Solving Linear Equations: From Balance to Algebra | 解一元一次方程：从平衡到代数

📚 Solving Linear Equations: From Balance to Algebra | 解一元一次方程：从平衡到代数

Linear equations are the foundation of algebra at Key Stage 3. They allow you to find unknown values by maintaining balance and applying inverse operations. Mastering them will prepare you for more advanced topics such as simultaneous equations, graphs, and functions.

线性方程是KS3阶段代数的基础。它们通过保持平衡和应用逆运算帮助你找到未知数。掌握它们将为你学习更高级的内容如联立方程、图像和函数做好准备。

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1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that says two expressions are equal. It contains an equals sign and often one or more unknown values, usually represented by letters such as x or y. For example, x + 3 = 10 is an equation because it states that when you add 3 to x, the result is 10.

方程是表明两个表达式相等的数学语句。它包含等号，通常有一个或多个未知数，常用字母如x或y表示。例如，x + 3 = 10 就是一个方程，因为它表明将x加3后等于10。

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2. The Balance Concept | 平衡的概念

Think of an equation as a perfectly balanced scale. Whatever operation you perform on one side, you must perform the same operation on the other side to keep the scale balanced. This is the golden rule of solving equations and ensures the equality remains true.

将方程想象为一架完美平衡的天平。你对方程一边做的任何运算，都必须对另一边做同样的运算，以保持天平平衡。这是解方程的黄金法则，确保等式始终成立。

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3. Inverse Operations | 逆运算

To isolate the variable, we use inverse operations. Addition and subtraction are inverses, as are multiplication and division. Squaring and square roots are also inverses, but at KS3 level we focus on the four basic operations. Applying an inverse operation undoes the original and helps reveal the unknown.

为了隔离变量，我们使用逆运算。加法和减法互为逆运算，乘法和除法也是如此。平方和开平方也是逆运算，但在KS3阶段我们主要关注四种基本运算。运用逆运算可以抵消原来的运算，从而求出未知数。

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4. Solving One-Step Equations: Addition and Subtraction | 解一步方程：加法和减法

When the unknown is combined with a number by addition or subtraction, simply undo that operation. For x + 5 = 12, subtract 5 from both sides to get x = 7. For y − 4 = 9, add 4 to both sides to obtain y = 13.

x + 5 = 12 → x = 7

当未知数与数字通过加法或减法结合时，只需撤销那个运算。对于 x + 5 = 12，两边同减5得到 x = 7；对于 y − 4 = 9，两边同加4得到 y = 13。

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5. Solving One-Step Equations: Multiplication and Division | 解一步方程：乘法和除法

If the variable is multiplied or divided by a number, reverse the operation. For 3x = 18, divide both sides by 3 to find x = 6. For z/5 = 3, multiply both sides by 5 to get z = 15.

3x = 18 → x = 6

如果变量被乘以或除以一个数，则反向操作。对于 3x = 18，两边同除以3得到 x = 6；对于 z/5 = 3，两边同乘5得到 z = 15。

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6. Solving Two-Step Equations | 解两步方程

Two-step equations involve a combination of operations, such as multiplication and addition. Undo them in reverse order of operations—undo addition or subtraction first, then multiplication or division. Example: 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

2x + 3 = 11 → 2x = 8 → x = 4

两步方程包含两种运算，如乘法和加法。按照运算顺序的逆序撤销——先处理加减，再处理乘除。例子：2x + 3 = 11。两边同减3得 2x = 8，再同除以2得 x = 4。

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7. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law. For instance, 3(x − 2) = 9 becomes 3x − 6 = 9. Then solve: add 6 to both sides (3x = 15) and divide by 3 to get x = 5.

3(x − 2) = 9 → 3x − 6 = 9 → x = 5

当方程含有括号时，先用分配律展开。例如 3(x − 2) = 9 展开为 3x − 6 = 9。然后求解：两边加6得 3x = 15，再除以3得 x = 5。

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8. Equations with Variables on Both Sides | 两边都有变量的方程

If variables appear on both sides, start by bringing all variable terms to one side. Subtract or add the smaller variable term to both sides. Example: 5x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, and divide by 2: x = 4.

5x + 2 = 3x + 10 → 2x = 8 → x = 4

如果变量同时出现在两边，先将所有含变量的项移到一边。两边减去或加上较小的变量项。例如：5x + 2 = 3x + 10。两边减 3x 得 2x + 2 = 10；再减2得 2x = 8，除以2得到 x = 4。

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9. Equations Involving Fractions | 涉及分数的方程

To eliminate fractions, multiply every term by the least common denominator (LCD). Solve x/3 + 2 = 5: multiply all terms by 3 to get x + 6 = 15. Then subtract 6 to find x = 9.

x/3 + 2 = 5 → x + 6 = 15 → x = 9

要去掉分数，可将每一项都乘以最小公分母。解 x/3 + 2 = 5：所有项乘3得 x + 6 = 15，再减6得 x = 9。

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

Read the problem carefully, define the unknown with a letter, and translate the words into an equation. Example: ‘Three more than twice a number is 11.’ Let n be the number. Then 2n + 3 = 11. Solve: subtract 3, 2n = 8, so n = 4.

仔细读题，用字母定义未知数，并将文字转化为方程。例子：“一个数的两倍多3是11。”设这个数为 n，则有 2n + 3 = 11。求解：减3得 2n = 8，所以 n = 4。

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11. Checking Your Solution | 检验你的解

Always substitute your answer back into the original equation to verify it works. For 2x + 3 = 11, substitute x = 4: 2(4) + 3 = 8 + 3 = 11, which matches the right side. This confirms the solution is correct.

一定要把求得的解代入原方程进行检验。对于 2x + 3 = 11，代入 x = 4：2(4) + 3 = 8 + 3 = 11，等于右边，说明解是正确的。

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12. Common Mistakes and Tips | 常见错误与提示

Typical errors include forgetting to do the same operation on both sides, mishandling negative signs, and misapplying the distributive law. Remember to expand brackets correctly, reverse the order of operations, and check your working step by step.

• Never only apply an operation to just one side.
• Be careful with signs: −3(x − 2) = −3x + 6.
• Always write the operation you are doing to both sides.
• Practice regularly to build speed and accuracy.

常见错误包括忘记在等号两边进行相同的操作、错误处理负号以及错误使用分配律。记住正确展开括号，逆序运算，并逐步检查你的解题过程。

• 绝对不要只对方程的一边施加运算。
• 注意符号：−3(x − 2) = −3x + 6。
• 始终写出你在等号两边进行的运算。
• 定期练习以提高速度和准确性。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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