📚 Solving Linear Equations with Unknowns on Both Sides | 解含两边未知数的一元一次方程
Mastering these equations builds confidence for Cambridge Checkpoint and later IGCSE work.
线性方程是 KS3 代数推理的基础。本文重点讲解未知数出现在等号两边的方程，例如 5x − 2 = 2x + 7。掌握这类方程能为 Cambridge Checkpoint 以及后续 IGCSE 学习建立信心。
1. What Are Linear Equations? | 什么是线性方程
A linear equation is an equation where the unknown appears only to the power of 1. There are no x², x³, √x or 1/x terms.
线性方程是指未知数只出现 1 次方的方程。其中没有 x²、x³、√x 或 1/x 项。
The graph of a linear equation is a straight line, which is where the name 'linear' comes from.
线性方程的图像是一条直线，'线性'这个名称正是由此而来。
3x + 5 = 2x + 12
In this example, x appears on both the left-hand side and the right-hand side.
在这个例子中，x 同时出现在等号左边和右边。
2. Key Vocabulary | 关键术语
| Unknown 未知数 | The letter that stands for an unknown value. 表示未知值的字母。 |
| Coefficient 系数 | The number multiplying the unknown, e.g. in 4x, 4 is the coefficient. 乘在未知数前面的数，例如 4x 中，4 是系数。 |
| Constant 常数项 | A number on its own, e.g. 3 in 2x + 3. 单独出现的数，例如 2x + 3 中的 3。 |
| Left-hand side / Right-hand side 左边 / 右边 | The expressions on either side of the equals sign. 等号两侧的表达式。 |
| Equation 方程 | A statement that two expressions are equal. 表示两个表达式相等的式子。 |
| Solution 解 | The value of the unknown that makes the equation true. 使方程成立的未知数的值。 |
3. Balancing Method | 平衡法
An equation is like a balance scale. Whatever you do to one side, you must also do to the other side, so the equation stays balanced.
方程就像一架天平。无论你对一边做什么运算，都必须对另一边做同样的运算，方程才能保持平衡。
The golden rule is: add, subtract, multiply or divide both sides by the same non-zero number.
黄金法则是：两边同时加、减、乘或除以同一个非零数。
Our goal is to collect all unknown terms on one side and all constant terms on the other.
我们的目标是把所有含未知数的项移到一边，把所有常数项移到另一边。
4. Solving with Unknowns on Both Sides | 解含两边未知数的方程
Consider 5x − 2 = 2x + 10.
例如 5x − 2 = 2x + 10。
Step 1: Subtract 2x from both sides to remove the smaller x-term.
第一步：两边同时减去 2x，消去较小的 x 项。
5x − 2 − 2x = 2x + 10 − 2x → 3x − 2 = 10
Step 2: Add 2 to both sides.
第二步：两边同时加上 2。
3x − 2 + 2 = 10 + 2 → 3x = 12
Step 3: Divide both sides by 3.
第三步：两边同时除以 3。
3x ÷ 3 = 12 ÷ 3 → x = 4
Finally, substitute x = 4 back into the original equation to check.
最后，把 x = 4 代回原方程检验。
5. Expanding Brackets First | 先展开括号
If an equation contains brackets, expand them before using the balancing method.
如果方程中含有括号，应先用乘法分配律展开，再使用平衡法。
3(x + 2) = 2(x + 5)
Expand both sides:
两边展开：
3x + 6 = 2x + 10
Subtract 2x from both sides:
两边同时减去 2x：
x + 6 = 10
Subtract 6 from both sides:
两边同时减去 6：
x = 4
6. Dealing with Negative Coefficients | 处理负系数
ive coefficient, such as 10 − x = 2x + 1, it is often easier to add x to both sides first.
当未知数的系数为负数时，例如 10 − x = 2x + 1，通常先两边同时加上 x 更容易。
10 − x + x = 2x + 1 + x → 10 = 3x + 1
Then subtract 1 from both sides:
然后两边同时减去 1：
10 − 1 = 3x + 1 − 1 → 9 = 3x
Divide both sides by 3:
两边同时除以 3：
9 ÷ 3 = 3x ÷ 3 → x = 3
Keeping the coefficient positive reduces sign errors.
保持系数为正能减少符号错误。
7. Fractional Coefficients | 分数系数
For equations like x/2 + 3 = 7, multiply every term by the denominator to clear fractions.
对于 x/2 + 3 = 7 这样的方程，可以将每一项都乘以分母，以清除分数。
x/2 + 3 = 7 → 2(x/2) + 2(3) = 2(7) → x + 6 = 14 → x = 8
If there is more than one fraction, multiply by the lowest common denominator.
如果有多个分数，则乘以最小公分母。
Always check whether the fraction is attached to the whole side or only to one term.
始终要检查分数是作用于整个一边，还是只作用于其中一项。
8. Equations with Decimals | 含小数的方程
Equations with decimals can be handled by multiplying both sides by 10, 100 or 1000.
含小数的方程可以通过两边同时乘以 10、100 或 1000 来处理。
0.5x + 0.25 = 0.75
Multiply every term by 100 to remove decimals:
每一项乘以 100，去掉小数点：
50x + 25 = 75
Subtract 25 from both sides:
两边同时减去 25：
50x = 50
Divide both sides by 50:
两边同时除以 50：
x = 1
9. Common Mistakes | 常见错误
- Forgetting to do the same operation on both sides. 忘记对两边同时进行相同的运算。
- Changing the sign incorrectly when moving a term across the equals sign. 把项移到等号另一边时符号处理错误。
- Combining unlike terms, such as adding 3x and 3 to get 6x. 合并不同类项，例如把 3x 和 3 相加得到 6x。
- Only multiplying part of an equation when clearing brackets or fractions. 清除括号或分数时，只乘以方程的一部分。
- Not checking the final solution by substitution. 没有通过代入检验最终的解。
10. Exam-Style Practice | 考试风格练习
Try these questions. Expand and solve where necessary.
尝试以下练习。必要时先展开再求解。
- 7x − 4 = 3x + 8
- 5(x + 2) = 3(x + 6)
- 12 − 2x = x + 3
- x/3 + 5 = 9
- 0.4x + 1.2 = 2.0
Answers: x = 3, x = 4, x = 3, x = 12, x = 2.
答案：x = 3，x = 4，x = 3，x = 12，x = 2。
11. Check Your Answers by Substitution | 代入检验答案
Substitution means replacing the unknown with your solution and confirming that both sides are equal.
代入法是指用你的解替换未知数，并确认两边相等。
For 5x − 2 = 2x + 10 with x = 4:
对于 5x − 2 = 2x + 10，当 x = 4 时：
Left-hand side: 5(4) − 2 = 20 − 2 = 18
Right-hand side: 2(4) + 10 = 8 + 10 = 18
Since left-hand side = right-hand side, the solution is correct.
因为左边 = 右边，所以解是正确的。
In an exam, substitution is a quick way to eliminate wrong answers in multiple-choice questions.
在考试中，代入法是快速排除选择题中错误答案的好方法。
12. Summary Checklist | 总结清单
By the end of this topic, you should be able to:
学完本主题后，你应当能够：
- Recognise linear equations with unknowns on both sides. 识别含两边未知数的线性方程。
- Use the balancing method confidently. 熟练运用平衡法。
- Expand brackets and clear fractions or decimals. 展开括号并清除分数或小数。
- Check solutions by substitution. 通过代入检验解。
Regular practice of these steps will make equation solving automatic.
经常练习这些步骤，方程求解就会变得自动化。
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