Simultaneous Equations: Elimination and Substitution | 联立方程组：消元法与代入法

📚 Simultaneous Equations: Elimination and Substitution | 联立方程组：消元法与代入法

Simultaneous equations are one of the highest-frequency algebra topics in the IGCSE Mathematics syllabus. You will meet them in pure algebra questions, in coordinate geometry and in word problems involving money, age, speed or mixtures. This revision guide explains the two core solution methods — elimination and substitution — and shows how to check answers carefully.

联立方程组是 IGCSE 数学大纲中最高频的代数专题之一。你会在纯代数题、坐标几何题以及涉及金额、年龄、速度或混合物的应用题中遇到它。本篇复习指南将讲解两种核心解法——消元法与代入法——并说明如何仔细检验答案。

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

A system of simultaneous equations consists of two or more equations that share the same unknown variables. In IGCSE, you usually work with two linear equations in two unknowns, often written as ax + by = c and dx + ey = f. A solution is a pair of values (x, y) that satisfies every equation at the same time.

联立方程组由两个或两个以上含有相同未知数的方程组成。在 IGCSE 中，你通常处理的是两个未知数的两个线性方程，常写作 ax + by = c 和 dx + ey = f。一个解就是一组数值 (x, y)，它能同时满足每一个方程。

x + y = 7

2x − y = 2

2. Checking a Solution | 检验解

To check whether a given pair is a solution, substitute the x- and y-values into both original equations. If both equations give true statements, the pair is correct. If one equation fails, the pair is not a solution of the system.

要检验某一组数是否为解，把 x 值和 y 值代入两个原方程。如果两个方程都得到成立的等式，这组数就是正确的。只要有一个方程不成立，这组数就不是方程组的解。

Example: For x + y = 7 and 2x − y = 2, the pair x = 3, y = 4 gives 3 + 4 = 7 and 2(3) − 4 = 2, so it is valid.

例如：对于 x + y = 7 和 2x − y = 2，x = 3, y = 4 代入得 3 + 4 = 7 和 2(3) − 4 = 2，因此它是有效的。

3. The Elimination Method: Basic Idea | 消元法的基本思路

Elimination works by adding or subtracting the two equations so that one unknown cancels out. You are left with a single equation in one variable, which is easy to solve. The same value can then be substituted back to find the other variable.

消元法的原理是将两个方程相加或相减，使其中一个未知数被消去。这样你就得到一个只含一个未知数的一元方程，很容易求解。然后再将该值代回原方程，求出另一个未知数。

4. Elimination When Coefficients Match | 系数相等时的消元法

If the coefficients of one variable are exactly the same in both equations, subtract one equation from the other. For example, in 3x + y = 9 and 2x + y = 7, the y-terms both have coefficient +1. Subtract the second equation from the first:

如果某个未知数在两个方程中的系数完全相同，就把两个方程相减。例如，在 3x + y = 9 和 2x + y = 7 中，y 项的系数都是 +1。用第一个方程减去第二个方程：

(3x + y) − (2x + y) = 9 − 7

x = 2

Then substitute x = 2 into 2x + y = 7 to get 4 + y = 7, so y = 3. The solution is (2, 3).

然后把 x = 2 代入 2x + y = 7，得到 4 + y = 7，所以 y = 3。解为 (2, 3)。

5. Elimination When Signs Differ | 系数异号时的消元法

If the coefficients have the same size but opposite signs, add the equations to cancel that variable. For example, x + 2y = 8 and 3x − 2y = 4 have +2y and −2y. Adding gives 4x = 12, so x = 3. Substitute back to find y = 2.5.

如果系数大小相同但符号相反，就把两个方程相加，从而消去该变量。例如，x + 2y = 8 和 3x − 2y = 4 中含有 +2y 和 −2y。相加得到 4x = 12，所以 x = 3。代回求得 y = 2.5。

6. Multiplying One Equation First | 先对方程进行倍数处理

When coefficients do not match, multiply one entire equation by a suitable number so that one pair of coefficients becomes equal or opposite. You must multiply every term on both sides to keep the equation balanced.

当系数不匹配时，可以把整个方程乘以一个适当的数，使某一对系数相等或相反。

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