📚 Solving Equations with Variables on Both Sides | 解含两边变量的方程
Welcome to this essential KS3 Cambridge Mathematics guide, based on the key skills developed around page 132 of your student book. Here we focus on one of the most important algebraic techniques: solving linear equations where the unknown appears on both sides of the equals sign. You will learn how to use the balance method, simplify expressions methodically and avoid common mistakes. By the end, you will be ready to tackle any equation of this type with confidence.
欢迎阅读这篇关键的 KS3 剑桥数学指南，内容基于学生用书第 132 页所培养的核心技能。这里我们聚焦于最重要的代数技巧之一：解未知数出现在等号两边的线性方程。你将学会如何运用天平法、有条理地化简表达式，并避开常见错误。读完后，你将能够自信地应对任何这类方程。
1. What Are Linear Equations? | 什么是线性方程？
A linear equation is an algebraic statement where the highest power of the variable is 1. They can be as simple as x + 3 = 7 or as challenging as 5(2x − 3) = 3x + 9. In KS3, we focus on equations that produce straight lines when graphed, hence the name ‘linear’. At this stage, you will often meet equations where the unknown x appears on both sides.
线性方程是变量的最高次幂为 1 的代数等式。它可以是简单的 x + 3 = 7，也可以是具有挑战性的 5(2x − 3) = 3x + 9。在 KS3 阶段，我们关注的是作图时会产生直线的方程，“线性”一名即由此而来。在这个阶段，你经常会遇到未知数 x 出现在等式两边的方程。
2. The Balance Method | 天平法
Think of an equation as a perfectly balanced set of scales. Whatever you do to one side, you must do exactly the same to the other side to maintain balance. This is the golden rule for solving any equation. If you add, subtract, multiply or divide on the left, you must carry out the identical operation on the right.
把方程想象成一架完全平衡的天平。你对一边所做的任何操作，都必须对另一边做同样的操作，以保持平衡。这是解任何方程的黄金法则。如果你在左边加、减、乘、除，就必须在右边执行完全相同的运算。
3. Solving One-Step and Two-Step Equations | 解一步和两步方程
Before tackling variables on both sides, it helps to master simpler forms. A one-step equation like x + 5 = 9 needs only subtraction of 5 from both sides: x = 4. A two-step equation like 2x + 1 = 7 involves two operations: subtract 1, then divide by 2. These foundations make the balance method second nature.
在处理两边都有变量之前，掌握更简单的形式很有帮助。像 x + 5 = 9 这样的一步方程只需两边减去 5：x = 4。像 2x + 1 = 7 这样的两步方程包含两种运算：先减 1，再除以 2。这些基础会让天平法成为你的第二天性。
4. Removing Brackets | 去括号
Equations often contain brackets. Always expand them before collecting like terms. For example, 2(x + 3) becomes 2x + 6. Remember to multiply the term outside the bracket by every term inside. This step turns a complicated-looking equation into a standard linear one that you can manage easily.
方程中经常含有括号。在合并同类项之前，一定要先将括号展开。例如，2(x + 3) 展开后得到 2x + 6。记得用括号外的项去乘括号内的每一项。这一步能将看似复杂的方程变成你容易处理的标准线性方程。
5. Introducing Variables on Both Sides | 引入两边变量
The real twist comes when x appears on both sides, such as 3x + 2 = x + 6. Your aim is to collect all x terms on one side and all numbers on the other. This is done by adding or subtracting the same amount of x to both sides. Never panic – the balance method still works perfectly.
真正的转折点在于 x 出现在等号两边，例如 3x + 2 = x + 6。你的目标是把所有含 x 的项集中到一边，把所有数字集中到另一边。这可以通过在两边加上或减去相同数量的 x 来实现。不必惊慌——天平法依然完美适用。
6. A Step-by-Step Approach: Move and Simplify | 步骤分解：移项与化简
Follow this strategy: first eliminate the smaller x term by subtracting it from both sides. Then simplify the equation to a two-step form. For 3x + 2 = x + 6, subtract x from each side to get 2x + 2 = 6. Now subtract 2 from both sides, giving 2x = 4, and finally divide by 2 to find x = 2.
遵循以下策略：先消去较小的 x 项，即从两边减去该项。然后将方程化简为两步形式。对于 3x + 2 = x + 6，两边减去 x 得到 2x + 2 = 6。接着两边减去 2，得到 2x = 4，最后除以 2 求出 x = 2。
7. Worked Example 1 | 示例详解一
Equation: 4x − 5 = 2x + 7. Step 1: subtract 2x from both sides → 2x − 5 = 7. Step 2: add 5 to both sides → 2x = 12. Step 3: divide by 2 → x = 6. Always check: substitute x = 6 back in: left side 4(6)−5=19, right side 2(6)+7=19, it balances correctly.
方程：4x − 5 = 2x + 7。步骤一：两边减去 2x → 2x − 5 = 7。步骤二：两边加上 5 → 2x = 12。步骤三：除以 2 → x = 6。永远要检验：将 x = 6 代回原方程：左边 4(6)−5=19，右边 2(6)+7=19，两边平衡正确。
8. Worked Example 2 | 示例详解二
Equation: 5x + 2 = 3x − 8. Subtract 3x from both sides: 2x + 2 = −8. Subtract 2: 2x = −10. Divide by 2: x = −5. Checking: left 5(−5)+2=−23, right 3(−5)−8=−23. Negative solutions are perfectly valid and just as common as positive ones.
方程：5x + 2 = 3x − 8。两边减去 3x：2x + 2 = −8。减去 2：2x = −10。除以 2：x = −5。检验：左边 5(−5)+2=−23，右边 3(−5)−8=−23。负数解完全有效，并且与正数解一样常见。
9. Handling Negative Numbers and Fractions | 处理负数和分数
When the variable coefficient becomes negative, you can multiply the entire equation by −1 to keep things tidy. For example, if you reach −x = 4, multiply both sides by −1 to get x = −4. For fractional coefficients like ½x, you can either multiply both sides by the denominator 2, or treat the fraction as division.
当变量系数变为负数时，你可以将整个方程乘以 −1，使式子保持整洁。例如，如果得出 −x = 4，将两边乘以 −1 得到 x = −4。对于像 ½x 这样的分数系数，你可以将两边乘以分母 2，或者把分数当作除法来处理。
10. Common Pitfalls and Check Your Answer | 常见陷阱与验算
A frequent error is forgetting to perform an operation on both sides. Another is mishandling signs when subtracting a negative term. Always write down each step clearly. Finally, always substitute your answer back into the original equation – this confirms your solution and builds solid exam habits.
一个常见错误是忘记对两边同时执行操作。另一个是在减去负数项时符号处理不当。始终要清晰地写出每一步。最后，总是将你的答案代回原方程——这能确认你的解，并养成扎实的考试习惯。
| Original equation | Operation | Result |
| 4x − 5 = 2x + 7 | −2x both sides | 2x − 5 = 7 |
| 2x − 5 = 7 | +5 both sides | 2x = 12 |
| 2x = 12 | ÷2 both sides | x = 6 |
Check: 4(6) − 5 = 19 and 2(6) + 7 = 19 ✔
11. Applying Equations to Problems | 解应用题
Linear equations with variables on both sides often appear in word problems. For instance: ‘Three times a number plus 2 is equal to the number plus 6. Find the number.’ Translate to 3x + 2 = x + 6, and solve as before. The ability to form an equation from text is a key KS3 skill.
两边含有变量的线性方程经常出现在文字题中。例如：“某数的三倍加 2 等于该数加 6。求这个数。”翻译成方程就是 3x + 2 = x + 6，然后像之前那样求解。从文字中建立方程是 KS3 的一项重要技能。
12. Summary and Page 132 Practice | 总结与第 132 页练习
You now have a structured approach to equations with unknowns on both sides: expand brackets, collect variable terms on one side, numbers on the other, simplify, and solve. The exercises on page 132 of your Cambridge coursebook are designed to reinforce exactly these steps. Work through them carefully, checking each solution.
现在你已经掌握了解含有两边未知数的方程的结构化方法：展开括号，将变量项集中到一边，数字集中到另一边，化简，然后求解。你的剑桥教材第 132 页上的练习正是为了强化这些步骤。认真完成这些练习，并检验每一道解。
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