Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a cornerstone of IGCSE Mathematics. They allow us to find the values of unknowns that satisfy multiple conditions at the same time. In this revision guide, we will work through the key methods – elimination, substitution and graphing – and extend these ideas to harder linear–quadratic systems and real-world problems.

联立方程组是 IGCSE 数学的核心内容之一。它让我们能够在同一时间满足多个条件，从而求出未知数的取值。在本复习指南中，我们将系统讲解消元法、代入法和图象法三大方法，并把思路延伸到更难的线性—二次方程组以及现实生活中的应用问题。

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

A pair of simultaneous equations is a set of two equations that contain the same two unknown variables, usually x and y. The solution of the system is the ordered pair (x, y) that makes both equations true at the same time.

一组联立方程组是由两个含有相同未知数（通常是 x 和 y）的方程组成的。方程组的解是能同时使两个方程成立的有序数对 (x, y)。

For example, consider x + y = 7 and x − y = 3. The pair x = 5 and y = 2 satisfies both equations, because 5 + 2 = 7 and 5 − 2 = 3. This solution is unique for two non-parallel straight lines.

例如，考虑 x + y = 7 与 x − y = 3。x = 5，y = 2 能同时满足两个方程，因为 5 + 2 = 7，且 5 − 2 = 3。对于两条不平行的直线，这个解是唯一的。

If the two lines are parallel, the system has no solution. If the two equations represent the same straight line, there are infinitely many solutions. In IGCSE exams you will normally deal with the unique-solution case.

若两条直线平行，方程组无解；若两个方程表示同一条直线，则有无穷多组解。在 IGCSE 考试中，通常只需要处理唯一解的情形。

2. The Elimination Method | 消元法

The elimination method is often the fastest approach when the coefficients of one variable are easy to match. The goal is to add or subtract the two equations so that one variable disappears, leaving a single equation in one unknown.

当一个变量的系数比较容易配成相等时，消元法往往是最快的方法。其目标是对方程进行相加或相减，使一个变量消去，从而得到一个只含一个未知数的方程。

• Multiply one or both equations by constants so that the coefficients of x (or y) are equal in magnitude. | 将一个或两个方程乘以适当的常数，使 x（或 y）的系数绝对值相等。
• Add the equations if the coefficients have opposite signs; subtract them if the coefficients have the same sign. | 若系数符号相反则把两个方程相加；若符号相同则相减。
• Solve the resulting linear equation for the remaining variable. | 解出这个一元一次方程，求出剩下的变量。
• Substitute this value back into either original equation to find the other variable. | 把这个值代回任意一个原方程，求出另一个变量。

Worked Example: Solve 2x + 3y = 8 and 3x + 2y = 7.

例题：解方程组 2x + 3y = 8 与 3x + 2y = 7。

2x + 3y = 8
3x + 2y = 7

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

将第一个方程乘以 3，第二个方程乘以 2：

6x + 9y = 24
6x + 4y = 14

Subtract the second equation from the first: 5y = 10, so y = 2.

用第一个方程减去第二个方程：5y = 10，所以 y = 2。

Substitute y = 2 into 2x + 3y = 8: 2x + 6 = 8, so 2x = 2 and x = 1.

将 y = 2 代入 2x + 3y = 8：2x + 6 = 8，所以 2x = 2，x = 1。

The solution is x = 1, y = 2. Always check both values in the original equations: 3(1) + 2(2) = 7 ✓.

因此解为 x = 1，y = 2。务必把两个值代回原方程检验：3(1) + 2(2) = 7，成立。

3. The Substitution Method | 代入法

Use substitution when one equation is already written with a single variable on one side, such as y = 2x + 1. Replace y in the other equation by the whole expression

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