Solving Linear Equations with Brackets and Fractions | 解含括号和分数的线性方程

📚 Solving Linear Equations with Brackets and Fractions | 解含括号和分数的线性方程

Linear equations are the foundation of algebra at KS3. Being able to solve equations with brackets and fractions helps you model real-world situations and prepares you for harder topics in Cambridge IGCSE and beyond.

线性方程是 KS3 代数的基石。能够解含括号和分数的方程可以帮助你建立现实情境的数学模型，并为剑桥 IGCSE 及更高阶段的学习做好准备。

---

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

A linear equation is a statement that two expressions are equal, where the unknown appears only to the power of 1. For example, 2x + 3 = 11 is linear because x is not squared or cubed.

线性方程是指两个表达式相等且未知数只出现 1 次方的等式。例如 2x + 3 = 11 就是线性方程，因为 x 没有平方或立方。

2x + 3 = 11

The goal is to find the value of the unknown that makes the equation true. We call this the solution or root of the equation. Examples of non-linear equations include x² = 9 and 1/x = 5, which are not covered at this stage.

我们的目标是求出使等式成立的未知数的值，这个值称为方程的解或根。非线性的例子包括 x² = 9 和 1/x = 5，这些不属于本阶段内容。

---

2. The Balance Method | 天平法

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

把方程想象成一个平衡的天平。无论你对一边做了什么，必须对另一边做同样的操作才能保持平衡。

To solve x + 5 = 12, subtract 5 from both sides. This gives x = 7. In a balanced equation, adding, subtracting, multiplying, or dividing both sides by the same non-zero number keeps it true.

解 x + 5 = 12 时，两边同时减去 5，得到 x = 7。在平衡等式中，两边同时加、减、乘或除以同一个非零数，等式仍然成立。

x + 5 − 5 = 12 − 5  →  x = 7

The balance method works because equality is preserved under the same operation. This is central to algebraic proof and to solving more difficult equations later.

天平法有效是因为等式在相同运算下保持不变，这正是代数证明以及日后解更复杂方程的核心。

---

3. Expanding Brackets First | 先去括号

When a bracket appears, expand it before applying the balance method. Use the distributive law: a(b + c) = ab + ac.

当方程中出现括号时，先用分配律 a(b + c) = ab + ac 展开括号，再使用天平法。

For 3(x + 2) = 21, first expand to get 3x + 6 = 21. Then subtract 6 from both sides to get 3x = 15. Finally divide both sides by 3 to get x = 5.

对于 3(x + 2) = 21，先展开得 3x + 6 = 21。然后两边减 6 得 3x = 15。最后两边除以 3 得 x = 5。

3(x + 2) = 21  →  3x + 6 = 21  →  x = 5

If the bracket has a negative coefficient, expand carefully: -2(x − 3) = -2x + 6. Many errors happen because the negative sign is not distributed to every term inside the bracket.

如果括号前是负系数，展开时要小心：-2(x − 3) = -2x + 6。很多错误是因为负号没有分配到括号内的每一项。

---

4. Equations with Unknowns on Both Sides | 未知数在等式两边

If the unknown appears on both sides, collect like terms. First remove the smaller unknown term from both sides, then solve as usual.

如果未知数出现在等式两边，要先合并同类项。通常先消去两边中较小的未知数项，再照常求解。

Solve 5x + 2 = 3x + 10. Subtract 3x from both sides to get 2x + 2 = 10. Then subtract 2 from both sides to get 2x = 8, so x = 4.

解 5x + 2 = 3x + 10。两边同时减去 3x 得 2x + 2 = 10。再两边减 2 得 2x = 8，因此 x = 4。

5x − 3x + 2 = 3x − 3x + 10  →  2x = 8  →  x = 4

If you have trouble choosing which side to collect on, collect the variable on the side with the larger coefficient to keep it positive. This reduces sign errors.

如果不确定把未知数合并到哪一边，可以把它移到系数较大的一边，使未知数项保持为正，这样可以减少符号错误。

---

5. Clearing Fractions | 去分母

To solve equations with fractions, multiply every term by the lowest common denominator (LCD). This clears the fractions and gives a simpler equation.

解含分数的方程时，将每一项乘以最小公分母 (LCD)，这样可以去掉分母，得到更简单的方程。

For x/2 + 3 = 7, the denominator is 2. Multiply every term by 2: 2(x/2) + 2(3) = 2(7), giving x + 6 = 14. Then x = 8.

对于 x/2 + 3 = 7，分母是 2。将每一项乘以 2：2(x/2) + 2(3) = 2(7)，得到 x + 6 = 14，所以 x = 8。

When there are two or more denominators, find the LCD. For example, in x/3 + x/4 = 7, the LCD is 12. Multiply through by 12 to obtain 4x + 3x = 84, so 7x = 84 and x = 12.

当有两个或多个分母时，要找到最小公分母。例如 x/3 + x/4 = 7，LCD 是 12。全式乘以 12 得 4x + 3x = 84，所以 7x = 84，x = 12。

x/3 + x/4 = 7  →  ×12  →  4x + 3x = 84  →  x = 12

Always check that you multiply both sides and every term by the LCD. For equations like x/2 + x/3 = 5, multiply by 6 to get 3x + 2x = 30.

始终确认将两边及每一项都乘以 LCD。对于 x/2 + x/3 = 5，乘以 6 得 3x + 2x = 30。

---

6. Common Mistakes to Avoid | 常见错误

A very common error is forgetting to multiply every term when clearing fractions. For example, in x/2 + 1 = 5, do not write x + 1 = 5; you must multiply the 1 as well.

一个常见错误是在去分母时忘记乘以每一个项。例如在 x/2 + 1 = 5 中，不能写成 x + 1 = 5；常数 1 也必须乘以 2。

Another mistake is to add or subtract before expanding brackets. Always deal with brackets first unless the equation is already in a form where brackets can be divided out.

另一个错误是在展开括号前就进行加减。除非方程已经可以约去括号外系数，否则应先处理括号。

• Do multiply every term by the LCD.
  确实要把每一个项都乘以 LCD。
• Do not forget to change signs when moving terms across the equals sign.
  移项时不要忘记变号。
• Do check your final answer in the original equation.
  务必把最终答案代回原方程检验。

---

7. Word Problems and Real-Life Contexts | 应用题与生活情境

Many KS3 questions ask you to form a linear equation from a word problem. Read the problem carefully, define the unknown with a letter, and write each relationship as an expression.

许多 KS3 题目要求你根据文字题列出线性方程。仔细读题，用字母设未知数，并把每个关系写成表达式。

Example: A rectangle has length 3 cm more than its width. The perimeter is 22 cm. Let the width be w. Then length is w + 3, and perimeter is 2(w + w + 3) = 22.

例题：一个矩形的长比宽多 3 cm，周长是 22 cm。设宽为 w，则长为 w + 3，周长为 2(w + w + 3) = 22。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）