📚 Student's Edition: IGCSE Mathematics Lesson 15-6 - Solving Linear Equations | 学生版：IGCSE 数学 第15-6课 - 解一元一次方程
nstants and a ≠ 0. The goal is to find the value of the unknown variable that makes the equation true. In this lesson, we will explore several reliable methods for solving linear equations, from basic one-step equations to those involving brackets, fractions and variables on both sides.
一元一次方程是 IGCSE 数学中最基础的主题之一。一个含一个未知数的一元一次方程可以写成 ax + b = c 的形式，其中 a、b、c 是常数且 a ≠ 0。我们的目标是求出使方程成立的未知数的值。在本课中，我们将探索解一元一次方程的几种可靠方法，从基本的一步方程到含有括号、分数以及两边都有变量的方程。
1. What Is a Linear Equation? | 什么是一元一次方程？
ment that contains an equals sign and exactly one variable raised to the power 1. The word 'linear' = 11 and 2(x + 3) = 14. The solution is the single value of x that balances the equation.
含一个未知数的一元一次方程是一个包含等号且只有一个变量的一次幂的代数表达式。“一元一次”意味着变量没有被平方、立方或放在根号下。例如 x + 5 = 12、3x − 4 = 11 和 2(x + 3) = 14。方程的解就是能使方程左右两边相等的唯一 x 值。
In IGCSE exams, you will be asked to solve such equations, often as part of a larger problem. You must be able to rearrange the equation systematically and check your answer by substitution.
在 IGCSE 考试中，你通常需要解这类方程，而且它常常是一个更大问题的一部分。你必须能够系统地整理方程，并通过代入检验你的答案。
2. The Golden Rule: Keep the Equation Balanced | 黄金法则：保持方程平衡
An equation is like a balance scale. Whatever you do to one side, you must do exactly the same to the other side. This is called the balance method. For example, 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 5, you must divide the right-hand side by 5 as well.
方程就像一架天平。你对一边做了什么，对另一边也必须做完全相同的操作。这种方法叫做平衡法。例如，如果你在左边加上 3，右边也必须加上 3。如果你把左边除以 5，右边也必须除以 5。
Always remember the two basic properties of equality: adding or subtracting the same number on both sides does not change the solution; multiplying or dividing both sides by the same non-zero number also keeps the equation true.
始终记住等式的两个基本性质：两边同时加上或减去同一个数不会改变方程的解；两边同时乘以或除以同一个非零数也能保持方程成立。
3. Solving Simple Linear Equations Using Inverse Operations | 用逆运算解简单的一元一次方程
se operations. The inverse of adding 7 is subtracting 7. So we subtract 7 from both sides:
要解一个简单的方程，例如 x + 7 = 15，我们使用逆运算。加 7 的逆运算是减 7。所以我们从两边同时减去 7：
x + 7 = 15
⇒ x = 15 − 7
⇒ x = 8
18, the inverse of multiplying by 3 is dividing by 3. Divide both sides by 3 to get x = 6.
对于乘法方程，例如 3x = 18，乘以 3 的逆运算是除以 3。两边同时除以 3，得到 x = 6。
, we get 8 + 7 = 15, which is true. This habit is especially important in exam situations.
始终通过将数值代回原方程来检验答案。对于 x = 8，我们得到 8 + 7 = 15，成立。这个习惯在考试中尤为重要。
4. Solving Equations with Brackets | 解含有括号的方程
law. For example, to solve 2(x + 3) = 14, we multiply both terms inside the bracket by 2:
当方程含有括号时，第一步通常是使用分配律将其展开。例如，要解 2(x + 3) = 14，我们将括号内的两项都乘以 2：
2(x + 3) = 14
⇒ 2x + 6 = 14
Now subtract 6 from both sides, then divide by 2:
现在两边同时减去 6，然后除以 2：
2x = 14 − 6 = 8
⇒ x = 8 ÷ 2 = 4
15 = 21. Add 15 to both sides to get 6x = 36. Then divide both sides by 6 to obtain x = 6.
另一个例子：解 3(2x − 5) = 21。展开得到 6x − 15 = 21。两边同时加上 15，得到 6x = 36。两边同时除以 6，得到 x = 6。
7, which simplifies to 11 − 2x = 7. Subtract 11 from both sides to get −2x = −4, so x = 2.
在 IGCSE 考题中，括号常与负号一起出现，例如 5 − 2(x − 3) = 7。要特别注意符号：−2(x − 3) = −2x + 6。然后方程变为 5 − 2x + 6 = 7，化简为 11 − 2x = 7。两边同时减去 11，得到 −2x = −4，所以 x = 2。
5. Equations with Variables on Both Sides | 两边都有变量的方程
get 3x + 3 = 12. Then subtract 3 from both sides to get 3x = 9. Divide by 3 to find x = 3.
有时未知数 x 会出现在方程的两边，例如 5x + 3 = 2x + 12。策略是将所有含变量的项移到一边，将所有常数项移到另一边。两边同时减去 2x，得到 3x + 3 = 12。两边同时减去 3，得到 3x = 9。除以 3，得到 x = 3。
btract 3x from both sides to get 4x − 4 = 8. Add 4 to both sides to get 4x = 12, so x = 3.
始终两边同时进行相同的运算以保持方程平衡。如果你先移除较大的变量项，就可以避免出现负系数。例如，在 7x − 4 = 3x + 8 中，两边同时减去 3x，得到 4x − 4 = 8。两边同时加上 4，得到 4x = 12，所以 x = 3。
6. Equations with Fractions | 含有分数的方程
tion (2x)/3 = 8. Multiply both sides by 3 to get 2x = 24. Then divide by 2 to find x = 12.
分数常常让方程看起来更难，但解题方法相似。目标是通过两边同时乘以最小公倍数来消去分母。考虑方程 (2x)/3 = 8。两边同时乘以 3，得到 2x = 24。然后除以 2，得到 x = 12。
f there are several fractions, find the LCM of all denominators and multiply every term by it.
对于像 (x + 1)/4 = 2 这样的方程，两边同时乘以 4，得到 x + 1 = 8。两边同时减去 1，得到 x = 7。如果有多个分数，先求出所有分母的最小公倍数，然后将每一项都乘以它。
sides to get 5x − 4 = 8. Add 4 to both sides to get 5x = 12. Divide by 5 to find x = 12/5.
一个常见的考题示例是 (3x)/2 − 1 = x/4 + 2。将每一项都乘以 4：4 × (3x)/2 − 4 × 1 = 4 × x/4 + 4 × 2，得到 6x − 4 = x + 8。两边同时减去 x，得到 5x − 4 = 8。两边同时加上 4，得到 5x = 12。除以 5，得到 x = 12/5。
7. Forming Equations from Word Problems | 根据文字题建立方程
Many IGCSE questions require you to translate a real-life situation into a linear equation. Let the unknown quantity be x, then write an equation based on the information given. Solve the equation and answer the question in context.
许多 IGCSE 题目要求你将实际情境转化为一元一次方程。设未知量为 x，然后根据所给信息写出方程。解方程并在具体情境中回答问题。
Example: The sum of three consecutive integers is 72. Find the integers. Let the first integer be x. Then the next two are x + 1 and x + 2. The equation is:
示例：三个连续整数的和是 72。求这三个整数。设第一个整数为 x。那么后两个分别是 x + 1 和 x + 2。方程为：
x + (x + 1) + (x + 2) = 72
⇒ 3x + 3 = 72
⇒ 3x = 69
⇒ x = 23
The three integers are 23, 24 and 25.
这三个整数分别是 23、24 和 25。
Another example: A rectangle has a length that is 4 cm longer than its width. The perimeter is 36 cm. Find the width. Let the width be x cm. Then the length is x + 4 cm. The perimeter equation is:
另一个例子：一个长方形的长比宽长 4 厘米。周长是 36 厘米。求宽。设宽为 x 厘米。则长为 x + 4 厘米。周长方程为：
2(x + x + 4) = 36
⇒ 2(2x + 4) = 36
⇒ 4x + 8 = 36
⇒ 4x = 28
⇒ x = 7
The width is 7 cm and the length is 11 cm.
宽为 7 厘米，长为 11 厘米。
8. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Students often lose marks in linear equations due to small but avoidable errors. Knowing these mistakes can help you avoid them in the exam.
学生在解一元一次方程时常常因为一些细小但可以避免的错误而丢分。了解这些错误可以帮助你在考试中避免它们。
- Sign errors when expanding brackets s : Always distribute the sign in front of the bracket. For example, −(x − 3) = −x + 3, not −x − 3.
- 括号展开时的符号错误：一定要将括号前的符号分配进去。例如，−(x − 3) = −x + 3，而不是 −x − 3。
- Forgetting to multiply every term: In (x + 2)/3 = 4, multiply both sides by 3 to get x + 2 = 12, not x/3 + 2 = 4.
- 忘记乘以每一项：在 (x + 2)/3 = 4 中，两边同时乘以 3，得到 x + 2 = 12，而不是 x/3 + 2 = 4。
- Incorrect inverse operation: To undo subtraction of 5, add 5. To undo division by 4, multiply by 4. Always use the opposite operation.
- 逆运算使用错误：要撤销减去 5，就加上 5。要撤销除以 4，就乘以 4。始终使用相反的运算。
- Not checking the answer: Substituting the solution back into the original equation takes only a few seconds and can catch arithmetic mistakes.
- 不检验答案：将解代回原方程只需几秒钟，却能发现计算错误。
first subtract 2 from both sides to get 4x = 8, then divide both sides by 4 to get x = 2.
一个常见错误是只除以一边的一项。例如，在 4x + 2 = 10 中，学生可能错误地只把 4x 项除以 4，写成 x + 2 = 2.5。正确的步骤是先两边同时减去 2，得到 4x = 8，然后两边同时除以 4，得到 x = 2。
9. Exam-Style Practice Questions | 考试风格练习题
Try these questions yourself before looking at the answers. They cover the main types of linear equations you will meet in IGCSE Mathematics.
在查看答案之前，请先尝试完成以下题目。它们涵盖了 IGCSE 数学中你会遇到的主要一元一次方程类型。
- Solve 5x − 7 = 18.
- 解方程 5x − 7 = 18。
- Solve 4(x − 2) = 3(x + 1).
- 解方程 4(x − 2) = 3(x + 1)。
- Solve (3x + 4)/2 = 5.
- 解方程 (3x + 4)/2 = 5。
- Solve 7x + 2 = 3x − 10.
- 解方程 7x + 2 = 3x − 10。
- The sum of two consecutive even numbers is 50. Find the numbers.
- 两个连续偶数的和是 50。求这两个数。
Answers: x = 5; x = 11; x = 2; x = −3; the numbers are 24 and 26.
答案：x = 5；x = 11；x = 2；x = −3；这两个数是 24 和 26。
+ (x + 2) = 50. This gives 2x + 2 = 50, so 2x = 48 and x = 24. The numbers are 24 and 26.
对于最后一题，设第一个偶数为 x。那么下一个连续偶数是 x + 2。方程为 x + (x + 2) = 50。化简得到 2x + 2 = 50，所以 2x = 48，x = 24。这两个数是 24 和 26。
10. Summary and Key Points | 总结与要点
Solving linear equations is a core skill in IGCSE Mathematics. The key steps are to simplify each side by expanding brackets and combining like terms, then use inverse operations to isolate the variable while keeping the equation balanced.
解一元一次方程是 IGCSE 数学的核心技能。关键步骤是：通过展开括号和合并同类项化简等式的两边，然后使用逆运算逐步分离变量，同时保持方程平衡。
Always perform the same operation on both sides. When fractions are involved, multiply every term by the LCM of the denominators. For word problems, define the unknown clearly and translate the information into an equation. Finally, check your answer by substituting it back into the original equation.
始终在方程两边同时进行相同的运算。当涉及分数时，将每一项都乘以分母的最小公倍数。对于文字题，要清楚地设出未知数，并将信息转化为方程。最后，将答案代回原方程进行检验。
With regular practice, you will be able to solve these equations quickly and accurately, giving you a strong foundation for harder algebra topics such as simultaneous equations and quadratic equations.
通过经常练习，你就能快速而准确地解出这些方程，为后续更难的代数主题，如联立方程组和二次方程，打下坚实的基础。
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