Solving Linear Equations for KS3 | KS3 线性方程求解技巧

📚 Solving Linear Equations for KS3 | KS3 线性方程求解技巧

A linear equation is one of the most important ideas in KS3 mathematics. It appears in almost every Cambridge checkpoint test, and the skill of solving linear equations confidently will help you in algebra, graphs, word problems and even later topics such as simultaneous equations. In this article, you will learn how to solve linear equations step by step using clear methods, and you will see how to avoid the common mistakes that many students make.

线性方程是 KS3 数学中最重要的内容之一。它几乎出现在每一次 Cambridge checkpoint 测试中，熟练掌握解线性方程的技巧将帮助你在代数、图像、应用题甚至日后的联立方程等主题中更加自信。本文将通过清晰的方法逐步教你如何解线性方程，并帮助你避免许多学生容易犯的常见错误。

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

A linear equation is an equation where the unknown value, usually written as x or y, is only raised to the power of 1. This means there are no terms like x², x³ or 1/x in a linear equation. For example, x + 3 = 7 and 2x – 5 = 11 are both linear equations. The word ‘linear’ comes from the fact that the graph of such an equation is always a straight line.

线性方程是指未知数（通常写作 x 或 y）只以一次方出现的方程。也就是说，线性方程中不会出现 x²、x³ 或 1/x 这样的项。例如 x + 3 = 7 和 2x – 5 = 11 都是线性方程。’线性’ 一词来源于这类方程的图像总是形成一条直线。

In KS3, you normally solve linear equations with one unknown. The goal is simple: find the value of the unknown that makes the equation true. The equation works like a balance, so whatever you do to one side, you must do to the other side. This is the key principle behind every method in this article.

在 KS3 阶段，你通常只解一个未知数的线性方程。目标很简单：找出使等式成立的未知数的值。方程就像一个天平，所以你对等式一边做什么运算，另一边也必须做同样的运算。这是本文所有方法背后的关键原则。

2. The Balance Method | 天平法

The balance method is the most reliable way to solve linear equations. Imagine the equation as a balance scale. The left side and the right side are perfectly balanced. If you add, subtract, multiply or divide by the same number on both sides, the balance is not broken. This allows you to rearrange the equation step by step until the unknown is alone on one side.

天平法是解线性方程最可靠的方法。把方程想象成一台天平。左边和右边完全平衡。如果你在两边同时加、减、乘或除以同一个数，天平不会失衡。这样你就可以一步一步地重新整理方程，直到未知数单独出现在一边。

For example, if x + 4 = 9, you can subtract 4 from both sides to undo the ‘+ 4’. This gives x + 4 – 4 = 9 – 4, which simplifies to x = 5. The key idea is to always do the opposite operation: addition is undone by subtraction, subtraction is undone by addition, multiplication is undone by division, and division is undone by multiplication.

例如，如果 x + 4 = 9，你可以在两边同时减去 4，以抵消 ‘+ 4’。这样得到 x + 4 – 4 = 9 – 4，化简后 x = 5。核心思想是始终进行相反的运算：加法用减法抵消，减法用加法抵消，乘法用除法抵消，除法用乘法抵消。

Many students find it helpful to write each step on a new line. This makes your working clear and reduces careless errors. Always check that the operation is applied to the whole side, not just one term.

许多学生发现每行只写一步会很有帮助。这样可以让解题过程更清晰，减少粗心错误。还要注意运算必须作用于整个一边，而不仅仅是其中的一项。

3. Solving One-Step Equations | 解一步方程

A one-step equation needs only one operation to find the solution. These are the building blocks for all harder equations. For addition and subtraction equations, you simply add or subtract the same number from both sides. For multiplication and division equations, you multiply or divide both sides by the same non-zero number.

一步方程只需要一次运算就能求出解。它们是所有更复杂方程的基础。对于加法和减法方程，你只需在两边同时加或减同一个数。对于乘法和除法方程，你需要在两边同时乘以或除以同一个非零数。

Example 1: x + 7 = 12. Subtract 7 from both sides: x = 5.

例 1：x + 7 = 12。两边同时减去 7：x = 5。
Example 2: x – 3 = 8. Add 3 to both sides: x = 11.

例 2：x – 3 = 8。两边同时加上 3：x = 11。
Example 3: 4x = 20. Divide both sides by 4: x = 5.

例 3：4x = 20。两边同时除以 4：x = 5。
Example 4: x ÷ 5 = 6. Multiply both sides by 5: x = 30.

例 4：x ÷ 5 = 6。两边同时乘以 5：x = 30。

Always remember that division and multiplication are opposite operations. If the unknown has been divided by a number, multiply by that number to undo it. If the unknown has been multiplied by a number, divide by that number to undo it.

始终记住除法和乘法是相反的运算。如果未知数被一个数除，就乘以这个数来抵消它。如果未知数被一个数乘，就除以这个数来抵消它。

4. Solving Two-Step Equations | 解两步方程

Two-step equations require two operations to isolate the unknown. A typical example is 2x + 3 = 11. You must first undo the addition or subtraction, and then undo the multiplication or division. The order is usually the reverse of the order of operations, also called BIDMAS or BODMAS.

两步方程需要两次运算才能将未知数分离出来。典型例子是 2x + 3 = 11。你必须先抵消加法或减法，然后再抵消乘法或除法。顺序通常与运算顺序（也叫 BIDMAS 或 BODMAS）相反。

2x + 3 = 11

Subtract 3 from both sides to undo the +3: 2x = 8. Then divide both sides by 2 to undo the ×2: x = 4. So the solution is x = 4.

两边同时减去 3 以抵消 +3：2x = 8。然后两边同时除以 2 以抵消 ×2：x = 4。所以解是 x = 4。

Another example is 5x – 7 = 18. First add 7 to both sides: 5x = 25. Then divide both sides by 5: x = 5. Notice how the addition or subtraction is always dealt with before the multiplication or division when you are trying to get x alone.

另一个例子是 5x – 7 = 18。首先两边同时加上 7：5x = 25。然后两边同时除以 5：x = 5。注意当你尝试将 x 分离出来时，总是先处理加法或减法，再处理乘法或除法。

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, such as 3(x + 2) = 15, the brackets mean multiplication. There are two common methods you can use. The first method is to expand the brackets using the distributive law. The second method is to divide both sides by the number in front of the bracket first.

当方程含有括号时，例如 3(x + 2) = 15，括号表示乘法。你可以使用两种常见方法。第一种方法是使用分配律展开括号。第二种方法是先在两边同时除以括号前的数字。

Using the expansion method, 3(x + 2) becomes 3x + 6. So the equation becomes 3x + 6 = 15. Subtract 6 from both sides: 3x = 9. Then divide both sides by 3: x = 3.

使用展开法，3(x + 2) 变为 3x + 6。所以方程变为 3x + 6 = 15。两边同时减去 6：3x = 9。然后两边同时除以 3：x = 3。

Using the division method, divide both sides by 3 first: x + 2 = 5. Then subtract 2 from both sides: x = 3. This method is often quicker when the number in front of the bracket divides evenly into the other side.

使用除法方法，先两边同时除以 3：x + 2 = 5。然后两边同时减去 2：x = 3。当括号前的数字能整除等式另一边时，这种方法通常更快。

Be careful with negative signs and subtraction inside brackets. For example, 2(x – 4) expands to 2x – 8, not 2x – 4. Always multiply every term inside the bracket by the number outside the bracket.

要注意括号内的负号和减法。例如，2(x – 4) 展开为 2x – 8，而不是 2x – 4。始终将括号内的每一项都乘以括号外的数字。

6. Equations with Unknowns on Both Sides | 未知数在等号两边

Some linear equations have the unknown on both sides of the equals sign. For example, 7x – 3 = 4x + 9. The first goal is to collect all the x terms on one side and all the number terms on the other side. You can do this by adding or subtracting the smaller x term from both sides.

有些线性方程的未知数出现在等号两边。例如，7x – 3 = 4x + 9。第一个目标是将所有含 x 的项移到一边，将所有数字项移到另一边。你可以通过在两边同时加上或减去较小的 x 项来实现。

7x – 3 = 4x + 9

Subtract 4x from both sides: 3x – 3 = 9. Then add 3 to both sides: 3x = 12. Finally divide both sides by 3: x = 4.

两边同时减去 4x：3x – 3 = 9。然后两边同时加上 3：3x = 12。最后两边同时除以 3：x = 4。

Always move the smaller x term so the coefficient of x stays positive. This reduces mistakes. You can move the larger x term too, but you will end up with a negative coefficient, which requires extra care.

始终移动较小的 x 项，这样 x 的系数保持为正。这可以减少错误。你也可以移动较大的 x 项，但最终会得到负系数，需要额外小心。

Once all the x terms are on one side, solve the remaining two-step equation as you did before. Check your answer by substituting it back into the original equation.

当所有含 x 的项都集中到一边后，像之前一样解剩下的两步方程。将答案代入原方程进行检验。

7. Equations with Fractions | 含分数的方程

Equations with fractions can look difficult, but they become much easier if you multiply every term by the lowest common multiple (LCM) of the denominators. This removes the fractions and turns the equation into a simpler linear equation. Always multiply every term on both sides.

含分数的方程看起来可能很难，但如果你将每一项都乘以所有分母的最小公倍数（LCM），问题就会变得简单得多。这样可以去掉分数，将方程转化为更简单的线性方程。一定要把两边每一项都乘以这个倍数。

x/3 + 2 = 5

Multiply every term by 3: x + 6 = 15. Then subtract 6 from both sides: x = 9.

将每一项乘以 3：x + 6 = 15。然后两边同时减去 6：x = 9。

For equations with more than one fraction, such as x/2 + x/3 = 5, the LCM of 2 and 3 is 6. Multiply every term by 6: 3x + 2x = 30. This simplifies to 5x = 30, so x = 6.

对于含有多个分数的方程，例如 x/2 + x/3 = 5，2 和 3 的最小公倍数是 6。将每一项乘以 6：3x + 2x = 30。化简为 5x = 30，所以 x = 6。

Be careful when a fraction has a negative sign in front of it. The negative sign applies to the whole fraction, not just the numerator or denominator. Use brackets if needed to avoid sign errors.

当分数前有负号时要特别小心。负号适用于整个分数，而不仅仅是分子或分母。必要时使用括号以避免符号错误。

8. Checking Your Solution | 检验解

After you have found a value for x, you should always check it by substituting it back into the original equation. You can do this mentally or on paper. For example, if you solved 2x + 3 = 11 and found x = 4, check by replacing x with 4: 2 × 4 + 3 = 8 + 3 = 11. Since the left side equals the right side, the solution is correct.

求出 x 的值后，你应该始终将它代入原方程进行检验。你可以在心里或纸上完成。例如，如果你解出 2x + 3 = 11 且得到 x = 4，用 4 替换 x 来检验：2 × 4 + 3 = 8 + 3 = 11。由于左边等于右边，解是正确的。

Checking is especially important in tests. It only takes a few seconds and can catch small mistakes in signs, arithmetic or rearranging. If the two sides do not match, trace back through your working to find where the error occurred.

在考试中检验尤为重要。它只需要几秒钟，就能发现符号、算术或移项中的小错误。如果两边不相等，就回头检查你的解题过程，找出错误所在。

You can also use a simple table to organise your checking steps. Write the left side, the right side and the result after substitution. This helps you show clear working and earns you valuable method marks in checkpoint papers.

你也可以用一个简单的表格来整理检验步骤。写出左边、右边和代入后的结果。这可以帮助你展示清晰的解题过程，并在 checkpoint 试卷中获得宝贵的方法分。

Left side	Right side	Equal?
2 × 4 + 3 = 11	11	Yes

9. Common Mistakes to Avoid | 常见错误

There are several mistakes that come up again and again when KS3 students solve linear equations. Knowing these mistakes will help you avoid them. The first mistake is forgetting to do the operation to both sides. If you subtract 3 from one side, you must subtract 3 from the other side as well.

KS3 学生在解线性方程时经常会犯一些相同的错误。了解这些错误将帮助你避免它们。第一个错误是忘记对两边同时进行运算。如果你在一边减去 3，那么另一边也必须减去 3。

Mistake 1: Adding or subtracting only one side. Example: x + 4 = 10 becomes x = 10 – 4? No, write x = 10 – 4 by subtracting 4 from both sides.

错误 1：只在一边加或减。例如 x + 4 = 10 变成 x = 10 – 4？错，应该是两边同时减去 4，写出 x = 10 – 4。
Mistake 2: Forgetting to multiply every term inside a bracket. Example: 3(x + 2) = 3x + 2 is wrong; it should be 3x + 6.

错误 2：忘记乘以括号内的每一项。例如 3(x + 2) = 3x + 2 是错误的；应该是 3x + 6。
Mistake 3: Losing the negative sign. Example: -2x = 10 gives x = -5, not x = 5.

错误 3：丢失负号。例如 -2x = 10 得出 x = -5，而不是 x = 5。
Mistake 4: Combining unlike terms. You cannot add 3x and 4 to get 7x.

错误 4：合并不同类项。你不能把 3x 和 4 相加得到 7x。

Another common error is dividing by a negative number and forgetting to change the sign. For instance, if -3x = 12, divide both sides by -3 to get x = -4. Always pay attention to the sign of the coefficient.

另一个常见错误是除以负数时忘记变号。例如，如果 -3x = 12，两边同时除以 -3 得到 x = -4。始终注意系数的符号。

Finally, many students try to solve equations by ‘moving’ terms without understanding the operation. It is better to write the operation you are performing on both sides at each step. This makes your working clear and helps you avoid sign errors.

最后，许多学生试图通过 ‘移项’ 来解方程而不理解运算过程。更好的做法是每一步都写出你在两边执行的运算。这样可以让你的解题过程更清晰，并帮助你避免符号错误。

10. Practice Questions | 练习题

Use the methods from this article to solve the following equations. Try to show each step clearly and check your answers by substituting back into the original equation. These questions are typical of the types found in KS3 Cambridge checkpoint papers.

使用本文中的方法解下列方程。尽量清楚地展示每一步，并通过代入原方程来检验答案。这些题目是 KS3 Cambridge checkpoint 试卷中常见的类型。

Solve x – 9 = 4.

解方程 x – 9 = 4。
Solve 6x = 48.

解方程 6x = 48。
Solve 3x + 5 = 20.

解方程 3x + 5 = 20。
Solve 4(x – 2) = 24.

解方程 4(x – 2) = 24。
Solve 5x – 7 = 3x + 9.

解方程 5x – 7 = 3x + 9。
Solve x/4 + 3 = 7.

解方程 x/4 + 3 = 7。

For each answer, write a checking line similar to the table in Section 8. This will help you build confidence and accuracy. If you make a mistake, find the exact step where the error occurred before moving on to the next question.

对于每个答案，写出类似第 8 节表格的检验过程。这可以帮助你建立信心并提高准确性。如果你犯了错误，在继续做下一题之前，找出错误发生的具体步骤。

Once you can solve these equations confidently, you will be well prepared for harder topics such as inequalities, straight-line graphs and simultaneous equations in later years. Keep practising and always show your working clearly.

当你能够自信地解出这些方程后，你就为日后更难的课题做好了准备，例如不等式、直线图像和联立方程。坚持练习，并始终清晰地展示你的解题过程。

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Solving Linear Equations for KS3 | KS3 线性方程求解技巧