Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the foundation of algebra. They appear in every corner of mathematics, from simple number puzzles to real‑world problems such as budgeting or calculating speeds. Mastering how to solve them at KS3 gives you a powerful toolkit for advanced topics like graphs, inequalities, and simultaneous equations. This article will guide you through the key methods, step by step, with plenty of examples and tips.

线性方程是代数的基础。它们出现在数学的各个角落，从简单的数字谜题到现实生活中的问题，比如预算或速度计算。在KS3阶段掌握如何解方程，能为你学习图像、不等式和联立方程等更高级的课题提供强大的工具。本文将一步一步带你了解关键方法，并配有丰富的例子和技巧。

1. Understanding Equations | 理解方程

An equation is a mathematical statement that two expressions are equal. It always contains an equals sign ‘=’. For example, x + 3 = 7 means that when you add 3 to an unknown number x, the result is 7. The goal of solving an equation is to find the value of the unknown that makes the statement true.

方程是一个数学陈述，表示两个表达式相等。它总是包含一个等号 ‘=’。例如，x + 3 = 7 意味着当你把一个未知数 x 加上 3 时，结果是 7。解方程的目标就是找到使这个陈述成立的未知数的值。

In any equation, the left‑hand side and the right‑hand side have the same value. Think of a balanced scale: whatever you do to one side, you must do exactly the same to the other side to keep it balanced. This idea is the heart of equation solving.

在任何方程中，左边和右边的值相同。想象一个平衡的天平：你对一边做了什么，就必须对另一边做完全相同的事，以保持平衡。这个想法是解方程的核心。

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2. The Balance Method | 平衡法

The balance method means performing the same operation on both sides of the equation. The operations you can use are addition, subtraction, multiplication, and division. Always keep the equation balanced: if you add 5 to the left, add 5 to the right. If you divide the left by 2, divide the right by 2.

平衡法意味着对方程的两边进行相同的运算。你可以使用的运算有加法、减法、乘法和除法。始终保持方程平衡：如果你在左边加5，右边也要加5；如果你把左边除以2，右边也要除以2。

For example, to solve x + 4 = 10, subtract 4 from both sides:

例如，要解 x + 4 = 10，两边同时减去4：

x + 4 − 4 = 10 − 4 → x = 6

The balance method ensures you never break the equation, just like keeping a pair of scales in perfect equilibrium.

平衡法确保你永远不会破坏方程，就像让一台天平保持完美的平衡一样。

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3. Solving Simple One‑Step Equations | 解简单的一步方程

One‑step equations require only one operation to isolate the unknown. Look at the operation being applied to the variable and do the opposite (inverse operation).

一步方程只需要一种运算就能把未知数分离出来。看看对变量施加了什么运算，然后做相反的运算（逆运算）。

• If the equation is x + a = b, subtract a from both sides.
• 如果方程是 x + a = b，两边减去 a。
• If the equation is x − a = b, add a to both sides.
• 如果方程是 x − a = b，两边加上 a。
• If the equation is ax = b, divide both sides by a.
• 如果方程是 ax = b，两边除以 a。
• If the equation is x/a = b, multiply both sides by a.
• 如果方程是 x/a = b，两边乘以 a。

Practice with these examples: solve x − 7 = 3 (add 7 to both sides to get x = 10); solve 5x = 20 (divide both sides by 5 to get x = 4); solve x/3 = 9 (multiply both sides by 3 to get x = 27).

用这些例子练习：解 x − 7 = 3（两边加7得到 x = 10）；解 5x = 20（两边除以5得到 x = 4）；解 x/3 = 9（两边乘以3得到 x = 27）。

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4. Solving Two‑Step Equations | 解两步方程

Two‑step equations involve two operations. A typical form is ax + b = c. The strategy is to undo the operations in reverse order: first get rid of the constant term (addition/subtraction), then deal with the coefficient (multiplication/division).

两步方程涉及两种运算。典型形式是 ax + b = c。策略是按照相反的顺序撤销运算：首先去掉常数项（加减法），然后处理系数（乘除法）。

Example: 2x + 3 = 11

例子：2x + 3 = 11

Step 1: subtract 3 from both sides → 2x = 8

第一步：两边减3 → 2x = 8

Step 2: divide both sides by 2 → x = 4

第二步：两边除以2 → x = 4

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

始终将解代回原方程进行检验：2(4) + 3 = 8 + 3 = 11。成立！

Avoid a common mistake: do not divide before removing the added constant. For 2x + 3 = 11, dividing first would give x + 1.5 = 5.5, which is messy. Remove the addition first.

避免一个常见错误：不要先除以系数再去掉常数。对于 2x + 3 = 11，如果先除以2，会得到 x + 1.5 = 5.5，很麻烦。要先去掉加法。

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5. Equations with Brackets | 带括号的方程

When an equation contains brackets, such as 3(x + 2) = 15, you must expand the brackets first, or, in some cases, treat the bracketed expression as a single item and divide first. Both methods work.

当方程中含有括号时，例如 3(x + 2) = 15，你必须先展开括号，或者在某些情况下，把括号里的表达式看成一个整体先除以系数。两种方法都行。

Method A (expand first): 3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3.

方法A（先展开）：3(x + 2) = 15 → 3x + 6 = 15 → 3x = 9 → x = 3。

Method B (divide first): Divide both sides by 3 → x + 2 = 5 → x = 3.

方法B（先除）：两边除以3 → x + 2 = 5 → x = 3。

Expanding is often safer, especially when the coefficient is not a factor of the other side. For 2(3x − 4) = 10, expand: 6x − 8 = 10 → 6x = 18 → x = 3.

展开往往更安全，特别是当系数不是另一边数的因数时。对于 2(3x − 4) = 10，展开：6x − 8 = 10 → 6x = 18 → x = 3。

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6. Equations with Unknowns on Both Sides | 未知数在方程两边的方程

For equations like 5x + 2 = 3x + 10, you need to collect all the x terms on one side and the numbers on the other. Choose to eliminate the smaller x term first to keep coefficients positive.

对于像 5x + 2 = 3x + 10 这样的方程，你需要把所有的 x 项移到一边，把数字移到另一边。选择先消去较小的 x 项，以保持系数为正。

Example: 5x + 2 = 3x + 10

例子：5x + 2 = 3x + 10

Subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 10 → 2x + 2 = 10

两边减3x：5x − 3x + 2 = 3x − 3x + 10 → 2x + 2 = 10

Subtract 2 from both sides: 2x = 8 → x = 4

两边减2：2x = 8 → x = 4

Always simplify each side before collecting terms. With practice, you can do these steps quickly, but writing them clearly helps avoid mistakes.

在移项前，先化简每一边。通过练习，你可以快速完成这些步骤，但清晰地写出来有助于避免错误。

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7. Equations Involving Fractions | 含分数的方程

Fractions can make equations look harder, but the strategy is to multiply every term by the lowest common denominator (LCD) to clear the fractions.

分数会让方程看起来更难，但策略是用最小公分母（LCD）乘以每一项，从而去掉分数。

Example: ½x + ⅓ = ¼

例子：½x + ⅓ = ¼

The LCD of 2, 3, and 4 is 12. Multiply every term by 12:

2、3、4的最小公分母是12。每一项都乘以12：

12 × (½x) + 12 × (⅓) = 12 × (¼) → 6x + 4 = 3

Then solve: 6x = −1 → x = −⅙

然后解：6x = −1 → x = −⅙

Another example: (2x)/3 = 4. Multiply both sides by 3: 2x = 12 → x = 6. Remember to multiply the entire side by the denominator, not just the fraction.

另一个例子：(2x)/3 = 4。两边乘以3：2x = 12 → x = 6。记住要对整个边乘以分母，而不仅仅是分数部分。

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8. Forming Equations from Word Problems | 从文字题建立方程

Many real‑life problems can be turned into equations. Read the problem carefully, identify the unknown (let it be x), and translate the words into an algebraic statement.

许多现实问题都可以转化为方程。仔细读题，确定未知数（用 x 表示），然后把文字翻译成代数语句。

Example: “I think of a number, multiply it by 3, subtract 7, and get 20. What is the number?”

例子：“我想一个数，乘以3，减7，得到20。这个数是多少？”

Let the number be x: 3x − 7 = 20 → 3x = 27 → x = 9.

设这个数为 x：3x − 7 = 20 → 3x = 27 → x = 9。

Key words to remember: ‘sum’ means add, ‘difference’ means subtract, ‘product’ means multiply, ‘quotient’ means divide. Practice forming equations from phrases like “five more than twice a number is 12” (2x + 5 = 12).

要记住的关键词：’和’表示加法，’差’表示减法，’积’表示乘法，’商’表示除法。练习用短语建方程，如“一个数的两倍再加5等于12”（2x + 5 = 12）。

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9. Checking Your Solution | 检验解

Always substitute your answer back into the original equation to verify it. This simple habit can catch arithmetic errors and build confidence.

一定要把你的答案代回原方程进行验证。这个简单的习惯可以发现计算错误，并建立信心。

Example: Solve 4(x − 2) = 2x + 4. Solution: 4x − 8 = 2x + 4 → 2x = 12 → x = 6. Check: left side = 4(6 − 2) = 4(4) = 16; right side = 2(6) + 4 = 12 + 4 = 16. Both sides equal, so x = 6 is correct.

例子：解 4(x − 2) = 2x + 4。解：4x − 8 = 2x + 4 → 2x = 12 → x = 6。检验：左边 = 4(6 − 2) = 4(4) = 16；右边 = 2(6) + 4 = 12 + 4 = 16。两边相等，所以 x = 6 正确。

If the check fails, retrace your steps. A common pitfall is forgetting to change signs when moving terms. Checking protects your grades!

如果检验不成立，就重新检查步骤。一个常见的陷阱是移项时忘记变号。检验能保护你的分数！

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10. Common Mistakes to Avoid | 常见错误要避免

Even strong students can slip up. Here are a few errors to watch out for:

即使优秀的学生也可能失误。以下是要留意的几个错误：

• Forgetting to do the same operation on both sides. Always keep the equation balanced.
• 忘记在两边做相同的运算。始终保持方程平衡。
• Incorrectly expanding brackets, such as 3(x + 2) = 3x + 2 instead of 3x + 6.
• 错误地展开括号，例如 3(x + 2) = 3x + 2 而不是 3x + 6。
• When collecting like terms, writing 5x − 3x as 2x². Remember, 5x − 3x = 2x, not 2x².
• 合并同类项时，把 5x − 3x 写成 2x²。记住，5x − 3x = 2x，而不是 2x²。
• Losing a negative sign when moving terms. For 2x + 5 = x − 3, subtracting x: x + 5 = −3, then subtract 5: x = −8.
• 移项时丢失负号。对于 2x + 5 = x − 3，减去 x：x + 5 = −3，再减5：x = −8。
• Dividing at the wrong time: in 2x + 3 = 7, divide first? Wrong: you get x + 1.5 = 3.5. Always undo addition/subtraction first.
• 在不恰当的时候做除法：在 2x + 3 = 7 中，先除？错误：会得到 x + 1.5 = 3.5。总是先撤销加减法。

Be patient, show your working, and you will reduce these mistakes dramatically.

耐心点，展示你的步骤，你会大大减少这些错误。

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11. Practice Makes Perfect | 熟能生巧

The only way to become fluent at solving equations is to practise regularly. Start with simple one‑step equations, then move to two‑step, brackets, and unknowns on both sides. Use online exercises or textbook questions, and always check your answers.

熟练解方程的唯一方法就是经常练习。从简单的一步方程开始，然后过渡到两步、带括号和未知数在两边的情况。使用在线练习或课本题目，并总是核对答案。

Equation Type 方程类型	Example 例子	Solution 解
One‑step 一步	x + 9 = 15	x = 6
Two‑step 两步	4x − 3 = 13	x = 4
Brackets 括号	5(2x − 1) = 25	x = 3
Unknowns both sides 两边有未知数	7x + 4 = 3x + 20	x = 4
Fractions 分数	⅔x = 8	x = 12

Each type builds on the previous one, so master them step by step. Soon, solving equations will feel as natural as regular arithmetic.

每一种类型都建立在前一种的基础之上，所以要一步一步地掌握它们。很快，解方程就会像普通的算术一样自然。

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12. Summary and Key Takeaways | 小结与要点

Solving linear equations is a skill you will use throughout your maths journey. Remember the balance principle: do the same to both sides. Isolate the unknown using inverse operations in the correct order. Expand brackets first, collect like terms, and always check your answer by substitution. With practice, you’ll handle any linear equation with confidence.

解线性方程是一项你将在整个数学学习过程中使用的技能。记住平衡原则：对两边做同样的操作。按照正确的顺序使用逆运算来隔离未知数。先展开括号，合并同类项，并始终通过代入来检验答案。通过练习，你将能充满信心地处理任何线性方程。

Keep a handy checklist: (1) Simplify both sides, (2) remove brackets, (3) collect x‑terms on one side and numbers on the other, (4) solve the simplified equation, (5) check your answer. Stick to this routine and you will excel.

记住一个实用的清单：(1) 化简两边，(2) 去括号，(3) 将 x 项移到一边，数字移到另一边，(4) 解简化后的方程，(5) 检验答案。坚持这个流程，你就会出类拔萃。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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