 27。

x – 5 = 12 → x = 12 + 5 → x = 17

8 × n = 56 → n = 56 ÷ 8 → n = 7

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5. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations. For example, 2x + 3 = 11. We first undo the addition of 3 by subtracting 3 from both sides, giving 2x = 8. Then we undo the multiplication by dividing by 2 to find x = 4. Always reverse the order of operations: tackle addition/subtraction before multiplication/division.

两步方程包含两种运算。例如，2x + 3 = 11。我们首先通过两边减 3 来抵消加 3，得到 2x = 8。然后再通过除以 2 来抵消乘法，得出 x = 4。始终要逆序运算：先处理加减法，再处理乘除法。

If the equation is written in a different order, such as 18 = 5p – 2, the method is the same. Add 2 to both sides to get 20 = 5p, then divide by 5 to obtain p = 4. It does not matter which side the variable is on; the goal is to isolate it.

如果方程的顺序不同，比如 18 = 5p – 2，方法是一样的。两边加 2 得到 20 = 5p，然后除以 5 得到 p = 4。变量出现在哪一边没关系；目标是将其单独分离出来。

Always check your solution by substituting it back into the original equation: 2(4) + 3 = 8 + 3 = 11 ✓

一定要将解代回原方程进行检验：2 × 4 + 3 = 8 + 3 = 11 ✓

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6. Equations with Brackets | 带括号的方程

When an equation contains brackets, you should expand them first using the distributive law. For example, solve 3(y + 2) = 21. Multiply the term outside the bracket by everything inside: 3 × y + 3 × 2 = 3y + 6. Now the equation is 3y + 6 = 21.

当方程含有括号时，你应该首先使用分配律将其展开。例如，解 3(y + 2) = 21。将括号外的项乘以括号内的每一项：3 × y + 3 × 2 = 3y + 6。现在方程变为 3y + 6 = 21。

Then solve as a two-step equation: subtract 6 from both sides to get 3y = 15, then divide by 3 to obtain y = 5. It is essential to expand correctly before you begin solving.

然后按两步方程来解：两边减 6 得到 3y = 15，再除以 3 得出 y = 5。开始解题前正确展开括号至关重要。

Some problems may have a negative number outside the bracket, such as 20 – 2(2x + 1) = 8. Expand carefully: –2 × 2x = –4x and –2 × +1 = –2. The equation becomes 20 – 4x – 2 = 8. Simplify to 18 – 4x = 8, then solve.

有些题目括号外可能是负数，例如 20 – 2(2x + 1) = 8。小心地展开：–2 × 2x = –4x，–2 × +1 = –2。方程变为 20 – 4x – 2 = 8。化简得到 18 – 4x = 8，然后再解。

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7. Equations with Unknowns on Both Sides | 两边均有未知数的方程

The next challenge is when variables appear on both sides of the equals sign, for instance 5x + 2 = 3x + 10. The strategy is to collect all variable terms on one side and all constant terms on the other. Subtract the smaller variable term from both sides to keep coefficients positive. Here, subtract 3x from both sides: 2x + 2 = 10.

下一个挑战是等号两边都出现变量，例如 5x + 2 = 3x + 10。策略是将所有变量项集中到一边，所有常数项集中到另一边。为了保持系数为正，从两边减去较小的变量项。在此，两边减 3x：2x + 2 = 10。

Then proceed as normal: subtract 2 from both sides to get 2x = 8, so x = 4. Always aim to have a positive coefficient for the variable on one side; this reduces sign errors.

然后照常进行：两边减 2 得到 2x = 8，因此 x = 4。总是要让一边的变量系数为正；这样可以减少符号错误。

For an equation like 4y – 11 = 7 – 2y, add 2y to both sides to get 6y – 11 = 7. Then add 11: 6y = 18, so y = 3. Remember that adding the variable term is often the easiest first move.

对于像 4y – 11 = 7 – 2y 这样的方程，两边加 2y 得到 6y – 11 = 7。然后加 11：6y = 18，所以 y = 3。请记住，加上变量项通常是最简单的第一步。

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8. Checking Your Solution and Understanding Identities | 检验解并理解恒等式

Substitution is the most reliable way to check your answer. Take your solution, put it back into the original equation, and evaluate both sides. If they are equal, your solution is correct. This habit will catch arithmetic mistakes and build confidence.

代入是检验答案最可靠的方法。将你的解代回原方程，计算两边。如果两边相等，你的解就是正确的。这个习惯能及时发现计算错误并建立信心。

Sometimes you may encounter an equation that is true for all values of x, like 2(x + 3) = 2x + 6. Expanding the left side gives 2x + 6, which is identical to the right side. This is called an identity. If you end up with a statement like 6 = 6, the equation has infinitely many solutions.

有时你可能会遇到一个对所有 x 值都成立的方程，比如 2(x + 3) = 2x + 6。展开左边得到 2x + 6，与右边完全相同。这被称为恒等式。如果你最终得到像 6 = 6 的陈述，那么该方程有无穷多个解。

On the other hand, if simplifying leads to a false statement like  = 8, then the equation has no solution. Being able to recognise these special cases is a valuable skill at KS3.

另一方面，如果化简得到一个像  = 8 这样的假命题，那么该方程无解。能够识别这些特殊情况是 KS3 一项很有价值的技能。

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9. Word Problems Leading to Equations | 列方程解应用题

Many real-life scenarios can be modelled with linear equations. The key is to read the problem carefully, define your variable clearly, and translate the words into an equation. For example: “Three times a number increased by 8 equals 20. Find the number.” Let n be the number. The equation is 3n + 8 = 20.

许多现实生活场景都可以用线性方程来建模。关键是仔细读题，清晰地定义你的变量，并将文字转化为方程。例如：”一个数的三倍再加 8 等于 20。求这个数。” 设这个数为 n，方程是 3n + 8 = 20。

Solve: 3n = 12, so n = 4. Always write a concluding sentence stating the answer in context, for instance “The number is 4.”

解方程：3n = 12，所以 n = 4。务必写出一个总结句，在上下文中陈述答案，例如 “这个数是 4。”

Another typical problem: “The perimeter of a rectangle is 54 cm. Its length is twice its width. Find the dimensions.” Let the width be w cm, so length is 2w cm. Perimeter = 2(length + width) = 2(2w + w) = 6w. Equation: 6w = 54, giving w = 9. Therefore the width is 9 cm and length is 18 cm.

另一个典型问题：”一个矩形的周长是 54 cm。它的长是宽的两倍。求它的尺寸。” 设宽为 w cm，则长为 2w cm。周长 = 2 × (长 + 宽) = 2 × (2w + w) = 6w。方程：6w = 54，得 w = 9。因此宽为 9 cm，长为 18 cm。

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10. Equations with Fractions | 含有分数的方程

Equations containing fractions can look intimidating, but they are manageable if you eliminate the denominators early. For an equation like x/4 + 2 = 5, subtract 2 first: x/4 = 3, then multiply by 4 to get x = 12.

含有分数的方程可能看起来令人生畏，但只要你尽快消去分母，它们就变得容易处理了。对于像 x/4 + 2 = 5 这样的方程，先减去 2：x/4 = 3，然后乘以 4 得到 x = 12。

When there are multiple fractions, find a common denominator and multiply every term by that number. For example, x/3 + 1 = x/2. Multiply everything by 6 (the LCM of 3 and 2): 6(x/3) + 6(1) = 6(x/2) → 2x + 6 = 3x. Subtract 2x from both sides: 6 = x, so x = 6.

当有多个分数时，找出公分母，并将每一项都乘以这个数。例如，x/3 + 1 = x/2。两边同乘 6（3 和 2 的最小公倍数）：6 × x/3 + 6 × 1 = 6 × x/2 → 2x + 6 = 3x。两边减 2x：6 = x，所以 x = 6。

Always simplify the resulting equation as usual, and check your answer in the original fractional form to be certain no mistakes were made.

始终照常化简得到的方程，并在原分数形式下检验答案，以确保没有出错。

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11. Common Mistakes and How to Avoid Them | 常见错误及如何避免

One of the most common errors is forgetting to perform an operation on both sides. For example, when solving x + 5 = 9, some students write x = 9 – 5 = 4 without showing the balancing step. While the answer might be right, in more complex equations this shortcut leads to mistakes. Always show your working.

最常见的错误之一是忘记对两边执行相同操作。例如，解 x + 5 = 9 时，有些学生直接写成 x = 9 – 5 = 4，而不呈现平衡的步骤。虽然答案可能是对的，但在更复杂的方程中这种捷径会导致错误。一定要展示你的计算过程。

Another frequent slip occurs when dealing with negative numbers, especially when expanding brackets like 5 – 3(x – 2). Remember to multiply the –3 by both x and –2, giving 5 – 3x + 6, which simplifies to 11 – 3x.

另一个常见失误出现在处理负数时，尤其是展开像 5 – 3(x – 2) 这样的括号时。记住要用 –3 去乘 x 和 –2，得到 5 – 3x + 6，化简后为 11 – 3x。

Students also sometimes misuse the equals sign by writing chains like 2x + 3 = 11 = 2x = 8 = x = 4. This is incorrect mathematically. Use separate lines for each step or write them horizontally but keep the meaning clear.

学生有时还会滥用等号，写出像 2x + 3 = 11 = 2x = 8 = x = 4 这样的长串。这在数学上是不正确的。每一步应该用单独一行来写，或者水平书写但保持意思清晰。

• Always do the same operation to both sides
• 始终对方程两边施加相同运算
• Expand brackets correctly, paying attention to signs
• 正确展开括号，注意正负号
• Show your working step by step
• 逐步展示你的运算过程
• Check your solution by substitution
• 通过代入来检验你的解

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12. Putting It All Together – Mixed Practice | 综合应用——混合练习

At KS3, exam-style questions often combine several of these skills. You might be asked to solve 2(3x – 1) – 4 = 7 – x. Start by expanding: 6x – 2 – 4 = 7 – x, which simplifies to 6x – 6 = 7 – x. Add x to both sides: 7x – 6 = 7. Add 6: 7x = 13, giving x = 13/7. Leave the answer as a fraction unless instructed otherwise.

在 KS3 阶段，考试型题目往往综合了这些技能中的几项。你可能会被要求解 2(3x – 1) – 4 = 7 – x。先展开括号：6x – 2 – 4 = 7 – x，化简为 6x – 6 = 7 – x。两边加 x：7x – 6 = 7。加 6：7x = 13，得到 x = 13/7。除非另有要求，答案保留为分数形式。

Regular practice with mixed problems helps you recognise which technique to apply first. Begin each problem by asking yourself: Are there brackets? Are there fractions? Are there variables on both sides? Then follow the logical order: simplify, collect like terms, isolate the variable.

定期进行混合练习有助于你识别哪项技术需要首先运用。开始每一道题目时先问自己：有括号吗？有分数吗？等号两边都有变量吗？然后按照逻辑顺序进行：化简、合并同类项、分离变量。

Above all, keep your working neat and aligned. A well-laid-out solution not only helps the examiner award marks for method but also reduces your own risk of confusion. Remember, with linear equations, patience and precision always pay off.

最重要的是，保持书写工整和对齐。版面清晰的解题过程不仅有助于阅卷老师给出方法分，也能减少你自己的困惑。请记住，在线性方程面前，耐心和精确永远会有回报。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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