Simultaneous Equations Explained | 联立方程精讲

📚 Simultaneous Equations Explained | 联立方程精讲

Simultaneous equations are a cornerstone of the Edexcel IGCSE Mathematics syllabus. Whether you are finding the intersection point of two lines or solving a quadratic and a straight line together, this skill appears in both Paper 1 and Paper 2, often worth 3 to 6 marks per question.

联立方程是 Edexcel IGCSE 数学考纲的基石。无论是求两条直线的交点，还是联立一个二次函数与一条直线，这个技能在 Paper 1 和 Paper 2 中都会出现，通常每题价值 3 到 6 分。

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

Simultaneous equations are a set of equations containing two or more unknown variables, all of which must be satisfied at the same time. In IGCSE Maths, you will most often work with two equations and two unknowns, usually x and y.

联立方程是一组含有两个或两个以上未知数的方程，所有方程必须同时成立。在 IGCSE 数学中，最常见的是两个方程、两个未知数，通常记为 x 和 y。

The solution to a linear pair is the single coordinate (x, y) that lies on both lines graphically. If the equations are inconsistent, there may be no solution; if they are multiples of each other, there are infinitely many solutions. In the IGCSE syllabus, the focus is on systems with exactly one solution.

一对线性方程组的解是图象上同时落在两条直线上的唯一坐标 (x, y)。如果方程相互矛盾，则无解；如果两个方程成倍数关系，则有无穷多解。IGCSE 考纲重点考查恰好有一个解的方程组。

2. The Elimination Method | 消元法

Elimination is often the fastest method for linear systems. The idea is to add or subtract the two equations so that one unknown disappears, leaving a single equation in one variable.

消元法通常是解线性方程组最快的方法。核心思想是通过将两个方程相加或相减，使其中一个未知数消去，从而得到只含一个未知数的方程。

Consider the system: 3x + 2y = 12 and x – 2y = 4. Adding these equations eliminates y: 4x = 16, so x = 4. Substituting x = 4 into x – 2y = 4 gives 4 – 2y = 4, so y = 0. The solution is (4, 0).

例如方程组：3x + 2y = 12 与 x – 2y = 4。将两式相加即可消去 y：4x = 16，所以 x = 4。将 x = 4 代入 x – 2y = 4，得 4 – 2y = 4，所以 y = 0。解为 (4, 0)。

When coefficients do not match, multiply one or both equations first. For example, with 2x + 3y = 7 and 5x + 4y = 14, multiply the first equation by 4 and the second by 3 to equalise the y coefficients before subtracting.

当系数不一致时，先给一个或两个方程乘以适当倍数。例如 2x + 3y = 7 与 5x + 4y = 14，可先将第一式乘以 4、第二式乘以 3，使 y 的系数相同后再相减。

3. The Substitution Method | 代入法

Substitution is especially useful when one equation is already solved for x or y, or when one variable has coefficient 1. Rearrange one equation in terms of one unknown, then replace that unknown in the other equation.

代入法特别适合其中一个方程已经解出 x 或 y，或者某个变量的系数为 1 的情况。先从一个方程中解出一个未知数，再把它代入另一个方程。

For instance: y = 2x + 1 and 3x + 2y = 23. Replace y in the second equation: 3x + 2(2x + 1) = 23, which simplifies to 7x + 2 = 23, so x = 3. Then y = 2(3) + 1 = 7. The solution is (3, 7).

例如：y = 2x + 1 与 3x + 2y = 23。将 y 代入第二式：3x + 2

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