Simultaneous Equations: Elimination, Substitution and Graphs | 联立方程：消元法、代入法与图像法

📚 Simultaneous Equations: Elimination, Substitution and Graphs | 联立方程：消元法、代入法与图像法

In IGCSE mathematics, simultaneous equations are one of the core algebra topics. A system of simultaneous equations is a set of two or more equations that share the same variables, and the solution is the pair of values that makes all the equations true at the same time. This topic is important because it appears not only as direct algebra questions but also in word problems, graphical problems and higher-tier questions involving non-linear equations.

在 IGCSE 数学中，联立方程是核心代数主题之一。联立方程组是由两个或更多共享相同变量的方程组成的集合，它的解就是能同时使所有方程成立的一组数值。这个主题非常重要，因为它不仅以直接代数题的形式出现，还会出现在文字题、图像题以及提高卷中涉及非线性方程的问题中。

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

Simultaneous equations are equations that must be satisfied by the same values of the unknowns. In IGCSE, you will usually work with two equations in two unknowns, often written as x and y. The general form of a linear equation in two variables is ax + by = c, where a, b and c are constants.

联立方程是指必须由相同的未知数值同时满足的方程。在 IGCSE 中，通常需要处理含有两个未知数的两个方程，未知数通常写作 x 和 y。二元一次方程的一般形式是 ax + by = c，其中 a、b 和 c 是常数。

For example, x + y = 5 and x – y = 1 form a linear system. The ordered pair (3, 2) satisfies both equations because 3 + 2 = 5 and 3 – 2 = 1. The solution can be found algebraically by elimination or substitution, or graphically by finding the point where the two lines intersect.

例如，x + y = 5 和 x – y = 1 构成一个线性方程组。有序数对 (3, 2) 同时满足两个方程，因为 3 + 2 = 5 且 3 – 2 = 1。解可以通过消元法或代入法进行代数求解，也可以通过找出两条直线交点的图像方法求得。

Understanding the structure of the equations helps you choose the most efficient method. If both equations are in standard form, elimination is often quicker. If one equation is already solved for

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