📚 Solving Linear Equations with Brackets and Fractions | 解含括号和分数的一元一次方程
At KS3 level, Cambridge mathematics focuses on building confidence with algebra. One of the most important skills is solving linear equations. In this article we will cover equations that contain brackets, fractions and unknowns on both sides. You will learn how to use the balance method, how to check your answers, and how to avoid common mistakes. This topic appears regularly in Checkpoint tests and provides the foundation for IGCSE algebra.
在 KS3 阶段，剑桥数学注重建立代数信心。最重要的技能之一是解一元一次方程。本文将涵盖含有括号、分数和两侧未知数的方程。你将学习如何使用天平法、如何检验答案以及如何避免常见错误。该主题在 Checkpoint 测试中经常出现，并为 IGCSE 代数打下基础。
1. What Is a Linear Equation? | 什么是一元一次方程？
A linear equation is an equation where the unknown variable appears only to the power of 1. It has no x², no x³ and no product of two variables. The graph of a linear equation is a straight line, which is why it is called linear.
一元一次方程是指未知数只出现一次幂的方程。它没有 x²、x³ 或两个变量的乘积。一元一次方程的图像是一条直线，因此被称为线性方程。
4(x + 2) = 24 and 3x + 1 = 2x + 9. Each of these can be solved using a systematic method.
常见的一元一次方程包括 2x + 3 = 11、x − 5 = 7、4(x + 2) = 24 和 3x + 1 = 2x + 9。每一个都可以用系统的方法求解。
2. The Balance Method | 天平法（等式平衡）
The balance method treats an equation like a pair of scales. Whatever you do to one side, you must do to the other to keep it balanced. If you add 5 to the left, add 5 to the right. If you divide the left by 3, divide the right by 3.
天平法将方程视为一架天平。你对一侧做的任何操作，必须对另一侧做同样的操作才能保持平衡。如果给左边加 5，右边也要加 5。如果左边除以 3，右边也要除以 3。
The aim is always to isolate the unknown variable on one side of the equals sign. You should perform inverse operations in the correct order, usually undoing addition or subtraction before multiplication or division.
目标始终是将未知数单独放在等号的一侧。你应当按照正确的顺序进行逆运算，通常先处理加减，再处理乘除。
x + 5 = 12 ⇒ x + 5 − 5 = 12 − 5 ⇒ x = 7
3. Solving Basic Two-Step Equations | 解基础两步方程
ariable x is first multiplied by 3 and then 4 is subtracted. To solve it, reverse the order: add 4 to both sides first, then divide by 3.
两步方程包含两个运算。例如，在 3x − 4 = 5 中，未知数 x 先乘以 3，然后减去 4。求解时要颠倒顺序：先给两边加 4，再除以 3。
Always write each new line directly underneath the previous line. This makes your working clear and helps you avoid missing a step.
每写一行新步骤都要直接写在上一行下面。这样能让你的解题过程清晰，并帮助你避免漏步。
3x − 4 = 5 ⇒ 3x = 9 ⇒ x = 3
4. Removing Brackets Using the Distributive Law | 用分配律去括号
When an equation contains brackets, expand them first using the distributive law. Multiply the term outside by every term inside the brackets. For example, 4(x + 2) becomes 4x + 8.
当方程含有括号时，先用分配律展开。将括号外的项乘以括号内的每一项。例如，4(x + 2) 展开为 4x + 8。
Be careful with negative signs. If the bracket is preceded by a minus sign, you must change the sign of every term inside the bracket when you remove it.
注意负号。如果括号前面是负号，去掉括号时必须改变括号内每一项的符号。
3(x − 2) = 12 ⇒ 3x − 6 = 12 ⇒ 3x = 18 ⇒ x = 6
5. Equations with Unknowns on Both Sides | 两侧都含未知数的方程
Some equations have the unknown on both sides of the equals sign. To solve these, collect all variable terms on one side and all constant terms on the other side. Choose the side that gives a positive number of x.
有些方程在等号两侧都有未知数。求解时，将所有含未知数的项移到一侧，将所有常数项移到另一侧。选择能保持未知数系数为正的一侧。
+ 1 = 2x + 9, subtract 2x from both sides to get x + 1 = 9, then subtract 1 to find x = 8.
例如，在 3x + 1 = 2x + 9 中，两边同时减去 2x 得到 x + 1 = 9，然后再减去 1 得到 x = 8。
3x + 1 = 2x + 9 ⇒ x + 1 = 9 ⇒ x = 8
6. Dealing with Fractions in Equations | 处理方程中的分数
Fractions can make an equation look harder than it really is. The best first step is to multiply every term on both sides by the lowest common denominator of all fractions. This clears the fractions and gives a simpler equation.
分数会让方程看起来比实际更难。最好的第一步是给两边每一项乘以所有分数的最小公分母。这样可以消去分数，得到更简单的方程。
the denominator. For example, x ÷ 4 = 3 becomes x = 12 after multiplying both sides by 4.
如果只有一项含分数，你也可以用分母乘以另一边。例如，x ÷ 4 = 3 在两边乘以 4 后变为 x = 12。
x ÷ 4 + 1 = 3 ⇒ x ÷ 4 = 2 ⇒ x = 8
7. Checking Your Solution by Substitution | 代入法检验答案
After solving an equation, it is good practice to substitute your answer back into the original equation. The left-hand side and the right-hand side should give the same value. If they do not, go back and find your mistake.
解完方程后，最好将答案代回原方程进行检验。左边和右边应当得出相同的值。如果不相等，就返回去查找错误。
ck: 2 × 4 + 3 = 8 + 3 = 11. Since the right-hand side is also 11, the solution is correct.
对于 2x + 3 = 11，我们得到 x = 4。检验：2 × 4 + 3 = 8 + 3 = 11。由于右边也是 11，解是正确的。
Check: 2(4) + 3 = 11 ⇒ 8 + 3 = 11 ⇒ 11 = 11
8. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common mistake is forgetting to apply an operation to both sides. Another is expanding brackets incorrectly when a negative number is involved. Students also sometimes divide before undoing addition or subtraction, which can lead to errors.
一个常见错误是忘记对两边同时进行运算。另一个是在涉及负数时展开括号出错。学生有时会在处理加减之前就进行除法，这也可能导致错误。
To avoid these mistakes, always write each step clearly, keep the equals signs aligned, and double-check any sign changes. Use the balance method one step at a time.
为了避免这些错误，始终清晰地写出每一步，保持等号对齐，并仔细检查符号变化。一次只进行一步天平法运算。
9. Word Problems Leading to Linear Equations | 由实际问题列一元一次方程
Many real-world problems can be modelled by a linear equation. Read the question carefully, define the unknown with a letter, and write an equation that describes the relationship given in the problem.
许多实际问题都可以用一元一次方程来建模。仔细阅读题目，用字母定义未知数，并写出描述题目关系的方程。
nd then 7 is added, giving 32. Let the number be n. The equation is 5n + 7 = 32, so n = 5.
例如：一个数乘以 5 后再加上 7，结果是 32。设这个数为 n。方程为 5n + 7 = 32，因此 n = 5。
5n + 7 = 32 ⇒ 5n = 25 ⇒ n = 5
10. Practice Questions and Answers | 练习题与答案
Try these questions to test your understanding. Solve each equation and then check your answer by substitution.
尝试以下题目来检验你的理解。解出每个方程，然后通过代入检验答案。
- 2x + 5 = 19
- 3(x − 4) = 15
- 5x − 2 = 3x + 10
- x ÷ 3 + 2 = 6
. Work through each step carefully and compare your method with the balance method shown above.
答案分别为 x = 7、x = 9、x = 6 和 x = 12。仔细完成每一步，并将你的方法与上文展示的天平法进行比较。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图