Solving Equations: Cambridge Checkpoint p.138 Revision | 解方程：剑桥Checkpoint p.138 复习

📚 Solving Equations: Cambridge Checkpoint p.138 Revision | 解方程：剑桥Checkpoint p.138 复习

Welcome to this revision guide on solving equations, designed to match the Cambridge Lower Secondary Mathematics syllabus and the practice exercises found on page 138. You will learn how to form, solve and check linear equations step by step.

欢迎阅读这份解方程复习指南，它贴合剑桥初中数学课程以及第138页的练习题。你将学会如何一步步建立、求解并检验线性方程。

1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal. It always contains an equals sign ‘=’.

方程是表明两个表达式相等的数学陈述。它总是包含一个等号 ‘=’。

Think of an equation as a balanced scale. Whatever you do to one side, you must also do to the other side to keep the balance.

把方程想像成一个平衡的天平。无论你对一侧做什么，都必须对另一侧做同样的事来保持平衡。

The unknown value in an equation is often represented by a letter, such as x, y or n. Your task is to find what that letter stands for.

方程中的未知数通常用一个字母表示，比如 x、y 或 n。你的任务就是找出这个字母代表什么数。

2. Solving Simple One-Step Equations | 解简单一步方程

A one-step equation can be solved by performing a single inverse operation. For example, if the equation is x + 5 = 12, subtract 5 from both sides.

一步方程可以通过一次逆运算来求解。例如，如果方程是 x + 5 = 12，两边同时减去 5。

x + 5 = 12 → x = 12 – 5 → x = 7

If the equation involves subtraction, such as y – 3 = 9, add 3 to both sides to get y = 12.

如果方程涉及减法，例如 y – 3 = 9，则两边加 3，得到 y = 12。

For multiplication or division, use the opposite operation. In 4z = 20, divide both sides by 4 to find z = 5. In p/6 = 3, multiply both sides by 6 to get p = 18.

对于乘法或除法，使用相反的运算。在 4z = 20 中，两边除以 4，得到 z = 5。在 p/6 = 3 中，两边乘以 6，得到 p = 18。

3. Two-Step Equations | 两步方程

Two-step equations require two inverse operations. Always undo addition or subtraction first, then multiplication or division.

两步方程需要两步逆运算。始终先去掉加法或减法，再去掉乘法或除法。

Take the equation 2x + 3 = 11. Start by subtracting 3 from both sides to get 2x = 8. Then divide both sides by 2 to obtain x = 4.

以方程 2x + 3 = 11 为例。先从两边减去 3，得到 2x = 8。然后两边除以 2，得到 x = 4。

Step	Working	中文步骤
1	2x + 3 = 11	原方程
2	Subtract 3: 2x = 8	两边减 3
3	Divide by 2: x = 4	两边除以 2

If the equation has subtraction, like 5y – 7 = 13, add 7 first to get 5y = 20, then divide by 5 to find y = 4.

如果方程带有减法，比如 5y – 7 = 13，先加 7 得到 5y = 20，再除以 5 求出 y = 4。

4. Equations with Brackets | 带括号的方程

When an equation contains brackets, expand them first using the distributive law, then solve as usual.

当方程含有括号时，先用分配律展开括号，然后像往常一样求解。

For instance, solve 3(x + 2) = 18. Expand to 3x + 6 = 18, then subtract 6 to get 3x = 12, and finally x = 4.

例如，解 3(x + 2) = 18。展开得 3x + 6 = 18，然后减 6 得到 3x = 12，最终 x = 4。

Another approach is to divide both sides by the coefficient outside the bracket first, but expanding is often simpler for KS3 learners.

另一种方法是先除以括号外的系数，但对于KS3学生来说，展开通常更简单。

Be careful with negative signs: –2(4 – y) = 6 becomes –8 + 2y = 6. Then add 8 to get 2y = 14, so y = 7.

注意负号：–2(4 – y) = 6 变为 –8 + 2y = 6。然后加 8 得到 2y = 14，因此 y = 7。

5. Unknowns on Both Sides | 两边有未知数的方程

When the unknown appears on both sides of the equation, collect all variable terms on one side and numbers on the other.

当未知数出现在等号两边时，把所有含变量的项移到一边，把数字移到另一边。

Consider 5x + 2 = 3x + 10. Subtract 3x from both sides to get 2x + 2 = 10. Then subtract 2 and divide by 2, giving x = 4.

考虑方程 5x + 2 = 3x + 10。两边减去 3x 得到 2x + 2 = 10。然后减 2 再除以 2，得到 x = 4。

Always aim to have a positive coefficient for the unknown. If you end up with –x = 5, multiply both sides by –1 to get x = –5.

始终要让未知数的系数为正。如果你得出 –x = 5，两边乘以 –1 得到 x = –5。

Move the smaller variable term to avoid negative coefficients early on. This reduces sign errors.

移动较小的变量项可以早些避免负系数，从而减少符号错误。

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

Read the problem carefully, identify the unknown, and express the given information as an equation.

仔细阅读题目，确定未知数，并用方程表达所给的信息。

If a question says “three times a number plus 5 equals 20”, let the number be n. The equation is 3n + 5 = 20.

如果题目说“一个数的 3 倍加 5 等于 20”，设这个数为 n，则方程为 3n + 5 = 20。

Use keywords to decide the operations: ‘sum’ means addition, ‘difference’ means subtraction, ‘product’ means multiplication, and ‘quotient’ means division.

利用关键词确定运算：’和’是加，’差’是减，’积’是乘，’商’是除。

Write an equation showing the balance, then solve it step by step. Check that your answer makes sense in the original context.

写出表示等量关系的方程，然后逐步求解。检查你的答案在原情境中是否合理。

7. Checking Your Solution | 检验解

Substitute your found value back into the original equation to verify both sides are equal.

把求出的值代回原方程，检验两边是否相等。

For x = 4 in 2x + 3 = 11, left side = 2(4) + 3 = 8 + 3 = 11, which equals the right side. So the solution is correct.

对于 x = 4 代入 2x + 3 = 11，左边 = 2(4) + 3 = 8 + 3 = 11，等于右边。因此解正确。

Always perform this check, especially in exams. It helps catch careless mistakes and boosts your confidence.

一定要进行检验，尤其是在考试中。这有助于发现粗心错误，并增强信心。

8. Common Mistakes to Avoid | 常见错误避免

Forgetting to perform the same operation on both sides is the most frequent error. Keep the balance.

最常见的错误是忘记在等号两边进行相同运算。务必保持平衡。

Misapplying signs, such as dropping a negative when distributing brackets, can lead to wrong answers.

符号使用不当，例如展开括号时漏掉负号，会导致错误答案。

Another mistake is solving in the wrong order: always reverse the order of operations (BIDMAS backwards) – undo addition/subtraction before multiplication/division.

另一个错误是运算顺序不对：始终逆用运算顺序（BIDMAS倒过来）– 先去掉加/减，再去掉乘/除。

Not checking the solution means you might miss an error. Always substitute back.

不检验解意味着你可能错过错误。一定要代回检验。

9. Practice Questions (p.138 Style) | 练习（138页风格）

Try these equations, which are similar to those on page 138. Solve each one step by step and check your answers.

试试这些与138页相似的方程。逐步求解每一个，并检验你的答案。

Solve x + 9 = 15

解 x + 9 = 15
Solve 4m – 3 = 17

解 4m – 3 = 17
Solve 2(a + 5) = 26

解 2(a + 5) = 26
Solve 7y – 4 = 2y + 11

解 7y – 4 = 2y + 11
Form and solve an equation: ‘Four more than twice a number is 20.’

列方程并求解：’一个数的两倍加 4 等于 20。’

After solving, check each solution by substituting it back into the original equation. For the word problem, define your variable clearly.

求解后，将每个解代回原方程检验。对于文字题，要清楚地定义变量。

10. Summary and Key Points | 总结与重点

Equations show that two expressions are equal. Solving means finding the value of the unknown that balances the equation.

方程表示两个表达式相等。求解意味着找出使方程平衡的未知数值。

Use inverse operations: addition undoes subtraction, multiplication undoes division, and vice versa. Always maintain balance.

使用逆运算：加法抵消减法，乘法抵消除法，反之亦然。始终保持平衡。

When brackets appear, expand first. When unknowns are on both sides, collect like terms. And never forget to check your answer.

出现括号时，先展开。未知数在等号两边时，合并同类项。永远不要忘记检验答案。

These skills match the core Cambridge Checkpoint objectives and the exercises on page 138. With practice, you will solve equations confidently.

这些技能符合剑桥Checkpoint的核心目标以及第138页的练习。通过练习，你将自信地解方程。

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Solving Equations: Cambridge Checkpoint p.138 Revision | 解方程：剑桥Checkpoint p.138 复习