Page 174 Exercise: Solving Linear Equations | 第174页练习：求解线性方程

📚 Page 174 Exercise: Solving Linear Equations | 第174页练习：求解线性方程

Welcome to the Page 174 exercise walkthrough, designed for Key Stage 3 learners following the Cambridge mathematics framework. In this article, you will explore how to solve linear equations step by step, starting from simple one-step operations and progressing to equations that contain brackets and unknowns on both sides. Each method is illustrated with clear examples, and common pitfalls are highlighted to help you build confidence and accuracy. Use this guide alongside your coursebook to master the techniques required in the Page 174 questions.

欢迎阅读第174页练习的详细讲解，专为遵循剑桥数学框架的KS3学生设计。在本文中，你将逐步学习如何解线性方程，从简单的一步运算开始，逐步深入到含有括号和两边均含未知数的方程。每一种方法都配有清晰的示例，并指出了常见错误，帮助你建立信心、提高准确性。请结合你的教材使用本指南，掌握第174页题目所需的技巧。

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1. Understanding Linear Equations | 理解线性方程

A linear equation is a mathematical statement that shows two expressions are equal, such as x + 5 = 12. It contains only numbers and variables raised to the power of 1, meaning there are no x² or x³ terms. The goal when solving any linear equation is to find the value (or values) of the unknown that makes the equation true. For example, in the equation 2n = 10, the solution is n = 5 because 2 × 5 equals 10. You will often see letters like x, y, a or n representing the unknown.

线性方程是表明两个表达式相等的数学语句，例如 x + 5 = 12。它只包含数字和次数为1的变量，即没有 x² 或 x³ 这样的项。解任何线性方程的目标都是求出使方程成立的未知数的值。例如，在方程 2n = 10 中，解为 n = 5，因为 2 × 5 等于 10。你经常会看到 x、y、a 或 n 等字母代表未知数。

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2. The Balancing Method | 平衡法原理

To solve an equation, you must always keep it balanced. Think of an equation as an old-fashioned pair of scales: whatever you do to one side, you must do exactly the same to the other side to maintain the balance. This is the golden rule of algebra. If you add 3 to the left-hand side, you must add 3 to the right-hand side. If you divide the left-hand side by 4, you must also divide the right-hand side by 4. By repeatedly applying the same operation to both sides, you can isolate the unknown and find its value.

解方程时，必须始终保持方程平衡。可以把方程想象成一架老式天平：无论你对一侧做什么操作，都必须对另一侧做完全相同的操作，才能保持平衡。这就是代数的黄金法则。如果你在左边加上3，就必须在右边也加上3；如果你将左边除以4，那么右边也必须除以4。通过对两边反复施加相同的运算，你就能分离出未知数并求出它的值。

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3. Solving One-Step Equations | 解一步方程

One-step equations require only a single inverse operation to reveal the unknown. If the equation involves addition, use subtraction; if it involves multiplication, use division. For instance, to solve m + 7 = 15, subtract 7 from both sides to get m = 8. Similarly, for 3y = 21, divide both sides by 3 to obtain y = 7. Remember to write the operation on both sides clearly, as this will help you when equations become more complicated.

一步方程仅需一次逆运算即可求出未知数。如果方程涉及加法，就用减法；如果涉及乘法，就用除法。例如，解 m + 7 = 15 时，两边同时减去7，得到 m = 8。类似地，对于 3y = 21，两边同时除以3，得到 y = 7。记得在两边明确写出运算过程，这会在方程变得更复杂时给你很大帮助。

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4. Solving Two-Step Equations | 解两步方程

Two-step equations involve a combination of operations, such as multiplication and addition. To solve them, you need to reverse the order of operations. The general strategy is to undo the addition or subtraction first, then tackle the multiplication or division. Consider the equation 2x + 5 = 17. Subtract 5 from both sides to get 2x = 12. Then divide both sides by 2 to find x = 6. Always check your solution by substituting it back into the original equation.

两步方程涉及运算的组合，例如乘法和加法。解这类方程时，你需要逆用运算顺序。一般策略是先消去加法或减法，再处理乘法或除法。以方程 2x + 5 = 17 为例。两边先减去5，得到 2x = 12。然后两边除以2，求得 x = 6。一定要将解代回原方程进行检验。

Example: 3a − 4 = 11 → add 4 to both sides: 3a = 15 → divide by 3: a = 5

示例：3a − 4 = 11 → 两边加4：3a = 15 → 除以3：a = 5

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5. Equations with Brackets | 含括号的方程

When brackets appear in an equation, you cannot simply ignore them. There are two common approaches: expanding the brackets first, or treating the bracketed expression as a single package and dividing later. For example, in 3(x + 2) = 18, you can expand to 3x + 6 = 18, then subtract 6 and divide by 3. Alternatively, you can divide both sides by 3 immediately to get x + 2 = 6, then subtract 2. Both routes lead to x = 4. The key is to follow the balancing principle at every step.

当方程中出现括号时，不能简单地忽略它们。通常有两种方法：先展开括号，或者把括号内的表达式看作一个整体稍后再除。例如，在 3(x + 2) = 18 中，你可以展开得到 3x + 6 = 18，然后减去6再除以3。另一种方法是直接将两边除以3，得到 x + 2 = 6，再减去2。两种途径都得出 x = 4。关键在于每一步都要遵循平衡原则。

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6. Expanding Brackets First | 先展开括号

Expanding brackets correctly is essential when the coefficient outside is a negative number or when the equation contains more than one bracket. Always multiply each term inside the bracket by the term outside. Look at the equation 5(2y − 3) = 35. Expand to get 10y − 15 = 35. Then add 15 to both sides: 10y = 50. Divide by 10 to find y = 5. Be especially careful with signs: multiplying a negative number outside by a term inside can change the sign, as in −2(t + 4) = −2t − 8.

当括号外的系数为负数或方程含有一个以上括号时，正确展开括号就非常重要。永远将括号外的项乘以括号内的每一项。以方程 5(2y − 3) = 35 为例。展开得到 10y − 15 = 35。然后两边加上15：10y = 50。再除以10，求得 y = 5。要特别注意符号：括号外的负数乘以括号内的项会改变符号，例如 −2(t + 4) = −2t − 8。

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

When the unknown appears on both sides of the equation, your first task is to collect all the variable terms on one side and the number terms on the other. For instance, solve 7p + 2 = 3p + 14. Subtract 3p from both sides to eliminate the smaller p-term: 4p + 2 = 14. Then subtract 2: 4p = 12, so p = 3. If you encounter an equation like 6x − 1 = 4x + 9, subtract 4x from both sides, then add 1 to both sides to isolate the x-term. Always aim to move the variable to the side where it becomes positive.

当方程两边都出现未知数时，首要任务是将所有含变量的项移到一边，将常数项移到另一边。例如，解方程 7p + 2 = 3p + 14。两边减去 3p 以消去较小的 p 项：4p + 2 = 14。接着减去2：4p = 12，得到 p = 3。如果遇到像 6x − 1 = 4x + 9 这样的方程，两边先减去 4x，然后两边加1就能分离出含 x 的项。始终尽量让变量移到使系数为正的一侧。

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8. Checking Your Solution | 检验你的解

After finding a solution, always substitute it back into the original equation to verify that it works. This habit catches simple arithmetic mistakes and ensures you haven’t broken the balance rule. For the equation 4(x − 5) = 12, the solution is x = 8. Check: left side 4(8 − 5) = 4 × 3 = 12, which equals the right side. If both sides match, your answer is correct. If they don’t, retrace your steps and look for an error, especially in signs or bracket expansion.

求出解之后，一定将它代回原方程检验是否成立。这个习惯能够发现简单的算术错误，并确保你没有违反平衡法则。对于方程 4(x − 5) = 12，解为 x = 8。检验：左边 4(8 − 5) = 4 × 3 = 12，等于右边。如果两边相等，答案就是正确的。如果不相等，就回头检查步骤，尤其要注意符号或括号展开中的错误。

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9. Forming Equations from Word Problems | 根据文字题列方程

Many real-life situations and exam questions ask you to form an equation before solving it. Read the problem carefully and let the unknown be represented by a letter, such as n. Translate the words into algebraic expressions. For example, “I think of a number, multiply it by 4, then subtract 7 and the result is 29” becomes the equation 4n − 7 = 29. Solve to get n = 9. Always define what your letter stands for and check whether the answer makes sense in the original context.

许多实际情境和考试题目都要求你先列出方程再求解。要仔细阅读问题，用一个字母（如 n）代表未知数，将文字翻译成代数表达式。例如，“我想一个数，乘以4，再减去7，结果是29”可以转化为方程 4n − 7 = 29。解方程得 n = 9。始终明确你使用的字母代表什么，并检查答案在原情境中是否合理。

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10. Common Mistakes to Avoid | 常见错误与提醒

Even experienced students can slip up when solving linear equations. One frequent error is forgetting to multiply every term inside the bracket. For 3(x + 2), some mistakenly write 3x + 2 instead of 3x + 6. Another mistake is performing an operation on one side only. If you add 5 to the left, you must add 5 to the right. Also, watch out for sign errors when dealing with negative coefficients. Finally, after dividing, ensure you simplify the fraction or integer correctly – for instance, −12 ÷ 4 = −3, not 3. Use the checking step to catch these errors before they cost you marks.

即使是经验丰富的学生在解线性方程时也可能出错。一个常见错误是忘记乘以括号内的每一项。对于 3(x + 2)，有人会错误地写成 3x + 2，而不是 3x + 6。另一个错误是只对一边进行运算。如果你在左边加5，那么右边也必须加5。此外，处理负系数时要注意符号错误。最后，在除法之后要确保正确化简分数或整数——例如，−12 ÷ 4 = −3，而不是3。在正式作答前利用检验步骤发现这些错误，以免丢分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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