Mastering Simultaneous Equations for IGCSE Mathematics | 掌握 IGCSE 数学联立方程

📚 Mastering Simultaneous Equations for IGCSE Mathematics | 掌握 IGCSE 数学联立方程

Simultaneous equations appear throughout the IGCSE Mathematics syllabus, from straightforward linear pairs to more demanding word problems and non-linear systems. They ask you to find values of unknowns that satisfy two or more conditions at the same time. A strong grasp of elimination, substitution, and graphical reasoning will save time in exams and improve accuracy.

联立方程贯穿 IGCSE 数学考纲，从简单的一次方程对到较难的文字题和非线性方程组。它们要求你求出同时满足两个或更多条件的未知数。扎实掌握消元法、代入法和图像推理，能在考试中节省时间并提高准确率。

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

An equation such as x + y = 5 has infinitely many possible solutions on its own. If we add a second condition, for example x − y = 1, we can pin down a unique pair of values. These two equations together are called simultaneous equations.

像 x + y = 5 这样的方程本身有无限多个解。如果我们再加一个条件，例如 x − y = 1，就可以确定唯一的一组值。这两个方程合在一起就称为联立方程。

In IGCSE, a system of simultaneous equations usually has two unknown variables, often x and y. A solution is an ordered pair (x, y) that makes both equations true at the same time.

在 IGCSE 中，联立方程组通常有两个未知数，常见为 x 和 y。解就是能同时使两个方程成立的有序数对 (x, y)。

Simultaneous equations can be linear-linear, linear-quadratic, or quadratic-quadratic. Linear means no powers higher than 1, while quadratic means the highest power is 2.

联立方程可以是一次-一次、一次-二次或二次-二次。一次表示未知数的最高次数为 1，二次表示最高次数为 2。

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2. Why Simultaneous Equations Matter in IGCSE | 为什么联立方程在 IGCSE 中重要

Many IGCSE papers include at least one direct question on solving simultaneous equations, usually worth three to five marks. The topic also appears inside other areas such as coordinate geometry, sequences, and real-world modelling.

许多 IGCSE 试卷至少包含一道直接求解联立方程的题，通常占 3 到 5 分。该主题还出现在坐标几何、数列和现实建模等其他领域。

• Direct questions: solve 2x + 3y = 12 and 4x − y = 5. — 直接考查：解 2x + 3y = 12 和 4x − y = 5。
• Word problems: tickets, mixtures, ages, money, and speed. — 文字题：票务、混合物、年龄、金钱和速度。
• Graph questions: two straight lines or a line and a curve. — 图像题：两条直线或一条直线与一条曲线。

Because the methods are highly structured, simultaneous equations are also one of the best places to collect reliable marks in algebra. Accuracy comes from careful arithmetic and clear setting out.

由于方法结构清晰，联立方程也是代数部分最容易稳定拿分的题型之一。准确率来自仔细的运算和清晰的书写过程。

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3. The Elimination Method Step by Step | 消元法逐步讲解

Elimination is often faster when both equations are linear and the coefficients of one variable are easy to match. The idea is to add or subtract the equations so that one variable cancels out.

当两个方程都是一次方程，并且某个变量的系数容易凑成相同或相反时，消元法通常更快。思路是把两个方程相加或相减，使其中一个变量被消去。

Step 1: Write both equations in the form ax + by = c.

第一步：把两个方程都写成 ax + by = c 的形式。

Step 2: Multiply one or both equations by suitable constants so the coefficients of y, or x, are equal in size.

第二步：给一个或两个方程乘以适当的常数，使 y 或 x 的系数大小相等。

Step 3: Add or subtract the equations to eliminate that variable. Solve the resulting one-variable equation.

第三步：将两个方程相加或相减，消去该变量。解剩下的一元方程。

Step 4: Substitute the value back into one original equation to find the other variable. Check in the second equation.

第四步：把求得的值代回原方程之一，求出另一个变量。再代入第二个方程检查。

Example: Solve the system

例题：解下列方程组

2x + 3y = 12

中文：第一个方程是 2x + 3y = 12。

4x − 3y = 6

中文：第二个方程是 4x − 3y = 6。

The y terms are +3y and −3y, so adding the equations eliminates y.

y 项分别为 +3y 和 −3y，因此把两个方程相加可以消去 y。

(2x + 3y) + (4x − 3y) = 12 + 6

中文：两个方程相加，左边合并，右边相加。

6x = 18 ⇒ x = 3

中文：得到 6x = 18，因此 x = 3。

Substitute x = 3 into the first equation:

把 x = 3 代入第一个方程：

2(3) + 3y = 12 ⇒ 6 + 3y = 12 ⇒ 3y = 6 ⇒ y = 2

中文：得到 6 + 3y = 12，接着 3y = 6，所以 y = 2。

So the solution is (3, 2). Check: 4(3) − 3(2) = 12 − 6 = 6, which is correct.

因此解为 (3, 2)。检查：4(3) − 3(2) = 12 − 6 = 6，结果正确。

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4. The Substitution Method | 代入法

Substitution works well when one equation is already solved for one variable, or when one equation is non-linear. Rearrange one equation to isolate x or y, then replace that variable in the other equation.

当其中一个方程已经解出某个变量，或其中一个方程是非线性时，代入法非常有效。把一个方程变形，使 x 或 y 单独出现在一边，然后替换另一个方程中的该变量。

The method is especially useful for systems where one equation is linear and one is quadratic. It reduces the whole system to a single equation in one unknown.

对于一个是线性方程、一个是二次方程的方程组，代入法尤其有用。它能把整个方程组化为只含一个未知数的单一方程。

Example: Solve y = x² + 1 and y = 2x + 1.

例题：解 y = x² + 1 和 y = 2x + 1。

Since both right-hand sides equal y, set them equal to each other:

因为两个右边都等于 y，所以可以令它们相等：

x² + 1 = 2x + 1

中文：得到 x² + 1 = 2x + 1。

Subtract 2x + 1 from both sides:

两边同时减去 2x + 1：

x² − 2x = 0

中文：得到 x² − 2x = 0。

Factorise and solve:

因式分解并求解：

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