Topic 214: Simultaneous Equations | 专题214：联立方程

📚 Topic 214: Simultaneous Equations | 专题214：联立方程

In Edexcel IGCSE Mathematics, simultaneous equations are equations that contain two or more unknown quantities. Solving them means finding the exact values that make all of the equations true at the same time. This article covers the substitution method, the elimination method, the graphical method, word problems, and non-linear cases, with exam-focused advice throughout.

在 Edexcel IGCSE 数学中，联立方程是指含有两个或两个以上未知量的方程。解联立方程，就是找到一组能使所有方程同时成立的未知数值。本文将系统讲解代入消元法、加减消元法、图像法、应用题以及非线性情形的解法，并贯穿考试要点。

1. What Are Linear Simultaneous Equations? | 1. 什么是线性联立方程？

A linear simultaneous equation pair has the form ax + by = c, where a, b and c are constants. Each equation represents a straight line on a coordinate grid. If the two lines intersect at one point, that point gives the unique solution to the pair of equations.

线性联立方程一般写成 ax + by = c 的形式，其中 a、b、c 为常数。每个方程在坐标平面中都表示一条直线。如果两条直线交于一个点，那么该交点的坐标就是这个方程组的唯一解。

Consider the simple example:

看一个简单例子：

x + y = 5
x – y = 1

Adding the two equations gives 2x = 6, so x = 3. Substituting x = 3 into the first equation gives 3 + y = 5, so y = 2. Therefore the solution is x = 3, y = 2.

两式相加得 2x = 6，所以 x = 3。把 x = 3 代入第一个方程，得 3 + y = 5，因此 y = 2。所以方程组的解是 x = 3，y = 2。

2. The Substitution Method | 2. 代入消元法

The substitution method is particularly useful when one equation is already arranged with one variable as the subject, or when one variable is easy to isolate.

代入消元法特别适合其中一个方程已经将某个变量写成“主语”的情况，或者其中一个变量很容易单独分离出来的情形。

Follow these steps:

解题步骤如下：

Rearrange one equation to express one variable in terms of the other. 把其中一个方程变形，使一个未知数用另一个未知数表示。
Substitute this expression into the other equation. 把这个表达式代入另一个方程。
Solve the resulting one-variable equation. 解这个只含一个未知数的新方程。
Substitute the value back to find the second variable. 再把求出的值代回去，求另一个未知数。

For example, solve y = 2x – 1 and 3x + y = 9.

例如，解方程组 y = 2x – 1 和 3x + y = 9。

Substitute y = 2x – 1 into 3x + y = 9:
3x + (2x – 1) = 9
5x – 1 = 9
5x = 10
x = 2

Then y = 2(2) – 1 = 3. Always check both values in the second equation: 3(2) + 3 = 9, which is correct.

代入后得 y = 2(2) – 1 = 3。务必把两个值代回第二个方程验证：3(2) + 3 = 9，成立。

3. The Elimination Method | 3. 加减消元法

The elimination method works by adding or subtracting equations so that one variable disappears. Sometimes you must multiply one or both equations first.

加减消元法的核心是通过对方程进行相加或相减，让其中一个变量被消去。有时需要先给一个或两个方程乘上适当倍数。

Solve the pair:

解方程组：

2x + 3y = 8
5x – 2y = 1

Multiply the first equation by 2 and the second equation by 3:

将第一个方程乘以 2，第二个方程乘以 3：

4x + 6y = 16
15x – 6y = 3

Now add the two equations to eliminate y: 19x = 19, so x = 1. Then substitute x = 1 into the first original equation: 2(1) + 3y = 8, so 3y = 6 and y = 2.

现在两式相加可以消去 y：19x = 19，所以 x = 1。将 x = 1 代回原第一个方程：2(1) + 3y = 8，得 3y = 6，y = 2。

The elimination method is often faster than substitution when the coefficients are simple or can be matched easily.

当系数比较简单或容易配平时，加减消元法通常比代入法更快。

4. Solving by Drawing Graphs | 4. 图像法求解

Graphical solution gives a visual interpretation of simultaneous equations. Each equation is drawn as a straight line, and the solution is the coordinates of the intersection point.

图像法为联立方程提供了直观的几何解释。把每个方程画成一条直线，两条直线交点的坐标就是方程组的解。

For example, draw y = x + 1 and y = -x + 3. The first line passes through (0,1) and (1,2); the second line passes through (0,3) and (3,0). The two lines cross at (1,2), so the solution is x = 1, y = 2.

例如，画出 y = x + 1 与 y = -x + 3。第一条直线经过 (0,1) 和 (1,2)；第二条直线经过 (0,3) 和 (3,0)。两条直线相交于点 (1,2)，所以解为 x = 1，y = 2。

Graphical methods can also show important cases: parallel lines mean no solution, while identical lines mean infinitely many solutions.

图像法还能展示特殊情形：两条直线平行则说明无解；两条直线完全重合则说明有无穷多组解。

5. Word Problems with Two Unknowns | 5. 含两个未知数的应用题

Exam questions often express simultaneous equations in words. The key is to define two variables clearly and write two equations from the given information.

考试中经常用文字叙述来给出联立方程。关键是要清楚定义两个变量，再根据题意列出两个方程。

Example: An adult ticket costs £a and a child ticket costs £b. Two adult tickets and three child tickets cost £34. One adult ticket and four child tickets cost £27. Find a and b.

例如：一张成人票价为 a 英镑，一张儿童票价为 b 英镑。两张成人票和三张儿童票共 34 英镑。一张成人票和四张儿童票共 27 英镑。求 a 和 b。

2a + 3b = 34
a + 4b = 27

Multiply the second equation by 2: 2a + 8b = 54. Subtract the first equation from this result: (2a + 8b) – (2a + 3b) = 54 – 34, so 5b = 20, hence b = 4. Substitute b = 4 into a + 4b = 27: a + 16 = 27, so a = 11.

将第二个方程乘以 2：2a + 8b = 54。用它减去第一个方程：(2a + 8b) – (2a + 3b) = 54 – 34，得 5b = 20，因此 b = 4。把 b = 4 代入 a + 4b = 27：a + 16 = 27，所以 a = 11。

Always write the final answer with units, and include a clear “let x = …” statement at the start.

作答时一定要带单位，并在开头写明“设 x = ……”，这样条理更清楚。

6. Equations with Fractions and Decimals | 6. 含分数与小数的方程

When simultaneous equations contain fractions, first multiply every term on both sides by the lowest common multiple of the denominators. This removes fractions and makes the equations easier to solve.

当联立方程中含有分数时，先给方程左右两边每一项都乘以分母的最小公倍数，这样能去掉分母，使方程更容易求解。

For example, solve:

例如，解方程组：

x/2 + y/3 = 5
x/3 + y/4 = 3

Multiply the first equation by 6 and the second by 12:

第一个方程乘以 6，第二个方程乘以 12：

3x + 2y = 30
4x + 3y = 36

Now multiply the first new equation by 3 and the second by 2:

将新的第一个方程乘以 3，第二个方程乘以 2：

9x + 6y = 90
8x + 6y = 72

Subtract: (9x + 6y) – (8x + 6y) = 90 – 72, so x = 18. Substitute into 3x + 2y = 30: 54 + 2y = 30, so 2y = -24 and y = -12.

两式相减：(9x + 6y) – (8x + 6y) = 90 – 72，得 x = 18。代入 3x + 2y = 30：54 + 2y = 30，所以 2y = -24，y = -12。

For decimals, multiply every term by a power of 10 until all numbers become integers.

对于小数方程，则将每一项乘以 10 的适当次幂，使所有数都变成整数。

7. Non-linear Simultaneous Equations | 7. 非线性联立方程

In harder IGCSE questions, one equation can be quadratic, for example y = x², while the other is linear. The substitution method is usually the best approach.

在 IGCSE 较难问题中，可能有一个方程是二次的，例如 y = x²，而另一个是线性的。这时通常使用代入消元法。

Solve y = x² and y = x + 2.

解方程组 y = x² 和 y = x + 2。

x² = x + 2
x² – x – 2 = 0
(x – 2)(x + 1) = 0

So x = 2 or x = -1. When x = 2, y = 4. When x = -1, y = 1. Therefore the solutions are (2,4) and (-1,1).

所以 x = 2 或 x = -1。当 x = 2 时，y = 4；当 x = -1 时，y = 1。因此解为 (2,4) 和 (-1,1)。

Remember that non-linear systems can have two, one, or no real solutions depending on whether the line cuts, touches, or misses the curve.

记住：非线性方程组可能有两个、一个或没有实数解，取决于直线和曲线相交、相切还是不相交。

8. Common Mistakes and Exam Tips | 8. 常见错误与考试技巧

Students often lose marks by making small sign errors or by forgetting to substitute back. Train yourself to check every answer in both original equations.

学生常因符号错误或忘记代回验证而失分。要养成把答案代回两个原方程检验的习惯。

Always multiply every term when removing fractions. 去分母时，一定要将每一项都乘以公倍数。
When subtracting equations, be careful with negative signs. 两式相减时，要注意负号的处理。
Check your final values in both equations. 把最终结果代入两个方程中进行验证。
For word problems, write down “let x = …” and “let y = …”. 应用题中先写“设 x = ……”“设 y = ……”。
When drawing graphs, use a ruler and label the intersection point clearly. 画图时要使用直尺，并标清交点坐标。

In the Edexcel IGCSE exam, show all working lines clearly. Mark schemes usually award a method mark for substitution or elimination, then a final accuracy mark.

在 Edexcel IGCSE 考试中，要清晰地写出每一步过程。评分标准一般会给代入或消元的过程分，最后再给结果分。

9. Practice Questions | 9. 练习题

Try these questions on your own, then check the answers below.

先独立尝试以下练习，再核对答案。

Question 1 Solve x + 2y = 7 and 3x – 2y = 5.	问题1 解方程组 x + 2y = 7 和 3x – 2y = 5。
Question 2 Solve y = 3x + 2 and y = 2x + 5.	问题2 解方程组 y = 3x + 2 和 y = 2x + 5。
Question 3 The sum of two numbers is 15 and their difference is 3. Find the numbers.	问题3 两个数之和为 15，之差为 3。求这两个数。

Answers: 1) x = 3, y = 2. 2) x = 3, y = 11. 3) The numbers are 9 and 6.

答案：1) x = 3，y = 2。2) x = 3，y = 11。3) 这两个数是 9 和 6。

10. Summary | 10. 小结

Simultaneous equations are a vital part of IGCSE Mathematics. You should be comfortable with substitution, elimination, and graphical methods, and know when to use each one.

联立方程是 IGCSE 数学的重要考点。你需要熟练运用代入法、加减消元法和图像法，并准确判断不同题型应该采用哪种方法。

The table below summarises the methods:

下表总结了三种主要方法：

Method	When to Use	方法	适用情形
Substitution	One variable is already the subject.	代入法	已经有一个变量被单独表示。
Elimination	Coefficients can be matched easily.	加减消元法	系数容易配平时。
Graphical	A rough check or visual interpretation is needed.	图像法	需要直观验证或分析图像位置关系。

Once you master these techniques, simultaneous equation questions become quick and reliable. Always finish by substituting your solution back into the original equations.

掌握这些技巧后，联立方程题会变得又快又稳。完成解题后，一定记得将答案代回原方程进行检验。

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Topic 214: Simultaneous Equations | 专题214：联立方程