Solving Simultaneous Equations | 联立方程的求解

📚 Solving Simultaneous Equations | 联立方程的求解

Simultaneous equations are a fundamental topic in IGCSE Mathematics. They appear in both Paper 1 and Paper 2, and they form the foundation for many higher-level topics such as coordinate geometry and calculus. In this revision guide, we will explore the three main methods of solving simultaneous equations, understand when to use each method, and learn how to avoid common pitfalls.

联立方程是 IGCSE 数学中的基础内容，出现在 Paper 1 和 Paper 2 两套试卷中，同时也是解析几何和微积分等更高阶话题的基础。在本复习指南中，我们将探讨求解联立方程的三种主要方法，理解各方法的适用时机，并学会避开常见错误。

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

Simultaneous equations are two or more equations that share the same set of unknown variables. A solution to a system of simultaneous equations is a set of values that satisfies every equation at the same time.

联立方程是指两个或两个以上共享相同未知数的方程。联立方程组的一个解就是同时满足每一个方程的一组数值。

For linear simultaneous equations in two variables, we typically have two equations of the form:

二元线性联立方程通常具有如下形式：

ax + by = c

where a, b and c are constants. The solution is the ordered pair (x, y) that makes both equations true simultaneously.

其中 a、b、c 为常数。解是有序数对 (x, y)，它同时使两个方程成立。

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2. The Substitution Method | 代入消元法

The substitution method involves rearranging one equation to express one variable in terms of the other, and then substituting that expression into the second equation. This produces a single equation in one variable, which can be solved normally.

代入消元法是指先将一个方程变形，用另一个变量表示其中一个变量，再将这个表达式代入第二个方程，从而得到只含一个变量的一元方程，然后按常规方法求解。

Example:

例题：

Solve: y = 2x + 1 and x – 2y = 5

求解：y = 2x + 1 与 x – 2y = 5

Step 1: The first equation already gives y in terms of x, so substitute y = 2x + 1 into the second equation:

第一步：第一个方程已经用 x 表示 y，因此将 y = 2x + 1 代入第二个方程：

x – 2(2x + 1) = 5

Step 2: Simplify and solve for x:

第二步：化简并求 x：

x – 4x – 2 = 5
-3x = 7
x = -7/3

Step 3: Substitute x = -7/3 back into y = 2x + 1:

第三步：将 x = -7/3 代回 y = 2x + 1：

y = 2(-7/3) + 1 = -14/3 + 3/3 = -11/3

Therefore the solution is x = -7/3, y = -11/3.

因此解为 x = -7/3，y = -11/3。

The substitution method is especially useful when one of the equations is already arranged with a single variable on one side, or when one of the coefficients is 1.

代入消元法尤其适合其中一个方程已经写成一边只有一个变量的形式，或者某个未知数的系数为 1 的情况。

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3. The Elimination Method | 加减消元法

The elimination method (also called the addition/subtraction method) works by adding or subtracting the two equations so that one of the variables cancels out. To achieve this, you may first need to multiply one or both equations by a suitable constant.

加减消元法（也称作消去法）通过将两个方程相加或相减，使其中一个变量消去。为了做到这一点，可能要先对其中一个方程或两个方程乘以适当的常数。

Example:

例题：

Solve: 3x + 2y = 12 and 2x – y = 1

求解：3x + 2y = 12 与 2x – y = 1

Step 1: Multiply the second equation by 2 so that the y coefficients are opposites:

第一步：将第二个方程乘以 2，使 y 的系数互为相反数：

4x – 2y = 2

Step 2: Add the two equations to eliminate y:

第二步：将两个方程相加以消去 y：

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