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

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

In IGCSE Mathematics, simultaneous equations are a core algebra skill. You will usually meet two equations with two unknowns, often x and y, and your task is to find the one pair of values that makes both equations true at the same time.

在 IGCSE 数学中，联立方程是代数核心技能。你通常会看到含有两个未知数（通常是 x 和 y）的两个方程，你的任务是找出同时使两个方程成立的一组值。

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

A simultaneous equation is one of a set of equations that share the same variables. For example, the pair below shares the variables x and y.

联立方程是一组共享相同变量的方程之一。例如，下面这对方程共享变量 x 和 y。

2x + y = 7 and x − y = 2

olution is x = 3 and y = 1 because substituting these values makes both equations correct.

解为 x = 3 和 y = 1，因为代入这些值后两个方程都成立。

2(3) + 1 = 7 and 3 − 1 = 2

GCSE answers, you can write the solution as x = 3, y = 1 or as the coordinate pair (3, 1).

在 IGCSE 答案中，你可以把解写成 x = 3，y = 1 或坐标 (3, 1)。

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2. Why Are Simultaneous Equations Important? | 为什么联立方程很重要？

Simultaneous equations model real-life situations with two related conditions, such as cost and quantity, speed and time, or mixture problems. They also develop logical algebra skills that appear throughout IGCSE papers.

联立方程可用于模拟具有两个相关条件的现实问题，例如成本与数量、速度与时间或混合物问题。它们还能培养逻辑代数能力，贯穿 IGCSE 试卷。

Questions may ask you to solve equations algebraically, interpret graphs, or form your own equations from written information.

题目可能要求你用代数方法解方程、解释图像，或者根据文字信息自己建立方程。

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3. The Elimination Method | 消元法

The elimination method works best when the coefficients of either x or y can be made the same or opposite. You then add or subtract the equations to remove one unknown.

当 x 或 y 的系数可以化为相同或相反时，消元法最有效。然后通过相加或相减消去一个未知数。

A useful sign rule is shown below.

下面是一条有用的符号规则。

Coefficients	Operation	Example
Opposite signs	Add the equations	+y and −y
Same signs	Subtract one equation from the other	+2y and +2y

Example 1: Solve 2x + y = 7 and x − y = 2.

例 1：解方程组 2x + y = 7 和 x − y = 2。

Step 1: Add the equations because the y coefficients are +1 and −1.

步骤 1：将两个方程相加，因为 y 的系数分别是 +1 和 −1。

(2x + y) + (x − y) = 7 + 2

3x = 9 → x = 3

Step 2: Substitute x = 3 into x − y = 2.

步骤 2：将 x = 3 代入 x − y = 2。

3 − y = 2 → y = 1

So the solution is x = 3, y = 1.

所以解为 x = 3，y = 1。

Example 2: Solve 3x + 2y = 12 and 2x − y = 1.

例 2：解方程组 3x + 2y = 12 和 2x − y = 1。

Multiply the second equation by 2 so the y coefficient becomes −2. This makes the y coefficients opposite to +2.

将第二个方程乘以 2，使 y 的系数变为 −2。这就使 y 的系数与 +2 相反。

2(2x − y) = 2(1) → 4x − 2y = 2

Add this to the first equation to eliminate y.

把它与第一个方程相加，以消去 y。

3x + 2y + 4x − 2y = 12 + 2

7x = 14 → x = 2

Substitute x = 2 into 2x − y = 1.

将 x = 2 代入 2x − y = 1。

4 − y = 1 → y = 3

The solution is x = 2, y =

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