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

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

In IGCSE Mathematics, simultaneous equations appear regularly in both core and extended papers. They require you to find values that satisfy two or more equations at the same time. This guide explains the main algebraic methods, graphical interpretation, special cases, and exam-style applications.

在 IGCSE 数学中，联立方程组是核心卷和扩展卷中的常见题型。题目要求你找到同时满足两个或更多方程的值。本指南将讲解主要的代数方法、图像含义、特殊情况以及考试型应用。

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

A system of simultaneous equations consists of two or more equations involving the same unknown variables. A solution is a pair of values (x, y) that satisfies every equation in the system. For two linear equations, the solution represents the point where their straight-line graphs intersect.

联立方程组由两个或更多含有相同未知数的方程组成。方程组的解是一组数值 (x, y)，它满足系统中的每一个方程。对于两个线性方程，解表示两条直线图像的交点坐标。

x + y = 7
2x − y = 5

The only solution is x = 4, y = 3 because substituting both values makes each equation true. In coordinate form, the two lines intersect at (4, 3).

唯一解是 x = 4, y = 3，因为代入这两个值后每个方程都成立。在坐标形式中，两条直线相交于点 (4, 3)。

2. The Elimination Method | 消元法

Elimination works best when the coefficients of one variable can be made equal or opposite. You add or subtract the two equations to remove that variable, then solve for the remaining unknown.

当某个变量的系数可以化为相同或相反数时，消元法最有效。你将两个方程相加或相减，消去该变量，然后解出剩下的未知数。

Example:

示例：

2x + y = 10
3x − y = 5

Adding the equations gives 5x = 15, so x = 3. Substituting back gives 2(3) + y = 10, hence y = 4.

将两个方程相加得到 5x = 15，所以 x = 3。代回原方程得 2(3) + y = 10，因此 y = 4。

If coefficients do not match, multiply one or both equations by suitable constants first. For example, multiplying 2x + 3y = 8 by 2 and 4x − y = 5 by 1 makes the x terms 4x and 4x, ready for subtraction.

如果系数不匹配，需要先给一个或两个方程乘上合适的常数。例如，将 2x + 3y = 8 乘以 2，4x − y = 5 保持不变，就可使 x 项都变为 4x，便于相减消元。

Another useful example is 5x + 2y = 11 and 3x + 4y = 13. Multiplying the first equation by 2 gives 10x + 4y = 22, while the second stays 3x + 4y = 13. Subtracting eliminates y, leaving 7x = 9.

另一个常用示例是 5x + 2y = 11 和 3x + 4y = 13。将第一个方程乘以 2 得到 10x + 4y = 22，而第二个方程仍为 3x + 4y = 13。相减后消去 y，得到 7x = 9。

3. The Substitution Method | 代入法

Substitution is useful when one equation has an isolated variable or can be rearranged easily. You substitute that expression into the other equation to create a one-v

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