📚 IGCSE Maths: Solving Simultaneous Equations by Substitution and Elimination | IGCSE 数学：用代入法和消元法解联立方程
Simultaneous equations are a core algebraic skill in the IGCSE Mathematics syllabus. They help us find values that satisfy two or more conditions at the same time. This article focuses on solving two linear equations in two unknowns, using both algebraic and graphical approaches, and shows you how to check your solution confidently.
联立方程是 IGCSE 数学大纲中的核心代数技能。它们帮助我们找到同时满足两个或多个条件的未知数的值。本文将重点讲解如何求解含有两个未知数的两个线性方程，包括代数方法和图解方法，并教你如何自信地检验答案。
1. What Are Simultaneous Equations? | 什么是联立方程？
must both be true at the same time. A solution is a pair of values for x and y that satisfies every equation in the system.
联立方程是指两个或两个以上共享相同变量的方程。例如，x + y = 7 和 2x − y = 2 必须同时成立。解就是一组 x 和 y 的值，能够同时满足方程组中的每一个方程。
- Linear simultaneous equations have variables raised only to the power 1.
- 线性联立方程中的变量指数仅为 1。
- The solution can be written as an ordered pair (x, y), such as (3, 4).
- 解可以写成有序数对 (x, y)，例如 (3, 4)。
- In IGCSE exams, you may need to solve algebraically or interpret a graph.
- 在 IGCSE 考试中，你可能需要用代数方法求解，或者解读图像。
2. What Does a Solution Mean? | 解的意义是什么？
If the lines cross at (2, 5), then x = 2 and y = 5 satisfy both equations simultaneously.
从几何角度看，每个线性方程都表示一条直线。联立方程的解就是两条直线相交的点。如果两条直线在 (2, 5) 处相交，那么 x = 2 和 y = 5 就同时满足两个方程。
Line 1: x + y = 7 Line 2: 2x − y = 2 → Intersection at (3, 4)
You can check a solution by substituting the values back into both original equations. If both sides balance, your solution is correct.
你可以将数值代回两个原方程来检验解。如果两边都相等，那么你的解就是正确的。
3. Graphical Method | 图解法
The graphical method involves plotting both equations on the same coordinate grid. The point where the two lines meet gives the solution. It is a useful visual check even when you later use algebra.
图解法是在同一坐标网格上绘制两个方程的图像。两条直线相交的点就是解。即使你之后使用代数方法，图解也是一种有用的可视化检验手段。
- ange each equation into y = mx + c form, where m is the gradient and c is the y-intercept.
- 将每个方程改写为 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。
- Plot both lines accurately on graph paper.
- 在方格纸上精确绘制两条直线。
- Read the coordinates of the intersection point.
- 读取交点坐标。
Graphical solutions may be less precise than algebraic ones, especially when the intersection is not at integer coordinates. That is why IGCSE often asks for algebraic methods.
图解法的精度可能不如代数方法，尤其是当交点不在整数坐标时。因此 IGCSE 考试通常要求使用代数方法。
4. Substitution Method | 代入法
when a variable can be isolated quickly. You replace that variable in the other equation.
当代入法适用于其中一个方程已经将某个变量单独表示出来时，例如 y = 2x + 1，或者某个变量可以快速分离时。你可以在另一个方程中替换该变量。
: 3x + (2x + 1) = 11. This gives 5x + 1 = 11, so 5x = 10 and x = 2. Then y = 2(2) + 1 = 5.
例题：解 y = 2x + 1 和 3x + y = 11。将 y = 2x + 1 代入第二个方程：3x + (2x + 1) = 11。得到 5x + 1 = 11，所以 5x = 10，x = 2。然后 y = 2(2) + 1 = 5。
Solution: x = 2, y = 5
Always substitute back into the original equation you did not use to avoid copying errors.
始终将结果代回你未使用的原方程，以避免复制错误。
5. Elimination Method | 加减消元法
The elimination method works by adding or subtracting the equations to remove one variable. You may need to multiply one or both equations by a constant first so that the coefficients of one variable match.
加减消元法通过将两个方程相加或相减来消去一个变量。你可能需要先给一个或两个方程乘以一个常数，使某个变量的系数匹配。
minates y: 6x = 24, so x = 4. Substitute x = 4 into 2x + 3y = 13: 8 + 3y = 13, so y = 5/3.
例题：解 2x + 3y = 13 和 4x − 3y = 11。将两个方程相加即可消去 y：6x = 24，所以 x = 4。将 x = 4 代入 2x + 3y = 13：8 + 3y = 13，所以 y = 5/3。
2x + 3y = 13 + 4x − 3y = 11 → 6x = 24
If the coefficients are not opposites, multiply the equations before adding or subtracting. The goal is to make one pair of coefficients equal in size but opposite in sign, or equal in size and same sign.
如果系数不是相反数，请先对方程乘以合适的数再相加或相减。目标是让某一对系数大小相等但符号相反，或者大小相等且符号相同。
6. Choosing the Best Method | 选择最佳方法
y = ... or x = ... . Use elimination when both equations are in standard ax + by = c form.
两种方法都能得到相同答案，但选择合适的方法可以节省时间。当一个方程已经是 y = ... 或 x = ... 的形式时使用代入法。当两个方程都是标准形式 ax + by = c 时使用加减消元法。
| Situation | Recommended method |
| y = 2x − 3 and x + y = 9 | Substitution |
| 3x + 2y = 5 and 4x − 2y = 10 | Elimination |
| 2x + y = 7 and x − y = 1 | Either method |
In exams, if you are not sure, elimination is often easier to keep neat and avoid substitution mistakes.
在考试中，如果你不确定，加减消元法通常更容易保持整洁并避免代入错误。
7. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
ay end up with a false statement such as 0 = 5. This means the equations are inconsistent.
当两条直线平行时，联立方程无解。在代数上，你可能最终得到 0 = 5 这样的矛盾式。这意味着方程组是不一致的。
, you may end up with 0 = 0 or y = y, which means one equation is a multiple of the other.
当两个方程表示同一条直线时，存在无穷多个解。在代数上，你可能最终得到 0 = 0 或 y = y，这意味着其中一个方程是另一个方程的倍数。
- No solution: lines are parallel, gradients equal but y-intercepts different.
- 无解：直线平行，斜率相同但 y 轴截距不同。
- Infinite solutions: lines coincide, one equation is a multiple of the other.
- 无穷多解：直线重合，一个方程是另一个方程的倍数。
- Unique solution: lines intersect at one point.
- 唯一解：直线相交于一点。
8. Word Problems Leading to Simultaneous Equations | 列联立方程解应用题
Many IGCSE questions are word problems. You need to define two variables, form two equations from the given information, and then solve them using substitution or elimination.
许多 IGCSE 题目是文字题。你需要定义两个变量，根据已知信息列出两个方程，然后用代入法或加减消元法求解。
Example: A pen and a notebook cost $5. Two pens and three notebooks cost $ e the price of a notebook. Then x + y = 5 and 2x + 3y = 13. Solving gives x = 2 and y = 3.
例题：一支笔和一个笔记本共 5 美元。两支笔和三个笔记本共 13 美元。设 x 为笔的价格，y 为笔记本的价格。则 x + y = 5 和 2x + 3y = 13。解得 x = 2，y = 3。
x + y = 5 and 2x + 3y = 13 → x = 2, y = 3
Always label your answers with units and check whether the values make sense in the original context.
始终为答案标注单位，并检查这些值在原始情境中是否合理。
9. Common Mistakes to Avoid | 常见错误
Students often lose marks by making small but avoidable mistakes in simultaneous equations. Watch out for sign errors when subtracting equations and for forgetting to multiply every term when scaling an equation.
学生在解联立方程时经常因为一些细小但可避免的错误而失分。减去方程时要注意符号错误，并且缩放方程时不要忘记乘以每一项。
- Forgetting to multiply all terms by the same constant.
- 忘记对所有项乘以相同的常数。
- Adding instead of subtracting when coefficients have the same sign.
- 当系数符号相同时应该相减却相加了。
- Substituting back into the equation that was already used.
- 将结果代回已经使用过的方程。
- Not simplifying the final answer fully.
- 没有完全化简最终答案。
To avoid these, write each step on a new line and keep your work organised. Check your answer in both original equations.
为避免这些问题，请每一步都写在新的行上，并保持书写工整。将答案代回两个原方程进行检验。
10. How to Check Your Answer | 如何检验答案
Checking your solution is fast and can catch many errors. Substitute your x and y values into both equations. If both sides of each equation are equal, the solution is correct.
检验答案很快，而且可以发现许多错误。将你的 x 和 y 值代入两个方程。如果每个方程的两边都相等，那么解就是正确的。
= 5 in x + y = 7 and 2x − y = 2, the first gives 2 + 5 = 7 and the second gives 4 − 5 = −1? correct. Let us use a valid pair: x = 3, y = 4 gives 3 + 4 = 7 and 6 − 4 = 2, so it works.
例题：将 x = 2 和 y = 5 代入 x + y = 7 和 2x − y = 2，第一个方程得到 2 + 5 = 7，第二个得到 4 − 5 = −1。等一下，这并不正确。实际检验必须匹配：2x − y = 2 得到 4 − 5 = −1，所以会失败。这意味着解不正确。我们使用有效的数对：x = 3, y = 4，得到 3 + 4 = 7 和 6 − 4 = 2，符合.
In the exam, always use the original equations, not any rearranged or multiplied version, because that may hide a mistake.
在考试中，始终使用原始方程，而不是任何整理过或乘以常数后的版本，因为那可能掩盖错误。
11. Worked Exam-Style Example | 考试风格例题精讲
e simultaneous equations 5x + 2y = 16 and 3x − y = 1. Give your answer as an ordered pair.
题目：解联立方程 5x + 2y = 16 和 3x − y = 1。将答案写成有序数对的形式。
Step 1: Label the equations: (1) 5x + 2y = 16 and (2) 3x − y = 1.
步骤 1：给方程编号：(1) 5x + 2y = 16 和 (2) 3x − y = 1。
Step 2: Multiply equation (2) by 2 to make the y coefficients opposites: 6x − 2y = 2.
步骤 2：将方程 (2) 乘以 2，使 y 的系数互为相反数：6x − 2y = 2。
Add this to equation (1): 5x + 2y + 6x − 2y = 16 + 2, which gives 11x = 18, so x = 18/11.
步骤 3：将其与方程 (1) 相加：5x + 2y + 6x − 2y = 16 + 2，得到 11x = 18，所以 x = 18/11。
nto (2): 3(18/11) − y = 1, so 54/11 − y = 1, giving y = 54/11 − 1 = 54/11 − 11/11 = 43/11.
步骤 4：将 x = 18/11 代入 (2)：3(18/11) − y = 1，所以 54/11 − y = 1，得到 y = 54/11 − 1 = 54/11 − 11/11 = 43/11。
Solution: (18/11, 43/11)
Even when answers are fractions, do not change them to decimals unless the question asks for a decimal answer. Simplified fractions are preferred in IGCSE.
即使答案是分数，也不要将其换成小数，除非题目要求用小数作答。在 IGCSE 中，化简后的分数更受欢迎。
12. Summary and Practice Tips | 总结与练习建议
Simultaneous equations are predictable and highly examinable. Master both substitution and elimination, and make checking your answer a habit. Practice with positive, negative, and fractional coefficients to build confidence.
联立方程题型规律性强，是考试重点。掌握代入法和加减消元法，并养成检验答案的习惯。练习正数、负数和分数系数的题目，以增强信心。
| Skill | Why it matters |
| Rearranging equations | Needed for substitution and graphing |
| Multiplying equations by constants | Core step in elimination |
| Checking solutions | Catches sign and arithmetic errors |
Work through past paper questions, especially those that combine simultaneous equations with fractions or word problems. Time yourself to improve accuracy under exam conditions.
练习历年真题，尤其是将联立方程与分数或文字题结合的题目。给自己计时，以提高考试环境下的准确性。
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