📚 Solving Linear Equations | 解线性方程
Linear equations are the simplest type of algebraic equation, where the highest power of the variable is 1. In IGCSE Mathematics, being able to solve these equations confidently is essential, as they form the foundation for more complex topics such as simultaneous equations, quadratic equations, and functions. This article breaks down the methods step by step, provides plenty of worked examples, and highlights common pitfalls to help you master solving linear equations.
线性方程是最简单的代数方程类型，其中变量的最高次幂为1。在IGCSE数学中，能够自信地解线性方程至关重要，因为它们是更复杂主题（如联立方程、二次方程和函数）的基础。本文逐步分解解法，提供了大量范例，并强调了常见易错点，帮助您掌握解线性方程。
1. Understanding Linear Equations | 认识线性方程
A linear equation is an equation that can be written in the form ax + b = c, where a, b and c are constants and a ≠ 0. The variable x has an exponent of 1, and the graph of such an equation is always a straight line. In IGCSE exams, you will encounter equations of varying complexity, from simple one‑step equations to those that involve brackets, fractions, or variables on both sides.
线性方程是可写成 ax + b = c 形式的方程，其中 a、b 和 c 是常数且 a ≠ 0。变量 x 的指数为1，这种方程的图像总是一条直线。在IGCSE考试中，你会遇到从简单的一步方程到涉及括号、分数或两边有变量的各种复杂程度的方程。
- General form: ax + b = c or ax + b = cx + d
- Key feature: the variable is not raised to any power other than 1.
- 一般形式：ax + b = c 或 ax + b = cx + d
- 关键特征：变量只出现一次幂，即指数为1。
2. The Balancing Method | 平衡法
The golden rule for solving any linear equation is the balancing method: whatever you do to one side of the equation, you must do exactly the same to the other side. This keeps the equation balanced, like a set of scales. The aim is to isolate the variable on one side and obtain its value.
解任何线性方程的金科玉律是平衡法：对方程的一边做什么，就必须对另一边做完全相同的操作。这就像天平一样保持方程平衡。目标是分离出变量，得到它的值。
To isolate x, we use inverse operations: addition undoes subtraction, multiplication undoes division, and vice versa. Always perform operations in the reverse order of BIDMAS/BODMAS (i.e., deal with addition/subtraction before multiplication/division).
为了分离 x，我们使用逆运算：加法消除减法，乘法消除除法，反之亦然。始终按照 BIDMAS/BODMAS 的相反顺序执行操作（即先处理加减，再处理乘除）。
x + 3 = 10 → x = 10 − 3 → x = 7
2x = 14 → x = 14 ÷ 2 → x = 7
3. Solving One‑Step Equations | 解一步方程
One‑step equations require exactly one operation to isolate the variable. They are the simplest type and build confidence in using inverse operations. For instance, if the equation is x + 5 = 13, subtract 5 from both sides to get x = 8. If the equation is 3x = 18, divide both sides by 3 to find x = 6.
一步方程只需一步运算即可分离出变量。它们是最简单的类型，有助于建立使用逆运算的信心。例如，若方程为 x + 5 = 13，两边减去5即可得到 x = 8。若方程为 3x = 18，两边除以3得到 x = 6。
Be careful with subtraction equations like x − 4 = 9: the inverse operation is addition, so x = 9 + 4 = 13. With division equations such as x/5 = 7, multiply both sides by 5 to get x = 35.
对减法方程如 x − 4 = 9 要小心：逆运算是加法，因此 x = 9 + 4 = 13。对于除法方程如 x/5 = 7，两边同乘以5，得 x = 35。
| Equation | Operation | Solution |
|---|---|---|
| y + 7 = 15 | Subtract 7 | y = 8 |
| t − 3 = 10 | Add 3 | t = 13 |
| 4m = 28 | Divide by 4 | m = 7 |
| p⁄6 = 12 | Multiply by 6 | p = 72 |
4. Solving Two‑Step Equations | 解两步方程
Two‑step equations involve two operations, such as a combination of multiplication/division and addition/subtraction. For example, in 2x + 3 = 11, the variable x is first multiplied by 2, then 3 is added. To solve, we reverse these steps: start by subtracting 3 from both sides (undo the addition), then divide by 2 (undo the multiplication).
两步方程包含两种运算，例如乘除与加减的组合。例如，在 2x + 3 = 11 中，变量 x 先乘以2，再加3。求解时，我们逆向操作：先两边减去3（消除加法），然后除以2（消除乘法）。
The order of undoing is crucial: always eliminate the constant term (addition/subtraction) before dealing with the coefficient of x. So for 2x + 3 = 11:
消除顺序至关重要：始终先处理常数项（加减），再处理 x 的系数。因此对于 2x + 3 = 11：
2x + 3 = 11
2x = 11 − 3 → 2x = 8
x = 8 ÷ 2 → x = 4
Another example: 5 − 3y = 14 , then divide by −3 to get y = −3. Note the negative coefficient often causes sign errors.
另一个例子：5 − 3y = 14。变量乘以 −3（因为 −3y 表示 −3 × y），然后加5。首先两边减5：−3y = 9，然后除以 −3 得到 y = −3。注意负系数常导致符号错误。
5. Equations with Brackets | 带括号的方程
When an equation contains brackets, your first task is to expand them using the distributive property. For example, 3(x + 2) = 15 becomes 3x + 6 = 15. Then proceed with the two‑ solving method: subtract 6 from both sides to get 3x = 9, then divide by 3 to find x = 3.
当方程含有括号时，首要任务是运用分配律展开括号。例如，3(x + 2) = 15 变为 3x + 6 = 15。然后按两步法求解：两边减去6得到 3x = 9，再除以3得到 x = 3。
Be especially careful when a negative sign or a number is in front of the bracket: −2(4x − 1) = 10 d −2 × (−1) = +2, giving −8x + 2 = 10. Then subtract 2: −8x = 8, and divide by −8: x = −1.
当括号前有负号或数字时要格外小心：−2(4x − 1) = 10。仔细展开：−2 × 4x = −8x，−2 × (−1) = +2，得到 −8x + 2 = 10。然后减2：−8x = 8，除以 −8：x = −1。
Sometimes it is easier to divide both sides by the coefficient of the bracket first. In 4(x + 5) = 32 t 5: x = 3. This method is valid as long as you divide the entire equation by the same non‑zero number.
有时先将两边除以括号前的系数会更简便。在 4(x + 5) = 32 中，可以两边先除以4得到 x + 5 = 8，再减5：x = 3。只要将整个方程除以相同的非零数，这种方法就是可行的。
6. Equations with Fractions | 含分数的方程
Fractions in equations often worry students, but they can be eliminated efficiently by multiplying every term on both sides by the least common denominator (LCD). For example, consider x/3 + 2 = 5 get x = 9. But when the fraction is part of a sum, clearing denominators is often easier.
方程中的分数常让学生担忧，但通过将两边每一项都乘以最小公分母（LCD），可以有效消去分母。例如，考虑 x/3 + 2 = 5。两边减2：x/3 = 3，然后乘以3得 x = 9。但当分数是求和的一部分时，消去分母通常更易处理。
In an equation like (x + 1)/2 = 4, think of it as 'the whole numerator divided by 2'. Multiply both sides by 2: x + 1 = 8, then x = 7. For more complex fractions such as (2x)/5 + 1/3 = 2, find the LCD of 5 and 3, which is 15, and multiply every term by 15:
在像 (x + 1)/2 = 4 这样的方程中，将它看作“整个分子除以2”。两边乘以2：x + 1 = 8，然后 x = 7。对于更复杂的分数如 (2x)/5 + 1/3 = 2，找出5和3的最小公分母，即15，并将每一项乘以15：
15 × (2x/5) + 15 × (1/3) = 15 × 2
3 × 2x + 5 = 30 → 6x + 5 = 30
6x = 25 → x = 25/6
Always simplify the fraction in the answer if possible; mixed numbers are acceptable but improper fractions are preferred in IGCSE.
若可能，始终化简答案中的分数；带分数也可接受，但在IGCSE中更推荐假分数形式。
7. Equations with Variables on Both Sides | 两边都有变量的方程
When an equation has the variable on both sides, you need to collect all variable terms on one side and all constant terms on the other. The rule is: remove the smaller variable term first to avoid negative coefficients if possible. Example: 5x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4.
当方程两边都含有变量时，需要将所有含变量的项移到一边，所有常数项移到另一边。原则是：优先去掉较小的变量项，以避免负系数（如果可能）。例子：5x + 2 = 3x + 10。两边减去 3x：2x + 2 = 10。然后减2：2x = 8，所以 x = 4。
If the coefficient of x becomes negative, like in 2x − 3 = 7x + 8 tively, you could subtract 7x, but that leads to negative x, which you then multiply by −1. The method is flexible as long as the equation stays balanced.
如果 x 的系数为负，如 2x − 3 = 7x + 8，两边减去 2x 得到 −3 = 5x + 8，再减8：−11 = 5x，最后除以5：x = −11/5。或者，也可以减7x，但那样会导致负 x，然后再乘以 −1。只要方程保持平衡，方法可以灵活选择。
A special variant: 4x − 6 = 2(2x − 1) 4x from both sides yields −6 = −2, which is false. This means the equation has no solution — the lines are parallel and never intersect. We'll look at this later.
一种特殊变式：4x − 6 = 2(2x − 1)。展开右边：4x − 6 = 4x − 2。两边减 4x 得到 −6 = −2，这是错误的。意味着方程无解——两条线平行且永不相交。我们稍后将探讨此情况。
8. Special Cases: No Solution and All Real Numbers | 特殊情况：无解与全体实数
Not all linear equations have a unique solution. Some have no solution (inconsistent), and some have infinitely many solutions (identity). Recognising these cases is an important skill for IGCSE.
并非所有线性方程都有唯一解。有些无解（矛盾方程），有些有无穷多解（恒等式）。识别这些情况是IGCSE的一项重要技能。
An equation with no solution 4 = −5, which is impossible. Such an equation is called inconsistent, and you should write 'no solution' or ∅.
当化简后得到一个错误命题，如 3 = 7，方程即无解。例如：2x + 4 = 2x − 5。两边减 2x：4 = −5，这是不可能的。这类方程称为矛盾方程，应答“无解”或 ∅。
An identity x − 6. Subtracting 3x from both sides gives −6 = −6, which is always true. The solution is 'all real numbers' or 'x ∈ ℝ'.
当任何 x 值都能满足方程时，即出现恒等式。例如：3(x − 2) = 3x − 6。左边展开：3x − 6 = 3x − 6。两边减 3x 得 −6 = −6，恒成立。此时解为“全体实数”或“x ∈ ℝ”。
| Case | After simplification | Conclusion |
|---|---|---|
| Unique solution | x = a number | ex. x = 5 |
| No solution | a false equality, like 2 = 3 | no solution |
| Infinite solutions | an identical equality, like 0 = 0 | all real numbers |
9. Checking Your Solution | 检验你的解
Substituting your answer back into the original equation is the best way to verify correctness. Take the equation 7 − 2x = 3x + 12 9. On the right: 3(−1) + 12 = −3 + 12 = 9. Both sides equal 9, so the solution is correct.
将答案代回原方程是验证正确性的最佳方法。以方程 7 − 2x = 3x + 12 和解 x = −1 为例。左边代入 −1：7 − 2(−1) = 7 + 2 = 9。右边：3(−1) + 12 = −3 + 12 = 9。两边都等于9，解正确。
If the sides do not match, you have made an error. Re‑check each step: did you apply the inverse operation correctly? Did you handle negative signs carefully? Common mistakes include forgetting to change the sign when moving a term, distributing incorrectly over brackets, or arithmetic slips. Checking helps catch these.
如果两边不相等，则存在错误。重新检查每一步：逆运算是否正确？负数符号是否处理得当？常见错误包括移项时忘记变号、括号分配不当或计算失误。检验有助于发现这些问题。
10. Solving Word Problems Using Linear Equations | 利用线性方程解文字题
Many IGCSE problems are set in real‑life contexts, requiring you to form a linear equation from a written description. Follow these steps: (1) Let the unknown be x (or another letter). (2) Express the given information as an algebraic expression. (3) Form an equation by linking the expressions with an equality. (4) Solve the equation. (5) Answer the question, including units if needed.
许多IGCSE题目基于实际情景，需要你从文字描述列出线性方程。遵循以下步骤：(1) 设未知数为 x（或其它字母）。(2) 将所给信息表示为代数式。(3) 通过等号将表达式联立形成方程。(4) 解方程。(5) 回答问题，需要时包括单位。
Example: 'The sum of three consecutive numbers is 72. Find the numbers.' n: x + (x+1) + (x+2) = 72 → 3x + 3 = 72 → 3x = 69 → x = 23. So the numbers are 23, 24, 25.
举例：“三个连续整数的和为72。求这三个数。”设最小数为 x，则另两个为 x+1 和 x+2。方程：x + (x+1) + (x+2) = 72 → 3x + 3 = 72 → 3x = 69 → x = 23。因此这三个数为 23, 24, 25。
Another typical problem: 'A rectangle's length is 3 cm more than twice its width. The perimeter is 36 cm. Find the dimensions.' w) = 2(3w+3) = 6w+6. Set equal to 36: 6w+6=36 → 6w=30 → w=5. Width = 5 cm, length = 13 cm.
另一典型题：“一矩形的长比宽的两倍多3厘米，周长为36厘米。求尺寸。”设宽为 w，则长 = 2w + 3。周长 = 2(长+宽) = 2(2w+3 + w) = 2(3w+3) = 6w+6。令其等于36：6w+6=36 → 6w=30 → w=5。宽5厘米，长13厘米。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Even learners who understand the method can lose marks due to careless errors. Watch out for: forgetting to apply an operation to both sides, incorrectly expanding −(x − 3) as −x − 3 instead of −x + 3, dividing by the coefficient without considering the sign, and mishandling fractions. Writing out every small step reduces these mistakes.
即使理解方法的学生也可能因粗心而失分。注意：忘记对两边进行同样操作、错误地将 −(x − 3) 展开为 −x − 3 而非 −x + 3、除以系数时忽略符号、以及分数处理不当。写出每一个小步骤可以减少这些错误。
A common pitfall: solving 5x = x + 8 x = 0 is a solution. In linear equations, x cannot be cancelled unless you know it is non‑zero, but we generally don't divide by the variable; we collect terms.
常见陷阱：解 5x = x + 8 时两边都减去5，这是无效的。正确步骤是两边减去 x：4x = 8，然后 x = 2。另一个：在不检查 x = 0 是否为解的情况下除以 x。在线性方程中，除非知道变量非零，否则通常不用除以变量；我们采用移项。
Always present your solution clearly, showing the variable equals a specific number or stating 'no solution' / 'all real numbers' as appropriate. In the exam, clear working can earn method marks even if the final answer is wrong.
务必清晰地展示解答，表明变量等于某个具体数值，或适当写明“无解”/“全体实数”。在考试中，清晰的步骤即使最终答案有误，也能赢得方法分。
12. Practice Questions and Summary | 练习题与总结
Now put your skills to the test with these equations. Try to solve each without peeking at the answers first. For each, remember to check your solution by substitution.
现在用以下方程检验你的技能。尽量不看答案先自行解答。每一题都要记得通过代入法检验你的解。
- 4x − 7 = 21
- 3(2y + 1) = 27
- (t + 3)/5 = 2
- 8 − 2z = 4z + 14
- 5(x − 2) = 3(x + 4)
- 6 − (2a + 1) = 3a − 5
Answers: x = 7, y = 4, t = 7, z = −1, x = 11, a = 2. If you didn't get these, re‑work the steps while referring to the relevant section above.
答案：x = 7, y = 4, t = 7, z = −1, x = 11, a = 2。如果你没有得到这些结果，参照上文相关小节重新演算。
Linear equations are a cornerstone of IGCSE Mathematics. Mastering them will build your confidence in algebra and pave the way for tackling simultaneous equations, inequalities, and quadratic equations. Practice regularly, learn from your mistakes, and you'll find that solving equations becomes second nature.
线性方程是IGCSE数学的基石。掌握它们将树立你代数方面的信心，并为处理联立方程、不等式和二次方程铺平道路。定期练习，从错误中学习，你将发现解方程变得驾轻就熟。
Published by TutorHao | Mathematics 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