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 everywhere in Key Stage 3 Mathematics and help us find the value of an unknown number. An equation is like a balanced set of scales – whatever you do to one side, you must do to the other to keep the balance. In this article, we will explore how to solve equations step by step, from simple one-step problems to more challenging ones with brackets and fractions. You will learn reliable methods, how to check your answer, and how to avoid common mistakes.

线性方程是代数的基础。它们在 Key Stage 3 数学中无处不在，帮助我们求出未知数的值。一个方程就像一个平衡的天平 —— 你对一边做什么，就必须对另一边做同样的事，以保持平衡。在本文中，我们将一步步探索如何解方程，从简单的一步问题到带有括号和分数的更具挑战性的题目。你将学到可靠的方法、如何检验答案，以及如何避免常见错误。

1. Understanding Equations | 理解方程

An equation is a mathematical statement that says two expressions are equal. It always contains an equals sign. For example, x + 3 = 7 is an equation. The letter x is called the variable or the unknown. Solving an equation means finding the value of the unknown that makes the statement true. In this case, x = 4 because 4 + 3 = 7. Equations are not just number puzzles – they model real-life situations, like calculating costs, distances, or quantities.

方程是一个数学陈述，表明两个表达式相等。它总是包含一个等号。例如，x + 3 = 7 就是一个方程。字母 x 被称为变量或未知数。解方程就是求出使这个陈述成立的未知数的值。在这种情况下，x = 4，因为 4 + 3 = 7。方程不仅仅是数字谜题——它们模拟现实生活中的情况，如计算成本、距离或数量。

Think of an equation as a balanced seesaw. The left side equals the right side. If we add or remove something from one side, the seesaw tips unless we do exactly the same to the other side. This idea is the golden rule of solving equations: always perform the same operation on both sides. Common operations include addition, subtraction, multiplication, and division. The goal is to isolate the variable, usually ending up with x = some number.

把方程想象成一个平衡的跷跷板。左边等于右边。如果我们从一边增加或拿走一些东西，跷跷板就会倾斜，除非我们对另一边做完全相同的操作。这个想法就是解方程的黄金法则：始终在等式的两边执行相同的运算。常见的运算包括加法、减法、乘法和除法。目标是单独分离出变量，通常最终得到 x = 某个数。

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2. Solving by Inspection | 通过观察求解

For very simple equations, you can often find the answer just by looking at the equation. This is called solving by inspection. For instance, 5 + y = 9. You might instantly know that y must be 4 because 5 + 4 = 9. Similarly, if you see 3x = 12, you can think ‘3 times what gives 12?’, which is 4. Inspection is quick, but it only works for straightforward problems. As equations get more complicated, we need a more structured method.

对于非常简单的方程，你常常只需看一眼方程就能找出答案。这称为通过观察求解。例如，5 + y = 9。你可能会立刻知道 y 一定是 4，因为 5 + 4 = 9。同样，如果你看到 3x = 12，你可以想“3 乘以什么数等于 12？”，答案是 4。观察法很快，但它只适用于简单直接的问题。当方程变得更复杂时，我们就需要一个更有条理的方法了。

Inspection is still a useful first step to make a guess, and then you can check it. However, for exams and more difficult problems, you are expected to show your working clearly. The methods described in the following sections will help you build up those algebra skills.

观察法仍然是做出猜测的一个有用的第一步，然后你可以检验它。但是，对于考试和更困难的题目，你需要清晰地展示出解题步骤。下面几节中描述的方法将帮助你构建这些代数技能。

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3. One-Step Equations: Addition & Subtraction | 一步方程：加法与减法

One-step equations involve only one operation to isolate the variable. When the operation is addition, you subtract the same number from both sides. For example, x + 5 = 11. To get x on its own, subtract 5 from both sides: x + 5 − 5 = 11 − 5, which simplifies to x = 6. The +5 and −5 cancel each other out on the left, leaving just x.

一步方程只涉及一步运算就能隔离出变量。当运算是加法时，你需要从两边减去同一个数。例如，x + 5 = 11。为了让 x 单独在一边，两边同时减去 5：x + 5 − 5 = 11 − 5，简化后得到 x = 6。左边的 +5 和 −5 相互抵消，只剩下 x。

When the equation involves subtraction, you add the same number to both sides. Consider x − 3 = 8. Add 3 to both sides: x − 3 + 3 = 8 + 3, so x = 11. Always write the operation you are doing on both sides to keep the equation balanced. This step-by-step recording is important for developing good mathematical habits.

当方程涉及减法时，你在两边加上同一个数。考虑 x − 3 = 8。两边同时加 3：x − 3 + 3 = 8 + 3，所以 x = 11。一定要写下你在两边所做的运算，以保持方程平衡。这种逐步记录的方式对于培养良好的数学习惯非常重要。

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4. One-Step Equations: Multiplication & Division | 一步方程：乘法与除法

If the variable is multiplied by a number, you divide both sides by that number to get the variable alone. For example, 4x = 20. Divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, giving x = 5. Remember that 4x means 4 times x. Dividing by 4 undoes the multiplication, leaving just x.

如果变量乘以一个数，你就把两边同时除以那个数，以便单独得到变量。例如，4x = 20。两边同时除以 4：4x ÷ 4 = 20 ÷ 4，得到 x = 5。记住，4x 表示 4 乘以 x。除以 4 会抵消乘法，只剩下 x。

If the variable is divided by a number, you multiply both sides by that number. Solve x ÷ 6 = 3. Multiply both sides by 6: (x ÷ 6) × 6 = 3 × 6, so x = 18. Again, the operation cancels out on the left side, freeing x.

如果变量除以一个数，你就把两边同时乘以那个数。解 x ÷ 6 = 3。两边同时乘以 6：(x ÷ 6) × 6 = 3 × 6，所以 x = 18。同样地，运算在左边抵消，释放了 x。

Pay attention to the notation: sometimes equations are written as x/4 = 5. This means exactly the same as x ÷ 4 = 5. Multiply both sides by 4 to find x = 20.

注意符号：有时方程写作 x/4 = 5。这和 x ÷ 4 = 5 的意思完全相同。两边乘以 4，求得 x = 20。

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5. Two-Step Equations | 两步方程

Two-step equations involve two operations. The classic form is something like 2x + 3 = 11. To solve, you need to undo the operations in reverse order – normally the addition or subtraction first, then the multiplication or division. So for 2x + 3 = 11, first subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

两步方程涉及两种运算。经典的形式类似于 2x + 3 = 11。要解这个方程，你需要按相反的顺序撤销运算——通常是先处理加法或减法，再处理乘法或除法。所以对于 2x + 3 = 11，首先两边减去 3：2x = 8。然后除以 2：x = 4。

Another example: 5x − 7 = 13. Add 7 to both sides to get 5x = 20. Then divide by 5 to get x = 4. Always do the additive step before the multiplicative step. This order ensures you isolate the term containing x first, then find x itself. Some students make the mistake of dividing first, which complicates the equation unnecessarily.

另一个例子：5x − 7 = 13。两边加 7 得到 5x = 20。然后除以 5 得到 x = 4。总是先做加减法步骤，再做乘除法步骤。这个顺序确保你先分离出含有 x 的项，然后再求出 x 本身。有些学生会犯先除的错误，这样会把方程不必要地复杂化。

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6. Equations with Brackets | 带括号的方程

When an equation contains brackets, such as 3(x + 2) = 18, you generally have two approaches. The first is to expand the brackets: multiply each term inside the bracket by the number outside. Here, 3 times x gives 3x, and 3 times 2 gives 6, so the equation becomes 3x + 6 = 18. Then solve as a two-step equation: subtract 6 (3x = 12), then divide by 3 (x = 4).

当方程含有括号时，比如 3(x + 2) = 18，你通常有两种方法。第一种是展开括号：用外面的数乘以括号内的每一项。这里，3 乘以 x 得 3x，3 乘以 2 得 6，所以方程变为 3x + 6 = 18。然后按两步方程来解：减 6 得 3x = 12，再除以 3 得 x = 4。

The second approach is to divide both sides by the number in front of the bracket first, if that number divides evenly into the right side. For 3(x + 2) = 18, dividing both sides by 3 gives x + 2 = 6. Then subtract 2 to get x = 4. This method is often quicker, but only works neatly when the division results in an integer. Choose whichever method you find most comfortable.

第二种方法是，如果括号前的数可以整除右边，先对两边除以那个数。对于 3(x + 2) = 18，两边除以 3 得到 x + 2 = 6。然后减去 2 得到 x = 4。这个方法通常更快，但只有在除法结果为整数时才方便。选择你最习惯的方法就行。

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7. Equations with Unknowns on Both Sides | 两边都有未知数的方程

In equations like 5x + 2 = 3x + 10, the variable appears on both sides. The strategy is to eliminate the smaller x term by subtracting it from both sides. Subtract 3x from both sides: 5x − 3x + 2 = 3x − 3x + 10, which simplifies to 2x + 2 = 10. Then subtract 2 and divide by 2 to get x = 4.

在像 5x + 2 = 3x + 10 这样的方程中，变量出现在两边。策略是通过从两边减去较小的 x 项来消除它。两边减去 3x：5x − 3x + 2 = 3x − 3x + 10，化简为 2x + 2 = 10。然后减去 2，再除以 2，得到 x = 4。

You could also collect the x terms on the right side, but it is usually easier to keep the variable on the side where the coefficient is larger to avoid negative coefficients. For instance, 7x − 4 = 2x + 11. Subtract 2x from both sides to get 5x − 4 = 11. Then add 4 to both sides: 5x = 15, so x = 3.

你也可以把 x 项收集到右边，但通常把变量放在系数较大的一边更容易，以避免出现负系数。例如，7x − 4 = 2x + 11。两边减去 2x 得到 5x − 4 = 11。然后两边加 4：5x = 15，所以 x = 3。

Sometimes the variable terms cancel out completely, leaving a false statement like 0 = 5, or a true statement like 0 = 0. If you end up with a false statement, the equation has no solution. If you get a true statement, the equation is true for all values of x – this is called an identity.

有时候变量项完全抵消，留下一个像 0 = 5 这样的错误陈述，或者像 0 = 0 这样的真实陈述。如果得到一个错误陈述，那么方程无解。如果得到一个真实陈述，那么方程对所有 x 值都成立——这称为恒等式。

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8. Solving Equations with Fractions | 解含有分数的方程

Equations with fractions can look scary, but they become much simpler if you eliminate the fractions early on. Multiply every term on both sides of the equation by the lowest common denominator (LCD) of all the fractions involved. For example, solve x/3 + 1 = 5. Multiply every term by 3: (x/3)×3 + 1×3 = 5×3, which gives x + 3 = 15. Then subtract 3 to find x = 12.

含有分数的方程可能看起来很吓人，但如果你尽早消掉分数，它们就会变得简单得多。将方程两边的每一项都乘以所有涉及分数的最小公分母 (LCD)。例如，解 x/3 + 1 = 5。把每一项都乘以 3：(x/3)×3 + 1×3 = 5×3，得到 x + 3 = 15。然后减去 3 得到 x = 12。

When there are different denominators, find the LCD first. Solve x/2 + x/3 = 5. The LCD of 2 and 3 is 6. Multiply every term by 6: (x/2)×6 + (x/3)×6 = 5×6. This simplifies to 3x + 2x = 30, so 5x = 30 and x = 6. This method eliminates fractions at the start, making further steps much easier.

当分母不同时，首先要找到最小公分母。解 x/2 + x/3 = 5。2 和 3 的最小公分母是 6。每一项乘以 6：(x/2)×6 + (x/3)×6 = 5×6。化简为 3x + 2x = 30，所以 5x = 30，x = 6。这个方法一开始就消掉了分数，使后面的步骤容易得多。

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

Many real-life problems require you to write your own equation before solving. Read the problem carefully and identify the unknown – often it is what the question asks you to find. Represent it with a letter, usually x. Then translate the words into mathematical operations. For example: ‘I think of a number, double it and add 5. The result is 23. What is the number?’ Let the number be x. Doubling it gives 2x, adding 5 gives 2x + 5. Set that equal to 23: 2x + 5 = 23. Solve to get x = 9.

许多现实生活中的问题要求你先写下自己的方程，然后再解。仔细阅读问题，找出未知数——通常就是题目要你求的东西。用一个字母表示它，通常是 x。然后把文字翻译成数学运算。例如：“我想一个数，把它翻倍再加 5。结果是 23。这个数是多少？”设这个数为 x。翻倍得到 2x，加 5 得到 2x + 5。让它等于 23：2x + 5 = 23。解得 x = 9。

You can also have problems with two quantities. For instance, ‘The perimeter of a rectangle is 30 cm. The length is twice the width. Find the length and width.’ Let the width be x cm, then the length = 2x cm. Perimeter = 2(length + width) = 2(2x + x) = 2(3x) = 6x. Set 6x = 30, so x = 5. Width = 5 cm, length = 10 cm. Always define your variable clearly before forming the equation.

你还可能遇到有两个量的问题。例如，“一个长方形的周长是 30 厘米。长是宽的两倍。求长和宽。”设宽为 x 厘米，那么长 = 2x 厘米。周长 = 2(长 + 宽) = 2(2x + x) = 2(3x) = 6x。令 6x = 30，所以 x = 5。宽 = 5 厘米，长 = 10 厘米。在建立方程之前，一定要清楚地定义你的变量。

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10. Checking Your Solutions | 检验解

After solving, always substitute your answer back into the original equation to check it. For example, if you solved 4x − 3 = 13 and got x = 4, check by calculating the left side: 4(4) − 3 = 16 − 3 = 13. This equals the right side, so the solution is correct. Checking helps catch arithmetic mistakes and ensures you didn’t accidentally perform an incorrect operation.

解完方程后，总是要把你的答案代回原方程进行检验。例如，如果你解了 4x − 3 = 13 并得到 x = 4，通过计算左边来检验：4(4) − 3 = 16 − 3 = 13。这等于右边，所以解是正确的。检验有助于发现计算错误，并确保你没有意外地进行错误的运算。

Checking is especially important in exams where you might be asked to ‘verify your answer’. Even if not asked, spending a few seconds on checking can boost your accuracy. If the check fails, retrace your steps to find where the error occurred. It’s a good habit to develop early in your algebra work.

检验在考试中尤其重要，考题可能要求你“验证你的答案”。即使没有要求，花几秒钟检验一下也能提高你的准确率。如果检验失败，倒推你的步骤，找到错误发生的地方。这是你在代数学习初期就应该养成的好习惯。

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11. Common Mistakes to Avoid | 常见错误避免

Many students lose marks by making the same common errors. One is forgetting to do the operation on both sides. For instance, solving x + 3 = 7, some students write x = 7 − 3, which is correct, but they might skip showing the −3 on both sides. Always show your working, and ensure balance is maintained. Another mistake is mishandling negative signs. In 4 − 2x = 10, after subtracting 4 you get −2x = 6. Some people forget the negative sign and end up with x = −3 instead of dividing properly: −2x ÷ (−2) = 6 ÷ (−2) gives x = −3.

许多学生因为犯同样的常见错误而丢分。一个是忘记在两边同时做运算。例如，解 x + 3 = 7，有些学生写 x = 7 − 3，这是对的，但他们可能跳过了在两边标明 −3 的步骤。一定要展示你的计算过程，并确保保持平衡。另一个错误是负号处理不当。在 4 − 2x = 10 中，减去 4 后得到 −2x = 6。有些人会忘记负号，最终错误地得到 x = 3，而正确的做法是除以 −2：−2x ÷ (−2) = 6 ÷ (−2)，得 x = −3。

A third common error is incorrect expansion of brackets: 3(x + 2) becomes 3x + 2, forgetting to multiply the 2 by 3. Always multiply each term inside the bracket by the factor outside. Also, when dealing with equations like 2(x + 3) = x + 8, some students subtract x from the left but not from the bracket term. Always treat the equation as a whole. Care and practice are the best remedies for these errors.

第三个常见错误是括号展开不正确：3(x + 2) 变成了 3x + 2，忘了把 2 也乘以 3。一定要把括号外的因数乘以括号内的每一项。另外，在解像 2(x + 3) = x + 8 这样的方程时，有些学生只从左边减去 x，却忽略了括号项。一定要把方程看作一个整体。细心和练习是纠正这些错误的最好方法。

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12. Practice Questions & Summary | 练习题与总结

Let’s try a few practice questions to reinforce your skills. Solve the following equations: (1) 5y − 7 = 18; (2) 4(a + 3) = 28; (3) 3m + 2 = m + 10; (4) (x/4) + 2 = 9. Work through each step, then check your answers. The solutions are: (1) y = 5, (2) a = 4, (3) m = 4, (4) x = 28. If you got them all right, well done! If not, revisit the relevant section.

让我们试做几道练习题来巩固你的技能。解下列方程：(1) 5y − 7 = 18；(2) 4(a + 3) = 28；(3) 3m + 2 = m + 10；(4) (x/4) + 2 = 9。逐步求出解，然后检验你的答案。答案是：(1) y = 5，(2) a = 4，(3) m = 4，(4) x = 28。如果你全做对了，太棒了！如果没有，重新看看相关的章节。

In summary, solving linear equations is all about keeping the balance while isolating the variable. Start with the simplest operation, work systematically, and always check your answer. As you progress, you’ll encounter more complex equations, but the core principles remain the same. With practice, equation solving becomes second nature. Good luck with your Key Stage 3 Mathematics!

总之，解线性方程就是在隔离变量的同时保持平衡。从最简单的运算开始，有条理地进行，并且始终检验你的答案。随着你的进步，你会遇到更复杂的方程，但核心原则是不变的。通过练习，解方程会变成你的第二天性。祝你在 Key Stage 3 数学中好运！

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