Simultaneous Equations | 联立方程

📚 Simultaneous Equations | 联立方程

Simultaneous equations are a pair (or more) of equations that share two or more unknown variables. In IGCSE Mathematics, you are expected to solve them using algebraic or graphical methods. This article covers the substitution method, elimination method, graphical approach, and real-world applications, with clear examples and exam-style tips.

联立方程是一组共享两个或更多未知数的方程（通常是两个方程）。在 IGCSE 数学中，你需要用代数方法或图像方法求解它们。本文将讲解代入法、消元法、图像法以及实际应用，并提供清晰的例题与考试技巧。

1. What Are Simultaneous Equations? | 什么是联立方程？

When two variables, such as x and y, appear in two different equations, the equations form a system. A solution of the system is a pair of values (x, y) that satisfies both equations at the same time. For example:

当两个变量（例如 x 和 y）同时出现在两个不同的方程中时，这些方程就构成一个方程组。方程组的解是一对数值 (x, y)，它能同时满足两个方程。例如：

2x + y = 10
x − y = 2

Here, x = 4 and y = 2 satisfy both equations because 2(4) + 2 = 10 and 4 − 2 = 2. A system of linear equations will usually have one unique solution, but may also have no solution or infinitely many solutions.

这里 x = 4 和 y = 2 同时满足两个方程，因为 2(4) + 2 = 10 且 4 − 2 = 2。一个线性方程组通常有唯一解，但也可能无解或有无限多组解。

2. Method 1: Substitution | 方法一：代入法

The substitution method is ideal when one equation has a variable with coefficient 1. You solve one equation for one variable, then substitute that expression into the other equation.

代入法特别适合其中一个方程中某个变量的系数为 1 的情况。先从一个方程解出一个变量，然后将该表达式代入另一个方程。

Example / 例题:

y = 3x − 5 ①
2x + y = 10 ②

Substitute ① into ②: 2x + (3x − 5) = 10. Simplify to 5x − 5 = 10, so 5x = 15, x = 3. Then y = 3(3) − 5 = 4. Solution: (3, 4).

将①代入②：2x + (3x − 5) = 10。整理得 5x − 5 = 10，所以 5x = 15，x = 3。再代回①得 y = 3(3) − 5 = 4。解为 (3, 4)。

Key steps for substitution: (1) Make one variable the subject; (2) Replace that variable in the other equation; (3) Solve the resulting linear equation; (4) Substitute back to find the second variable; (5) Check your answer.

代入法的关键步骤：(1) 将一个变量表示成另一个变量的表达式；(2) 在另一个方程中替换该变量；(3) 解所得的一元一次方程；(4) 代回求第二个变量；(5) 检查答案。

3. Method 2: Elimination | 方法二：消元法

The elimination method involves adding or subtracting the equations to cancel one variable. First, align like terms vertically. Then multiply one or both equations so that the coefficients of one variable are equal in magnitude.

消元法是通过两个方程相加或相减来消去一个变量。首先将同类项上下对齐，然后使某个变量的系数绝对值相等，再相加或相减。

Example / 例题:

3x + 2y = 13 ①
2x + 2y = 10 ②

Subtract ② from ①: (3x − 2x) + (2y − 2y) = 13 − 10, so x = 3. Substitute into ②: 2(3) + 2y = 10, therefore 2y = 4, y = 2. Solution: (3, 2).

① − ②：(3x − 2x) + (2y − 2y) = 13 − 10，得 x = 3。代入②：2(3) + 2y = 10，所以 2y = 4，y = 2。解为 (3, 2)。

If coefficients are not equal, multiply one or both equations first. For example, to eliminate y from 2x + 3y = 8 and 5x − 2y = 1, multiply the first by 2 and the second by 3.

若系数不同，需先乘一个数或两个方程。例如要消去 y，在 2x + 3y = 8 和 5x − 2y = 1 中，可将第一个方程乘以 2，第二个方程乘以 3。

Original 原方程	Multiply by 乘以	New 新方程
2x + 3y = 8	× 2	4x + 6y = 16
5x − 2y = 1	× 3	15x − 6y = 3

Adding gives 19x = 19, so x = 1. Then y = 2. Always verify by substituting both values back into the original equations.

两式相加得 19x = 19，所以 x = 1。再代入原方程得 y = 2。务必把两个值代回原方程验证。

4. Method 3: Graphical Solution | 方法三：图像法

A linear equation can be drawn as a straight line on a coordinate grid. The solution of a pair of simultaneous equations is the point where the two lines intersect. To use this method, first rearrange each equation into the form y = mx + c or find two points on each line.

二元一次方程在坐标平面上对应一条直线。联立方程组的解是两条直线的交点。使用图像法时，先把每个方程改写为 y = mx + c 的形式，或者在每条直线上找到两个点。

Example / 例题:

x + y = 6
y = 2x

For x + y = 6, you can plot (0, 6) and (6, 0). For y = 2x, plot (0, 0) and (2, 4). The two lines meet at (2, 4), which is the solution.

对于 x + y = 6，可描点 (0, 6) 和 (6, 0)。对于 y = 2x，描点 (0, 0) 和 (2, 4)。两条直线相交于点 (2, 4)，即为方程组的解。

Graphical solutions are useful for estimating answers, but algebraic methods are usually preferred for exact values. In an exam, if you use a graph, you must draw the lines carefully and write down the intersection point clearly.

图像法适合估算答案，但要想得到精确值通常还是用代数方法更好。考试中若使用图像法，必须精确画线，并清楚写出交点坐标。

5. Choosing the Best Method | 如何选择最优方法

Deciding whether to use substitution or elimination can save time in an exam. The table below summarises the main criteria.

在考试中选择代入法还是消元法可以节省时间。下表总结了主要的判断依据。

Situation 情况	Recommended method 推荐方法
One equation has y = or x = already 某个方程已经是 y = 或 x = 形式	Substitution 代入法
Coefficients of x or y are equal 某个变量的系数相同	Elimination by subtraction 相减消元
Coefficients are opposites 某个变量的系数互为相反数	Elimination by addition 相加消元
Equations are already in y = mx + c form 方程已是 y = mx + c 形式	Graphical method 图像法

In many cases, elimination is faster when both equations are in the form ax + by = c. Substitution is easier when one variable is already isolated. Practise both until you can choose instantly.

当两个方程都是 ax + by = c 的形式时，消元法通常更快。当某个变量已被单独表示时，代入法更方便。多练习两种方法，直到能迅速判断为准。

6. Word Problems with Simultaneous Equations | 联立方程解决应用题

Real-world problems often give two pieces of information that can be translated into equations. The first step is always to define your variables clearly.

实际应用题通常给出两个信息，可转化为两个方程。第一步永远是清晰地定义未知数。

Example / 例题: A shop sells apples and bananas. Two apples and three bananas cost $2.90. One apple and four bananas cost $2.45. Find the price of each fruit.

例题：一家商店出售苹果和香蕉。两个苹果和三个香蕉共 2.90 元，一个苹果和四个香蕉共 2.45 元。求每种水果的单价。

Let a be the price of an apple and b be the price of a banana. Then:

设 a 为苹果单价，b 为香蕉单价。则：

2a + 3b = 2.90 ①
a + 4b = 2.45 ②

Multiply ② by 2: 2a + 8b = 4.90. Subtract ①: (2a + 8b) − (2a + 3b) = 4.90 − 2.90, so 5b = 2.00, b = 0.40. Substitute into ②: a + 4(0.40) = 2.45, so a = 0.85. Therefore an apple costs $0.85 and a banana costs $0.40.

将②乘以 2：2a + 8b = 4.90。再与①相减：(2a + 8b) − (2a + 3b) = 4.90 − 2.90，得 5b = 2.00，b = 0.40。代入②：a + 4(0.40) = 2.45，所以 a = 0.85。因此苹果单价 0.85 元，香蕉单价 0.40 元。

When solving word problems, always include units in your final answer and check that values make sense in the original context. For example, a negative price would be impossible.

解应用题时，最终答案要带上单位，并检验数值在实际情况中是否合理。例如，价格为负数就不合理。

7. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解

Not every pair of simultaneous linear equations has a unique solution. If the two lines are parallel, they never intersect, and the system has no solution. If the two equations represent the same line, they intersect everywhere, giving infinitely many solutions.

并非所有联立线性方程都有唯一解。若两条直线平行，它们永不相交，方程组无解。若两个方程表示同一条直线，则它们处处相交，有无限多组解。

Example of no solution / 无解例子:

x + y = 5
2x + 2y = 9

Multiplying the first equation by 2 gives 2x + 2y = 10, which contradicts 2x + 2y = 9. The lines are parallel with different intercepts, so no solution exists.

将第一个方程乘以 2 得 2x + 2y = 10，与 2x + 2y = 9 矛盾。两条直线平行且截距不同，因此无解。

Example of infinite solutions / 无穷多解例子:

x + y = 3
3x + 3y = 9

The second equation is just the first multiplied by 3. They are the same line, so every point on the line is a solution.

第二个方程只是第一个方程乘以 3 的结果。它们表示同一条直线，所以直线上的每一个点都是解。

8. Non-Linear Simultaneous Equations | 非线性联立方程

IGCSE Extended Mathematics also covers cases where one equation is linear and the other is quadratic. These are solved primarily by substitution.

IGCSE 扩展数学还会考察一个线性方程与一个二次方程联立的情况，通常使用代入法求解。

Example / 例题:

y = x² − 2x + 1 ①
y = x + 3 ②

Substitute ② into ①: x + 3 = x² − 2x + 1. Rearrange: x² − 3x − 2 = 0? Wait, check: x + 3 = x² − 2x + 1 → 0 = x² − 3x − 2. Actually, subtract x and 3 from both sides: x² − 2x + 1 − x − 3 = 0, so x² − 3x − 2 = 0. However this quadratic does not factor nicely; we can solve using the quadratic formula:

将②代入①：x + 3 = x² − 2x + 1。移项得 x² − 3x − 2 = 0。这个二次方程不易因式分解，我们用求根公式来解：

x = (3 ± √(9 + 8)) / 2 = (3 ± √17) / 2

Then find y by substituting each x into y = x + 3. Such solutions may be left in surd form. In the exam, they will usually choose examples that factorise, but you should still be comfortable with the formula.

然后分别把每个 x 代入 y = x + 3 求出对应的 y。这类解可保留根号形式。考试通常选择能因式分解的例题，但你仍需熟练使用公式法。

9. Exam Tips and Common Mistakes | 考试技巧与常见错误

Below are common pitfalls and ways to avoid them.

以下是一些常见误区和相应的避免方法。

Sign errors when subtracting / 相减时符号错误: When you subtract one equation from another, be careful with negative terms. Write each row clearly.

相减时符号错误：一个方程减去另一个方程时，要注意负项。把每一行写清楚。
Forgetting to substitute back / 忘记代回求另一个变量: Some students solve for x and stop. Always substitute to find y.

忘记代回求另一个变量：有些学生求出 x 就停止了。一定要代回求 y。
Not checking answers / 不检查答案: Substitute both values into the original equations to verify your solution. This catches many careless errors.

不检查答案：把两个值代回原方程组验证。这能发现许多粗心错误。
Using LaTeX-style notation / 使用 LaTeX 形式的记号: In an exam, just write normal numbers and symbols, not code. Use a neat layout.

使用 LaTeX 风格的记号：考试中只需要正常书写数字和符号，不要写代码。版面要整洁。

A recommended layout: number your equations, show each step on a new line, and circle or box your final answer. This also helps the examiner award partial marks.

一个推荐的排版方式：给每个方程编号，每一步换行书写，最后把答案圈出或框出。这样也便于阅卷老师按步骤给分。

10. Practice Questions | 练习题目

Try the following questions on your own before checking the answers.

请先独立尝试以下题目，再对照答案。

Q1 / 题1: Solve by elimination: 2x + 5y = 16 and 3x − 5y = 4.

用消元法解：2x + 5y = 16 和 3x − 5y = 4。

Answer / 答案: Adding gives 5x = 20, x = 4. Then 2(4) + 5y = 16, so 5y = 8, y = 1.6. Solution: (4, 1.6).

Q2 / 题2: Solve by substitution: y = 2x + 1 and x + y = 10.

用代入法解：y = 2x + 1 和 x + y = 10。

Answer / 答案: x + (2x + 1) = 10 → 3x = 9, x = 3. Then y = 7. Solution: (3, 7).

Q3 / 题3: The sum of two numbers is 21 and their difference is 5. Let the numbers be a and b. Find a and b.

题3：两个数的和为 21，差为 5。设这两个数为 a 和 b，求 a 和 b。

Answer / 答案: a + b = 21, a − b = 5. Adding gives 2a = 26, a = 13, then b = 8.

For each question, always check the solution in both original equations.

每道题都请把解代入两个原方程进行验证。

11. Conclusion | 结语

Simultaneous equations are a fundamental tool in algebra and appear in many IGCSE exam papers. Mastering substitution, elimination, and the graphical method gives you a flexible skill set. With consistent practice, you will quickly recognise which method to use.

联立方程是代数中的基础工具，在 IGCSE 考试中频繁出现。熟练掌握代入法、消元法和图像法，你就拥有了一套灵活的解题技能。经过持续练习，你会很快判断出该用哪种方法。

Remember to define variables in word problems, check your answers, and present your working neat and clear. Good luck!

记住在应用题中先定义变量、检查答案、书写工整清楚。祝你顺利！

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Simultaneous Equations | 联立方程