Solving Equations with Variables on Both Sides | 解含双边变量的线性方程

📚 Solving Equations with Variables on Both Sides | 解含双边变量的线性方程

Equations are at the heart of algebra, and the ability to solve them confidently builds a solid foundation for all future mathematics. In this topic, derived from p172.pdf of the Cambridge KS3 Mathematics series, we focus specifically on linear equations that contain the unknown variable on both sides of the equals sign. You will learn to use the balance method, collect like terms, and systematically isolate the variable to find the solution.

方程是代数的核心，能熟练解方程会为今后的数学学习打下坚实基础。本章内容取材自剑桥KS3数学系列教材的p172.pdf，重点讨论等号两边都含有未知变量的线性方程。你将学习如何使用平衡法、合并同类项，并系统地分离变量以求出解。

---

1. Understanding the Equation Format | 理解方程的结构

An equation is a mathematical statement that two expressions are equal. In linear equations with variables on both sides, the unknown (often denoted by x) appears on the left and the right of the equals sign. For example: 3x + 2 = x + 10. The goal is to find the value of x that makes this statement true.

方程是说明两个表达式相等的数学陈述。在两边都有变量的线性方程中，未知数（通常用x表示）同时出现在等号的左边和右边。比如：3x + 2 = x + 10。我们的目标就是找到使该等式成立的x值。

A typical KS3 equation will involve integers, and possibly brackets or simple fractions. Always look at the entire equation before starting: note the variable terms on both sides, the constant terms, and any operations that need to be undone.

典型的KS3方程会涉及整数，也可能含括号或简单分数。在开始求解前，务必先通览整个方程：注意两边含变量的项、常数项以及需要逆运算的运算。

---

2. The Golden Rule: Keep the Balance | 黄金法则：保持平衡

Think of an equation as a perfectly balanced set of scales. Whatever you do to one side, you must also do to the other side to maintain equality. This balance principle allows you to add, subtract, multiply, or divide on both sides without changing the solution.

把方程想象成一个完全平衡的天平。无论你对一边做了什么，都必须对另一边做同样的操作，才能维持相等关系。这一平衡原理允许你在两边同时加、减、乘、除，而不改变方程的解。

For example, if you want to remove +5 from the left side, you subtract 5 from both sides. If you need to eliminate a coefficient of 4, you divide both sides by 4. This rule is the foundation of all algebraic equation solving.

例如，如果你想从左式去掉+5，就要在两边同时减去5。如果需要消去系数4，就要两边同时除以4。这条法则是所有代数方程求解的基础。

---

3. Collecting Like Terms Before Moving | 先合并同类项再移项

Sometimes each side of the equation contains like terms that can be simplified first. Look for terms with the variable and constant terms on each side and combine them. This makes the equation shorter and reduces the risk of mistakes.

有时，方程两边各自含有可以合并的同类项。先找到各边的含变量项和常数项，并把它们合并。这样做能让方程更简洁，减少出错概率。

Example: 2x + 3x + 4 = 5 + 2x – 1. On the left, 2x + 3x simplifies to 5x. On the right, 5 – 1 simplifies to 4. The equation becomes 5x + 4 = 2x + 4. Now it is ready for moving variables.

例题：2x + 3x + 4 = 5 + 2x – 1。左边2x+3x可合并为5x；右边5–1合并为4。方程变为5x+4=2x+4，此时便适合开始移项。

5x + 4 = 2x + 4

---

4. Moving Variables to One Side | 将变量移到同一边

To solve an equation with variables on both sides, we gather all variable terms on one side and all constants on the other. Usually, we eliminate the smaller variable term first to avoid negative coefficients, but either side works.

要解两边均有变量的方程，我们可以把所有含变量的项集中到等号一边，所有常数项集中到另一边。通常，我们会先消去系数较小的变量项以避免负系数，但选哪边其实都可以。

Using the equation 5x + 4 = 2x + 4, subtract 2x from both sides to move the variable terms to the left: 5x – 2x + 4 = 2x – 2x + 4, which simplifies to 3x + 4 = 4.

就方程5x+4=2x+4而言，两边同时减去2x，把变量项移到左边：5x–2x+4=2x–2x+4，化简得3x+4=4。

3x + 4 = 4

---

5. Isolating the Variable | 隔离变量

After moving all variable terms to one side, you are left with a simple one-step or two-step equation. Next, undo any addition or subtraction around the variable term, and then deal with the coefficient.

把所有变量项移到同一边后，你面对的就是简单的一步或两步方程。接下来，先把变量项旁边的加减法逆运算掉，再处理系数。

From 3x + 4 = 4, subtract 4 from both sides: 3x = 0. Finally, divide both sides by 3: x = 0. The solution is x = 0.

由3x+4=4，两边同时减4得3x=0。最后两边同除以3，得x=0。方程的解为x=0。

3x = 0 → x = 0

---

6. Solving Equations with Brackets | 解含括号的方程

When brackets are present, always expand them before starting to move terms. Use the distributive property: a(b + c) = ab + ac. Remember to multiply the term outside the bracket by every term inside, paying close attention to signs.

当方程含有括号时，务必先展开括号再移项。利用分配律：a(b + c) = ab + ac。注意用括号外的项乘以括号内的每一项，并留意正负号。

Example: 3(x – 2) = 2x + 5. Expand the left side: 3x – 6 = 2x + 5. Then subtract 2x from both sides: x – 6 = 5. Add 6 to both sides: x = 11.

例题：3(x – 2) = 2x + 5。展开左边：3x – 6 = 2x + 5。然后两边同时减2x得x – 6 = 5。两边加6得x = 11。

3(x – 2) = 2x + 5 → x = 11

---

7. Dealing with Negative Coefficients | 处理负系数

Sometimes after moving terms you might end up with a negative variable term, such as –2x = 10. You can multiply both sides by –1 to make the variable positive, or simply divide by the negative coefficient directly.

移项后有时你会得到负的变量项，比如–2x = 10。你可以将两边同时乘以–1使其变成正变量项，也可以直接除以该负系数。

For –2x = 10, dividing both sides by –2 gives x = –5. Always perform the same operation on both sides, and keep track of negative signs to avoid errors.

以–2x = 10为例，两边同除以–2得x = –5。始终在两边执行相同的操作，并仔细处理负号，避免出错。

–2x = 10 → x = –5

---

8. Solving Equations with Simple Fractions | 解含简单分数的方程

Equations may include fractions such as ½ or ¼. A reliable method is to eliminate fractions early by multiplying every term by the common denominator. This turns the equation into an equivalent integer equation.

方程中可能出现像½或¼这样的分数。一种可靠的方法是，先找出公分母，然后用它乘以每一项，从而消去分母，将方程转变为等价的整数方程。

Example: ½x + 1 = x – 2. Multiply every term by 2: x + 2 = 2x – 4. Then subtract x from both sides: 2 = x – 4. Add 4: x = 6.

例题：½x + 1 = x – 2。每项乘以2得：x + 2 = 2x – 4。两边减x：2 = x – 4。再加4得x = 6。

½x + 1 = x – 2 → x = 6

---

9. Checking Your Solution | 检验解的正确性

Substitute your found value back into the original equation, both on the left-hand side and the right-hand side, and compare the results. If the two sides are equal, your solution is correct.

把你求出的值代回到原方程中，分别计算等号左边和右边的值，然后比较两边结果。如果两边相等，说明你的解正确。

For x = 6 in ½x + 1 = x – 2: left side = ½×6 + 1 = 3 + 1 = 4; right side = 6 – 2 = 4. Both sides equal 4, confirming the solution.

方程½x+1=x–2，将x=6代入：左边 = ½×6+1 = 3+1 = 4；右边 = 6–2 = 4。两边都是4，确认解正确。

This habit is especially useful during exams to catch arithmetic mistakes.

这个习惯在考试中尤其有用，能帮你发现计算错误。

---

10. Word Problems Leading to Equations | 从文字题到方程

Many real-life situations can be modelled by linear equations with variables on both sides. Read the problem carefully, define your variable, and build the equation by translating relationships into algebraic language.

很多实际情境都可以用带有双边变量的线性方程来建模。仔细阅读题目，设出变量，并将关系翻译成代数语言，从而建立方程。

Example: ‘Three times a number increased by 5 is equal to twice the number decreased by 2.’ Let the number be x. Equation: 3x + 5 = 2x – 2. Solve: subtract 2x → x + 5 = –2 → subtract 5 → x = –7.

例题：“一个数的3倍加5等于这个数的2倍减2。”设这个数为x，则方程为3x+5=2x–2。求解：两边减2x得x+5=–2；再减5得x=–7。

3x + 5 = 2x – 2 → x = –7

---

11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

One frequent error is forgetting to perform an operation on both sides, especially when moving terms. Another is mishandling negative signs when expanding brackets. Always write each step clearly and check signs.

常见错误之一是忘记在方程两边同时进行操作，尤其在移项时。另一个是在展开括号时处理负号不当。务必清楚地写出每一步，并认真核对符号。

Using a table to organise steps can help. Below is a modelled solution layout:

用表格来整理步骤会有所帮助。下面是示范的解题布局：

Step	Equation	操作
1	2(x + 3) = x + 8	展开括号
2	2x + 6 = x + 8	两边减 x
3	x + 6 = 8	两边减 6
4	x = 2	解

By following a structured method, you reduce slip-ups significantly.

采用有条理的方法能大幅减少疏漏。

---

12. Practice Examples and Summary | 练习题与总结

Now try these equations on your own, applying the balance method and checking your solutions:

现在请独立求解下面这些方程，运用平衡法并检验你的解：

(a) 4x + 3 = 2x + 11

(b) 5(y – 1) = 3y + 7

(c) ⅓ t + 2 = t – 4

Answers: (a) x = 4; (b) y = 6; (c) t = 9. Revisit any step you found tricky and re-check your work using the original equation.

答案：(a) x=4；(b) y=6；(c) t=9。如果某个步骤觉得困难，请重新回顾，并用原方程验算你的结果。

In this revision lesson, we have covered the essential techniques for solving linear equations with variables on both sides, from initial simplification and expansion to isolating the variable and checking answers. Mastery of these skills will serve you well not only in KS3 assessments but also in more advanced algebra topics ahead.

在这节复习课中，我们从最初的化简和展开，到隔离变量并验算答案，全面讲解了求解双边变量线性方程的核心技巧。熟练掌握这些技能不仅对KS3测评大有帮助，对后续更高阶的代数学习也至关重要。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）