Solving Linear Simultaneous Equations | 解线性联立方程组

📚 Solving Linear Simultaneous Equations | 解线性联立方程组

Simultaneous equations are one of the most important topics in IGCSE Mathematics. They appear in nearly every exam paper and form the foundation for many advanced topics such as coordinate geometry, calculus, and physics applications. In this article, we will explore the three main methods for solving linear simultaneous equations: substitution, elimination, and graphical representation.

联立方程组是 IGCSE 数学中最重要的考点之一。几乎每次考试都会出现，并且它是坐标几何、微积分以及物理应用等许多高级课题的基础。在本文中，我们将探讨求解线性联立方程组的三种主要方法：代入法、消元法和图像法。

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

A linear equation in two variables, such as x and y, represents a straight line on a Cartesian plane. When we have two such equations together, we call them a pair of simultaneous equations. The solution is the point (x, y) where both lines intersect.

含有两个变量（如 x 和 y）的线性方程在笛卡尔平面上表示一条直线。当我们把两个这样的方程放在一起时，就称为联立方程组。方程组的解就是两条直线交点的坐标 (x, y)。

2x + y = 7
x – y = 2

In this system, the solution must satisfy BOTH equations at the same time. Let’s verify: if x = 3 and y = 1, then 2(3) + 1 = 7 ✓ and 3 – 1 = 2 ✓. Therefore, the solution is x = 3, y = 1.

在这个方程组中，解必须同时满足两个方程。让我们验证一下：如果 x = 3，y = 1，那么 2(3) + 1 = 7 ✓，且 3 – 1 = 2 ✓。因此，解为 x = 3，y = 1。

2. The Substitution Method | 代入法

The substitution method is ideal when one variable has a coefficient of 1. We rearrange one equation to make one variable the subject, then substitute this expression into the other equation.

代入法适合某个变量系数为 1 的情况。我们先将一个方程变形，用另一个变量表示该变量，然后将这个表达式代入另一个方程。

Worked Example | 典型例题：

y = 3x – 4 (1)
2x + y = 11 (2)

Step 1: Equation (1) already gives y in terms of x. Substitute y = 3x – 4 into equation (2).

步骤一：方程 (1) 已经用 x 表示出了 y。将 y = 3x – 4 代入方程 (2)。

2x + (3x – 4) = 11
5x – 4 = 11
5x = 15
x = 3

Step 2: Substitute x = 3 back into equation (1): y = 3(3) – 4 = 9 – 4 = 5. Therefore, the solution is x = 3, y = 5.

步骤二：将 x = 3 代回方程 (1)：y = 3(3) – 4 = 9 – 4 = 5。因此解为 x = 3，y = 5。

Always check: 2(3) + 5 = 6 + 5 = 11 ✓

务必检验：2(3) + 5 = 6 + 5 = 11 ✓

3. The Elimination Method | 消元法

The elimination method works by adding or subtracting the two equations to eliminate one variable. This requires that the coefficients of one variable are the same (or opposites) in both equations.

消元法通过将两个方程相加或相减来消除一个变量。这需要两个方程中某个变量的系数相同（或互为相反数）。

Worked Example | 典型例题：

3x + 2y = 13 (1)
2x – 2y = 2 (2)

Here, the coefficients of y are +2 and -2. They are already opposites, so we ADD the equations.

这里，y 的系数分别为 +2 和 -2。它们已经是相反数，因此我们将两个方程相加。

(3x + 2y) + (2x – 2y) = 13 + 2
5x = 15
x = 3

Now substitute x = 3 into equation (1): 3(3) + 2y = 13 → 9 + 2y = 13 → 2y = 4 → y = 2. The solution is x = 3, y = 2.

将 x = 3 代入方程 (1)：3(3) + 2y = 13 → 9 + 2y = 13 → 2y = 4 → y = 2。解为 x = 3，y = 2。

Important: When the coefficients are the same sign, SUBTRACT; when they are opposite signs, ADD.

注意：当系数同号时，两式相减；当系数异号时，两式相加。

4. When Coefficients Are Not Matched | 当系数不同时

Sometimes the coefficients do not match at all. In such cases, we must multiply one or both equations by suitable constants before adding or subtracting.

有时两个方程中系数组完全不相同。这时，我们需要在相加或相减之前，先将一个或两个方程乘以适当的常数。

Worked Example | 典型例题：

3x + 2y = 8 (1)
2x + 5y = 9 (2)

Multiply equation (1) by 2 and equation (2) by 3 so that both have 6x:

将方程 (1) 乘以 2，方程 (2) 乘以 3，使两式中 x 的系数都为 6：

(1) × 2: 6x + 4y = 16
(2) × 3: 6x + 15y = 27

Now subtract: (6x + 15y) – (6x + 4y) = 27 – 16 → 11y = 11 → y = 1.

现在相减：(6x + 15y) – (6x + 4y) = 27 – 16 → 11y = 11 → y = 1。

Substitute y = 1 into (1): 3x + 2(1) = 8 → 3x = 6 → x = 2. Solution: x = 2, y = 1.

将 y = 1 代入 (1)：3x + 2(1) = 8 → 3x = 6 → x = 2。解为 x = 2，y = 1。

Exam Tip: Always multiply by the coefficient of the variable in the OTHER equation. This is the most systematic approach.

考试技巧：总是乘以另一个方程中该变量的系数。这是最系统的方法。

5. Rearranging Equations First | 先整理方程

In the exam, equations are not always neatly aligned. You may need to expand brackets or rearrange terms before applying any method.

在考试中，方程并非总是整齐排好。你可能需要先展开括号或重新排列各项，才能使用任何方法。

Worked Example | 典型例题：

4x + 3(y + 2) = 25 (1)
5x – 2y = 14 (2)

First expand the bracket in (1): 4x + 3y + 6 = 25 → 4x + 3y = 19.

首先展开方程 (1) 中的括号：4x + 3y + 6 = 25 → 4x + 3y = 19。

Now we have: 4x + 3y = 19 (1) and 5x – 2y = 14 (2). To eliminate y, multiply (1) by 2 and (2) by 3:

现在得到：4x + 3y = 19 (1) 和 5x – 2y = 14 (2)。为了消去 y，将 (1) 乘以 2，(2) 乘以 3：

(1) × 2: 8x + 6y = 38
(2) × 3: 15x – 6y = 42

Adding: 23x = 80 → x = 80/23. Wait, this doesn’t give a nice integer. Let’s check our arithmetic… Actually, adding gives 23x = 80, so x = 80/23. Substitute back to find y.

相加：23x = 80 → x = 80/23。等一下，这样得不到一个漂亮的整数。让我们检查一下计算……实际上，相加得到 23x = 80，所以 x = 80/23。代回求 y。

This example shows that not all solutions are integers. In IGCSE, fractional answers are perfectly acceptable.

这个例子说明并非所有解都是整数。在 IGCSE 中，分数答案完全可以接受。

6. The Graphical Method | 图像法

The graphical method involves plotting both lines on the same set of axes. The point where the two lines cross is the solution.

图像法需要将两条线绘制在同一坐标轴上。两条线相交的点就是方程组的解。

Example | 示例：

y = 2x + 1
y = -x + 4

For the first line, create a table of values:

对于第一条线，建立值表：

x	0	1	2
y = 2x + 1	1	3	5

For the second line:

对于第二条线：

x	0	1	2
y = -x + 4	4	3	2

Plotting these points and drawing the lines, we can see they intersect at (1, 3). Therefore, x = 1, y = 3.

在坐标纸上标出这些点并画线，可以看出两条线相交于 (1, 3)。因此，x = 1，y = 3。

Note: The graphical method is useful for estimating solutions but is less precise than algebraic methods. In exams, use it when specifically asked.

注意：图像法适合估算解，但精度不如代数方法。在考试中，仅在题目特别要求时使用。

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

Not all simultaneous equations have a unique solution. There are two special cases you must recognise:

并非所有联立方程组都有唯一解。有两种你必须识别的特殊情况：

Case 1: Parallel Lines (No Solution) | 情况一：平行线（无解）

y = 2x + 3
y = 2x – 1

Both lines have a gradient of 2, so they are parallel and never intersect. Such a system is called inconsistent.

两条线的斜率都是 2，所以它们是平行的，永远不会相交。这样的方程组称为不相容方程组。

Case 2: Identical Lines (Infinite Solutions) | 情况二：重合线（无穷多解）

2x + y = 5
4x + 2y = 10

The second equation is simply twice the first, so they represent the same line. Every point on the line is a solution.

第二个方程只是第一个方程的两倍，所以它们表示同一条线。线上的每一个点都是解。

How to identify algebraically: When using elimination, if all variables disappear and you are left with a false statement like 0 = 5, there is no solution. If you are left with a true statement like 0 = 0, there are infinitely many solutions.

如何用代数方法识别：使用消元法时，如果所有变量都消失了，剩下一个假命题如 0 = 5，则无解。如果剩下真命题如 0 = 0，则有无穷多解。

8. Real-World Applications | 实际应用

Simultaneous equations are used to solve many everyday problems involving two unknown quantities. Here is a typical IGCSE exam-style question:

联立方程组常用于解决涉及两个未知量的日常问题。下面是一道典型的 IGCSE 考试风格题目：

Problem | 问题： A shop sells pencils and pens. Two pencils and three pens cost $4.10. Three pencils and two pens cost $3.90. Find the price of one pencil and one pen.

题目：一家商店出售铅笔和钢笔。两支铅笔和三支钢笔共 4.10 美元。三支铅笔和两支钢笔共 3.90 美元。求一支铅笔和一支钢笔的价格。

Let p = price of one pencil and q = price of one pen. Set up the equations:

设 p 为铅笔单价，q 为钢笔单价。建立方程组：

2p + 3q = 4.10 (1)
3p + 2q = 3.90 (2)

Multiply (1) by 3 and (2) by 2:

将 (1) 乘以 3，(2) 乘以 2：

6p + 9q = 12.30
6p + 4q = 7.80

Subtract: 5q = 4.50 → q = 0.90. Substitute q = 0.90 into (1): 2p + 2.70 = 4.10 → 2p = 1.40 → p = 0.70.

相减：5q = 4.50 → q = 0.90。将 q = 0.90 代入 (1)：2p + 2.70 = 4.10 → 2p = 1.40 → p = 0.70。

Answer: A pencil costs $0.70 and a pen costs $0.90.

答案：一支铅笔 0.70 美元，一支钢笔 0.90 美元。

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

Here are the most frequent errors students make, along with advice for avoiding them.

以下是最常见的错误以及避免它们的建议。

Sign errors when subtracting: When subtracting one equation from another, be careful with negative terms. For example, (2x + 3y) – (2x – y) is NOT 2y; it is 4y, because 3y – (-y) = 4y. 减去时发生符号错误：一个方程减去另一个方程时，注意负项。例如 (2x + 3y) – (2x – y) 不是 2y，而是 4y，因为 3y – (-y) = 4y。
Forgetting to substitute back: Solving for one variable is only half the work. Always substitute back to find the other variable. 忘记回代：求出一个变量只完成了一半工作。务必回代求出另一个变量。
Not checking the answer: A few seconds of verification can save you from losing valuable marks. 不检查答案：花几秒钟验证可以避免失分。
Mixing up the multiplication: When matching coefficients, multiply each term in the equation, not just the variable term. 乘错项：匹配系数时，要乘以方程中的每一项，而不仅仅是变量项。

Exam Tip: In a calculator paper, always check your solution by substituting both values into both original equations.

考试技巧：在用计算器的试卷中，务必通过将两个值代入两个原方程来检验解。

10. Practice Questions | 练习题目

Try these problems on your own, then look at the answers below.

自己尝试解决以下问题，然后查看答案。

Question 1 | 题 1： Solve 2x + y = 10 and x – y = 2.

Question 2 | 题 2： Solve 5x + 2y = 16 and 3x + 4y = 18.

Question 3 | 题 3： The sum of two numbers is 15, and their difference is 3. Find the two numbers.

题 3：两个数的和为 15，差为 3。求这两个数。

Answers | 答案：

Q1: x = 4, y = 2. Using elimination: (2x + y) + (x – y) = 10 + 2 → 3x = 12 → x = 4. Then 2(4) + y = 10 → y = 2.

题 1 解：x = 4，y = 2。用消元法：(2x + y) + (x – y) = 10 + 2 → 3x = 12 → x = 4。然后 2(4) + y = 10 → y = 2。

Q2: x = 2, y = 3. Multiply (1) by 2: 10x + 4y = 32. Subtract (2): 7x = 14 → x = 2. Substitute: 5(2) + 2y = 16 → 2y = 6 → y = 3.

题 2 解：x = 2，y = 3。将 (1) 乘以 2：10x + 4y = 32。减去 (2)：7x = 14 → x = 2。代入：5(2) + 2y = 16 → 2y = 6 → y = 3。

Q3: The numbers are 9 and 6. Let a + b = 15 and a – b = 3. Adding: 2a = 18 → a = 9, so b = 6.

题 3 解：两个数为 9 和 6。设 a + b = 15，a – b = 3。相加：2a = 18 → a = 9，所以 b = 6。

11. Summary | 总结

Solving linear simultaneous equations is a skill that requires practice. Choose the method that works best for the given equations: substitution when one variable has a coefficient of 1, elimination when coefficients can be easily matched, and graphs when estimation is acceptable.

解线性联立方程组是一项需要练习的技能。根据题目选择合适的解法：当某个变量系数为 1 时用代入法；当系数容易配平时用消元法；当题目允许估算时用图像法。

Remember these key points:

记住以下要点：

Always check your solution in BOTH equations. 始终在两个方程中检验你的解。
When using elimination, add if the signs are different, subtract if they are the same. 用消元法时，符号不同则相加，符号相同则相减。
Substitute back to find the second variable. 回代求第二个变量。
Keep working neatly — algebraic mistakes often come from messy layouts. 书写整洁——代数错误往往源于版面混乱。

Mastering this topic will not only earn you marks in the exam but also prepare you for A-Level mathematics, where systems of equations become even more important.

掌握这个考点不仅能在考试中帮助你得分，还能为 A-Level 数学做好准备，在那里方程组将变得更加重要。

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Solving Linear Simultaneous Equations | 解线性联立方程组