Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are at the heart of KS3 algebra. Being able to solve them confidently unlocks topic after topic, from graphs to problem‑solving. This article walks you through the key skills step by step, with plenty of examples and tips to help you master the balance method.

线性方程是 KS3 代数的核心。能熟练解方程会让你在图表、应用题等一个又一个主题中游刃有余。本文将通过大量实例和技巧，一步步带你掌握天平法，打好解方程的基础。

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

A linear equation is a mathematical statement where two expressions are equal and the highest power of the unknown (often x) is 1. For example, 2x + 3 = 11 is linear, but x² + 5 = 14 is not because of the x² term.

线性方程是含有等号、且未知数（通常用 x 表示）的最高次数为 1 的等式。例如 2x + 3 = 11 是一个线性方程，而 x² + 5 = 14 不是，因为它含有 x² 项。

Think of an equation as a pair of balanced scales: whatever you do to one side, you must do to the other to keep them level. This balance idea is the golden rule for all equation solving.

可以把方程想象成一座平衡的天平：你在一边所做的任何操作，必须在另一边同样操作，才能保持平衡。这个天平思想是解所有方程的金科玉律。

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2. The Balance Method | 天平法

The balance method means performing identical operations on both sides of the equation until the unknown is isolated. Common operations include adding, subtracting, multiplying, or dividing by the same non‑zero number.

天平法就是对等号两边进行相同的运算，直到把未知数单独留在一边。常见的运算有加、减、乘、除同一个非零数。

Example: x + 7 = 15. Subtract 7 from both sides to keep the scales balanced: x + 7 − 7 = 15 − 7, so x = 8.

例如：x + 7 = 15。两边同时减去 7：x + 7 − 7 = 15 − 7，得到 x = 8。

x + 7 = 15 → x = 8

Always show your working clearly, writing each new line below the previous one. This reduces mistakes and gains marks in exams.

一定要把计算步骤清楚地写在下一行，这能减少失误，考试时也能拿到步骤分。

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3. One‑Step Equations | 一步方程

One‑step equations require just one operation to isolate the variable. They may involve addition, subtraction, multiplication, or division.

一步方程只需要一次运算就能把变量分离出来，涉及加法、减法、乘法或除法。

Operation	Example	Opposite step	Solution
Addition	x + 5 = 12	Subtract 5	x = 7
Subtraction	y − 3 = 9	Add 3	y = 12
Multiplication	4a = 20	Divide by 4	a = 5
Division	b ÷ 6 = 3	Multiply by 6	b = 18

Remember: the opposite of addition is subtraction, the opposite of multiplication is division, and vice versa.

记住：加法与减法互为逆运算，乘法与除法互为逆运算。

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4. Two‑Step Equations | 两步方程

Two‑step equations contain two operations. Always undo addition or subtraction first, then undo multiplication or division. This order ensures you reverse the order of operations correctly.

两步方程中包含两种运算。一定要先处理加减，再处理乘除。这个顺序能让你正确地反向使用运算顺序。

Example: 3x + 2 = 14. Step 1: subtract 2 from both sides → 3x = 12. Step 2: divide both sides by 3 → x = 4.

例如：3x + 2 = 14。第一步：两边减 2 → 3x = 12。第二步：两边除以 3 → x = 4。

3x + 2 = 14 → 3x = 12 → x = 4

Another example: 5y − 7 = 23. Add 7 first → 5y = 30, then divide by 5 → y = 6.

再比如：5y − 7 = 23。先加 7 → 5y = 30，再除以 5 → y = 6。

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5. Equations with Brackets | 带括号的方程

When an equation contains brackets, expand them first using the distributive law, then solve as usual.

当方程含有括号时，先用分配律展开，再按常规方法求解。

Example: 2(x + 3) = 16. Expand: 2x + 6 = 16. Subtract 6 → 2x = 10. Divide by 2 → x = 5.

例如：2(x + 3) = 16。展开：2x + 6 = 16。减 6 → 2x = 10。除以 2 → x = 5。

Alternative method: divide both sides by 2 first, giving x + 3 = 8, then subtract 3 → x = 5. Both ways are valid; choose the one you find easier.

另一种方法：先两边除以 2，得 x + 3 = 8，再减 3 → x = 5。两种方法都对，选你感觉更容易的那种。

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6. Variables on Both Sides | 两边含变量的方程

When the unknown appears on both sides of the equation, use addition or subtraction to collect all variable terms on one side and constant terms on the other.

当未知数出现在等号两边时，用加减法把所有含变量的项移到一边，所有常数项移到另一边。

Example: 7x + 4 = 3x + 20. Subtract 3x from both sides: 4x + 4 = 20. Then subtract 4: 4x = 16. Divide by 4: x = 4.

例如：7x + 4 = 3x + 20。两边减 3x：4x + 4 = 20。再减 4：4x = 16。除以 4：x = 4。

It is often easier to move the smaller variable term to avoid negative coefficients, but if negatives appear, handle them carefully.

通常把较小的变量项移过去可以避免负系数，但如果出现负数，仔细处理就行，不必害怕。

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7. Equations Involving Fractions | 含分数的方程

If an equation contains fractions, multiply every term on both sides by the lowest common denominator (LCD) to clear the fractions, then solve the resulting integer equation.

如果方程含有分数，先在等式两边每一项都乘以最小公分母，把分母消掉，再解得到的整数方程。

Example: x/2 + 3 = 7. LCD is 2. Multiply every term by 2: x + 6 = 14. Then x = 8.

例如：x/2 + 3 = 7。最小公分母是 2。每项乘以 2：x + 6 = 14。然后 x = 8。

For equations like (2x)/3 = 8, multiply both sides by 3 → 2x = 24 → x = 12.

对于 (2x)/3 = 8 这类方程，两边乘 3 → 2x = 24 → x = 12。

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8. Checking Your Solution | 检验你的解

Always substitute your answer back into the original equation. The left‑hand side (LHS) should equal the right‑hand side (RHS). This habit catches arithmetic errors instantly.

一定要把答案代回原方程检验。左边应等于右边。这个习惯能立刻发现计算错误。

For x = 5 in 2(x + 3) = 16: LHS = 2(5 + 3) = 2 × 8 = 16 = RHS. Correct.

以 x = 5 代入 2(x + 3) = 16：左边 = 2(5 + 3) = 2 × 8 = 16 = 右边。正确。

If LHS ≠ RHS, re‑check your steps; a small sign error is often the culprit.

如果左边不等于右边，回头检查步骤，小小的符号错误往往是罪魁祸首。

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

• Forgetting the balance rule: Always do the same to both sides, including when subtracting a term.
• 犯错：忘记天平规则。 一定要对两边做相同操作，移项时也不例外。
• Expanding brackets incorrectly: Remember, 2(x + 3) = 2x + 6, not 2x + 3.
• 括号展开错误： 记住 2(x + 3) = 2x + 6，而不是 2x + 3。
• Reversing the order: For 3x + 2, undo the +2 first, then the ×3.
• 运算顺序搞反： 对 3x + 2，先处理 +2，再处理 ×3。
• Dropping negative signs: Be careful when moving terms; −x from one side added to the other becomes +x.
• 遗漏负号： 移项时要小心，比如把 −x 移到另一边就变成 +x。

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10. Practice Questions | 练习题

Try solving these equations step by step, then check your answers.

试着逐步解以下方程，然后检验答案。

• (a) x + 9 = 21  (b) y − 4 = 15  (c) 6p = 42  (d) a ÷ 5 = 7
• (e) 4m + 3 = 19  (f) 5k − 8 = 27  (g) 3(2x − 1) = 21  (h) 9n + 5 = 4n + 30
• (i) y/3 + 2 = 10  (j) (5x)/2 = 15

Answers: (a) 12 (b) 19 (c) 7 (d) 35 (e) 4 (f) 7 (g) 4 (h) 5 (i) 24 (j) 6. How many did you get right?

答案：(a) 12 (b) 19 (c) 7 (d) 35 (e) 4 (f) 7 (g) 4 (h) 5 (i) 24 (j) 6。你做对了多少？

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11. From Solving to Applying | 从求解到应用

Once you can solve equations fluently, you can tackle word problems. Translate the written scenario into an algebraic equation, solve it, and interpret the result in context.

一旦你能熟练解方程，就可以挑战应用题了。把文字情境翻译成代数方程，解出来，再结合情境解释结果。

Example: ‘Three times a number added to 7 gives 22.’ Let the number be n → 3n + 7 = 22 → 3n = 15 → n = 5.

例如：“某个数的 3 倍加 7 等于 22。”设这个数为 n → 3n + 7 = 22 → 3n = 15 → n = 5。

Practice forming equations from real‑life situations — it strengthens logical thinking and prepares you for more advanced problem‑solving.

多练习从实际情境列方程，这能强化逻辑思维，为更高级的问题解决打下基础。

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12. Summary and Key Tips | 总结与关键技巧

Mastering linear equations comes down to three things: understanding the balance method, practising step‑by‑step solutions, and checking your answers. Keep your working neat, and don’t rush — accuracy matters more than speed at KS3.

掌握线性方程归结起来就是三点：理解天平法、坚持分步求解、每次检验答案。保持书写工整，不要图快——在 KS3 阶段，准确比速度更重要。

• Write each new line directly below the previous one.
• 每写一个新的算式，都放在上一行的正下方。
• Use opposite operations to isolate the variable.
• 使用逆运算来分离变量。
• Tackle addition/subtraction before multiplication/division.
• 先处理加减，再处理乘除。
• Expand brackets first if they are present.
• 如果有括号，先展开。
• Clear fractions by multiplying by the LCD.
• 乘以最小公分母消去分数。
• Always plug your solution back into the original equation.
• 始终把解代回原方程检验。

With consistent practice, solving linear equations will soon feel as natural as balancing a seesaw — and you’ll be ready for graphs, inequalities, and simultaneous equations with confidence.

通过持续练习，解线性方程很快会像平衡跷跷板一样自然——你将有信心迎接图像、不等式和联立方程的挑战。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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