Solving Linear Equations with One Variable | 求解一元一次方程

📚 Solving Linear Equations with One Variable | 求解一元一次方程

Linear equations are the foundation of algebra at the KS3 level. In this topic, you will learn how to solve equations that contain one unknown variable using systematic methods. The key goal is to isolate the variable on one side of the equation while keeping the balance. Whether the variable appears on one side or both sides, the same balancing rules apply. Mastering this skill will prepare you for more advanced algebra in GCSE and beyond. Let us explore step-by-step techniques, examples, and common pitfalls.

线性方程是 KS3 阶段代数的基石。在这一主题中，你将学习如何运用系统的方法求解含有一个未知变量的方程。核心目标是将变量单独移到方程的一边，同时保持等式的平衡。无论变量出现在一边还是两边，平衡法则始终适用。掌握这一技能将为你学习 GCSE 以及更高级的代数做好准备。让我们逐步探索求解技巧、示例和常见错误。

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1. Understanding Linear Equations | 理解线性方程

A linear equation is an algebraic statement where the highest power of the variable is 1. For example, 2x + 3 = 9 is a linear equation. The equation contains an expression on the left, an expression on the right, and an equals sign in the middle. The letter x represents the unknown number we need to find. Linear equations are called ‘linear’ because if you graph them, they produce a straight line. In KS3, we solve these equations without drawing graphs; we use algebraic manipulation.

线性方程是一种代数表达式，其中变量的最高次幂为 1。例如，2x + 3 = 9 就是一个线性方程。方程左边有一个表达式，右边有一个表达式，中间用等号连接。字母 x 代表我们需要寻找的未知数。之所以称为“线性”方程，是因为如果画出其图像，会得到一条直线。在 KS3 阶段，我们不借助图像，而是通过代数变形来求解这类方程。

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2. The Balancing Method | 平衡法

Think of an equation as a balanced scale. Whatever you do to one side, you must do exactly the same to the other side to keep the scale balanced. This is the most important rule in solving equations. If you add 5 to the left side, add 5 to the right side. If you divide the left side by 3, divide the right side by 3. The equals sign represents the pivot of the scale. By applying inverse operations step by step, we can isolate the variable and find its value.

把方程想象成一架平衡的天平。你对一边所做的任何操作，都必须对另一边完全同样地执行，以保持天平的平衡。这是解方程最重要的法则。如果你在左边加上 5，右边也必须加上 5。如果你将左边除以 3，右边同样要除以 3。等号代表天平的中轴。通过逐步应用逆运算，我们可以分离出变量并求出它的值。

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3. One-Step Equations | 一步方程

The simplest linear equations require just one operation to solve. For instance, x + 7 = 12. To undo the addition, subtract 7 from both sides: x + 7 − 7 = 12 − 7, giving x = 5. Another example is 4x = 20. Since x is multiplied by 4, we divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, so x = 5. Practising one-step equations builds confidence before moving to more complex types.

最简单的线性方程只需一步运算即可求解。例如，x + 7 = 12。要消去加法，就两边同时减去 7：x + 7 − 7 = 12 − 7，得出 x = 5。另一个例子是 4x = 20。因为 x 与 4 相乘，我们将两边同时除以 4：4x ÷ 4 = 20 ÷ 4，于是 x = 5。练习一步方程可以在进入更复杂类型前建立信心。

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4. Two-Step Equations | 两步方程

Two-step equations involve two operations. Consider 3x + 2 = 11. First, subtract 2 from both sides to undo the addition: 3x = 9. Then divide both sides by 3 to undo the multiplication: x = 3. The order of undoing operations follows the reverse of BIDMAS: we deal with addition/subtraction first, then multiplication/division. Always check your solution by substituting it back into the original equation.

两步方程包含两次运算。考虑方程 3x + 2 = 11。首先，两边同时减去 2，消去加法：3x = 9。然后将两边同时除以 3，消去乘法：x = 3。逆运算的顺序与 BIDMAS 相反：先处理加减法，再处理乘除法。务必通过将解代回原方程来验证正确性。

3x + 2 = 11 → x = 3

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5. Equations with Variables on Both Sides | 变量在两侧的方程

When the variable appears on both sides of the equation, we first collect all variable terms on one side and constant terms on the other. For example, 5x − 3 = 2x + 9. Subtract 2x from both sides: 3x − 3 = 9. Then add 3 to both sides: 3x = 12. Finally, divide by 3: x = 4. The principle of balance remains the same; we just need an extra step to bring like terms together.

当变量出现在方程的两侧时，我们首先将所有含变量的项移到一边，把常数项移到另一边。例如，5x − 3 = 2x + 9。两边同时减去 2x：3x − 3 = 9。然后两边同时加上 3：3x = 12。最后除以 3：x = 4。平衡原则保持不变；我们只需额外一步把同类项合并在一起。

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6. Combining Like Terms | 合并同类项

Equations often contain several like terms that need to be simplified first. Like terms are those that have the same variable raised to the same power. For example, 4x + 3 + 2x − 5 = 20 can be simplified on the left side by combining 4x and 2x to get 6x, and 3 − 5 to get −2. This gives 6x − 2 = 20, which is then easier to solve. Never skip the simplification step; it reduces mistakes.

方程中通常含有多个需要先进行化简的同类项。同类项是指那些具有相同变量且相同指数的项。例如，方程 4x + 3 + 2x − 5 = 20 可以通过将左边的 4x 与 2x 合并为 6x，3 和 −5 合并为 −2 来进行简化。得到 6x − 2 = 20，然后求解就容易多了。切勿跳过化简步骤；这能减少错误。

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7. Equations Involving Brackets | 带有括号的方程

If an equation contains brackets, expand them as the first step. Use the distributive law: multiply each term inside the bracket by the factor outside. For instance, 3(x + 4) = 21 becomes 3x + 12 = 21 after expansion. Then solve as a two-step equation: subtract 12 (3x = 9), divide by 3 (x = 3). If there are brackets on both sides, expand both, then simplify and solve. Brackets can hide the structure of an equation, so always remove them first.

如果方程中含有括号，第一步就要将其展开。运用分配律：将括号外的因数乘给括号内的每一项。例如，3(x + 4) = 21 展开后变为 3x + 12 = 21。然后按照两步方程来解：减去 12（3x = 9），再除以 3（x = 3）。如果两边都有括号，两边同时展开，然后化简并求解。括号会掩盖方程的结构，因此一定要优先去掉它们。

2(y − 5) + 3y = 4(y + 1)

Step 1: Expand → 2y − 10 + 3y = 4y + 4
Step 2: Combine like terms → 5y − 10 = 4y + 4
Step 3: Subtract 4y → y − 10 = 4
Step 4: Add 10 → y = 14

第一步：展开 → 2y − 10 + 3y = 4y + 4
第二步：合并同类项 → 5y − 10 = 4y + 4
第三步：减 4y → y − 10 = 4
第四步：加 10 → y = 14

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8. Equations with Fractions | 含有分数的方程

Fractions in equations can be cleared by multiplying every term by the lowest common denominator (LCD). For example, in the equation x/3 + 1/2 = 5/6, the LCD of 3, 2, and 6 is 6. Multiply all terms by 6: 6(x/3) + 6(1/2) = 6(5/6), which simplifies to 2x + 3 = 5. Then solve: subtract 3 (2x = 2), divide by 2 (x = 1). Removing fractions early makes the equation much friendlier to work with.

方程中的分数可以通过将每一项乘以最小公分母(LCD)来消除。例如，在方程 x/3 + 1/2 = 5/6 中，3、2 和 6 的最小公分母是 6。将所有项乘以 6：6(x/3) + 6(1/2) = 6(5/6)，化简为 2x + 3 = 5。然后求解：减 3（2x = 2），除以 2（x = 1）。尽早消去分数会让方程的解答过程简单得多。

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9. Checking Your Solution | 检验你的解

Always substitute your found value back into the original equation to verify it works. For x = 4 in 5x − 3 = 2x + 9: left side gives 5(4) − 3 = 20 − 3 = 17; right side gives 2(4) + 9 = 8 + 9 = 17. Both sides equal, so the solution is correct. If the two sides do not match, you have made a mistake along the way. Checking not only confirms accuracy but also reinforces your understanding of the balance method.

一定要将求得的值代回原方程，验证它是否正确。对于 x = 4 代入 5x − 3 = 2x + 9：左边得到 5(4) − 3 = 20 − 3 = 17；右边得到 2(4) + 9 = 8 + 9 = 17。两边相等，说明解是正确的。如果两边不相等，就说明你在解题过程中犯了错误。检验不仅能确认准确性，还能加深你对平衡法的理解。

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10. Real-Life Word Problems | 实际生活中的应用题

Linear equations often model real-world situations. For example: ‘Three more than twice a number is 15. Find the number.’ Translate into algebra: let the number be n; equation: 2n + 3 = 15. Solve: 2n = 12, n = 6. Another example: ‘The perimeter of a rectangle is 30 cm. Its length is twice its width. Find the dimensions.’ Let width = w, length = 2w; perimeter = 2(w + 2w) = 6w; so 6w = 30, w = 5 cm, length = 10 cm. Translating words into equations is a key skill.

线性方程常常用于模拟现实世界的情况。例如：“一个数的两倍加三等于 15。求这个数。”将其转化为代数：设这个数为 n；列出方程：2n + 3 = 15。求解：2n = 12，n = 6。另一个例子：“一个长方形的周长是 30 cm，长是宽的两倍。求它的尺寸。”设宽 = w，长 = 2w；周长 = 2(w + 2w) = 6w；于是 6w = 30，w = 5 cm，长 = 10 cm。将文字转化为方程是一项关键技能。

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11. Common Mistakes to Avoid | 应避免的常见错误

Many students lose marks through careless errors. Common mistakes include: forgetting to apply an operation to both sides; incorrectly expanding brackets (e.g. 3(x + 2) becomes 3x + 2); mishandling negative signs; dividing in the wrong order; and not checking the final answer. Always write down each step clearly. If the equation involves negatives, use extra care: −2x = 8 means x = −4, not x = 4. Regular practice and self-checking are the best cures for these errors.

许多学生因粗心而丢分。常见错误包括：忘记对两边同时进行同一操作；错误展开括号（例如 3(x + 2) 错成 3x + 2）；处理负号不当；除法顺序出错；以及不检查最终答案。务必清晰写出每一步。如果方程中包含负数，要格外小心：−2x = 8 意味着 x = −4，而不是 x = 4。经常练习并自我检查是纠正这些错误的最好方法。

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12. Practice Makes Perfect | 熟能生巧

To become confident in solving linear equations, you must practise a variety of problems. Start with one-step and two-step equations, then move to equations with variables on both sides, brackets, and fractions. Use the balance method consistently. Over time, the steps will become automatic. Remember to always verify your solutions. The skills you build now form the backbone of GCSE algebra, where you will solve simultaneous equations, quadratic equations, and more.

要自信地求解一元一次方程，你需要练习各种不同类型的题目。先从一步和两步方程开始，再逐步过渡到变量在两侧的方程、带括号的方程以及含分数的方程。始终如一地运用平衡法。久而久之，解题步骤将变得自然而然。记住要经常验证你的解。你现在建立的技能是 GCSE 代数的核心，届时你将要解联立方程、二次方程等更复杂的题目。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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