2. Global Systems in A-Level Mathematics: Simultaneous Equations and Interconnected Modelling | 全局系统：A-Level 数学中的联立方程与关联建模

📚 2. Global Systems in A-Level Mathematics: Simultaneous Equations and Interconnected Modelling | 全局系统：A-Level 数学中的联立方程与关联建模

In A-Level Mathematics, the term “global systems” may not appear as a standalone chapter, but the core idea is central: many real-world and mathematical problems involve several variables that must satisfy multiple conditions at the same time. These interconnected conditions form a system of equations or inequalities. Mastering the techniques for solving such systems—especially simultaneous equations—is a key Edexcel assessment objective, appearing in Pure Mathematics, Mechanics, and Statistics.

在 A-Level 数学中，“全局系统”或许不是独立章节的名称，但其核心思想至关重要：许多现实世界和数学问题都涉及多个变量同时满足多个条件。这些相互关联的条件构成了方程组或不等式组。掌握求解此类系统（尤其是联立方程）的技巧，是 Edexcel 考试的重要评估目标，出现在纯数学、力学和统计中。

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1. What Are Global Systems in Mathematics? | 什么是数学中的全局系统？

A mathematical system is a collection of equations or inequalities involving the same set of variables. When we say “global”, we mean that the conditions apply across the whole problem, not just locally. For example, in a market model, the supply equation and the demand equation together form a global system that determines equilibrium price and quantity.

数学系统是涉及同一组变量的方程或不等式的集合。当我们说“全局”时，指的是这些条件对整个问题都适用，而不仅限于局部。例如，在市场经济模型中，供给方程和需求方程共同构成一个全局系统，决定均衡价格与数量。

In Edexcel A-Level Mathematics, you will meet global systems in three main contexts: simultaneous equations in Pure Mathematics, force equilibrium in Mechanics, and constraints in linear programming. In all cases, the task is to find values of the variables that satisfy every condition at once.

在 Edexcel A-Level 数学中，你会在三个主要情境中遇到全局系统：纯数学中的联立方程、力学中的力平衡以及线性规划中的约束条件。无论哪种情况，任务都是找出同时满足每一个条件的变量值。

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2. Linear Simultaneous Equations | 线性联立方程

The simplest global systems are linear simultaneous equations. Edexcel expects you to solve two linear equations in two unknowns, such as 2x + y = 7 and x − y = 2. These equations are both linear, meaning each graph is a straight line. The solution is the pair of values (x, y) that satisfies both equations at once.

最简单的全局系统是线性联立方程。Edexcel 要求你会求解两个未知数的两个线性方程，例如 2x + y = 7 和 x − y = 2。这两个方程都是线性的，意味着每个图像都是直线。解就是同时满足这两个方程的一组值 (x, y)。

Two algebraic methods are required at A-Level: elimination and substitution. Both rely on the fact that if two expressions are equal to the same quantity, you can add, subtract, or replace them without changing the solution set.

A-Level 要求掌握两种代数方法：消元法和代入法。两者都基于一个事实：如果两个表达式等于同一个量，你可以对它们进行加、减或替换，而不会改变解集。

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3. Solving by Elimination | 消元法求解

Elimination works by adding or subtracting equations to remove one variable. In the system above, adding 2x + y = 7 and x − y = 2 gives 3x = 9, so x = 3. Substituting x = 3 into x − y = 2 gives 3 − y = 2, hence y = 1. The solution is (3, 1). This method is efficient when coefficients of one variable are equal or opposite.

消元法通过将方程相加或相减来消去一个变量。在上面的系统中，将 2x + y = 7 与 x − y = 2 相加得到 3x = 9，因此 x = 3。将 x = 3 代入 x − y = 2 得 3 − y = 2，所以 y = 1。解为 (3, 1)。当某个变量的系数相等或互为相反数时，这种方法非常高效。

If the coefficients are not equal, you can multiply one or both equations by suitable constants first. For example, to solve 3x + 2y = 12 and 4x − y = 5, multiply the second equation by 2 to get 8x − 2y = 10, then add to the first to eliminate y. This gives 11x = 22, so x = 2 and y = 3.

如果系数不相等，你可以先给一个或两个方程乘以适当的常数。例如，求解 3x + 2y = 12 和 4x − y = 5，将第二个方程乘以 2 得到 8x − 2y = 10，再与第一个方程相加消去 y。得到 11x = 22，所以 x = 2，y = 3。

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4. Solving by Substitution | 代入法求解

Substitution involves rearranging one equation to express one variable in terms of the other, then substituting into the second equation. For example, from x − y = 2 we get x = y + 2. Substituting into 2x + y = 7 gives 2(y + 2) + y = 7, so 3y + 4 = 7, giving y = 1 and x = 3. Substitution is particularly useful when one coefficient is 1 or −1.

代入法先将一个方程变形，用另一个变量表示某个变量，再将其代入第二个方程。例如，由 x − y = 2 可得 x = y + 2。代入 2x + y = 7 得 2(y + 2) + y = 7，即 3y + 4 = 7，解得 y = 1，x = 3。当某个系数为 1 或 −1 时，代入法尤为适用。

In exam questions, substitution often produces a quadratic equation when one equation is non-linear. Always simplify carefully and check both solutions in the original equations, as extraneous roots can appear if you square expressions.

在考试题中，当一个方程是非线性时，代入法常常会得到一个二次方程。务必仔细化简，并在原方程中检验两个解，因为平方表达式可能会产生增根。

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5. Graphical Solutions and Intersections | 图形解法与交点

Each linear equation represents a straight line on the Cartesian plane. Solving the system is equivalent to finding the point where the two lines intersect. If the lines are parallel and distinct, there is no solution; if they are coincident, there are infinitely many solutions. Edexcel questions may ask you to interpret the number of intersections graphically.

每个线性方程在笛卡尔平面上表示一条直线。求解方程组等价于找到两条直线的交点。如果两条直线平行但不重合，则无解；如果两条直线重合，则有无穷多解。Edexcel 考题可能要求你从图形上解释交点的个数。

The following table summarises the three possible outcomes when solving a pair of linear equations.

下表总结了解一对线性方程时可能出现的三种结果。

Algebraic result	Graphical meaning	Number of solutions
Unique value for x and y	Two lines intersect at one point	Exactly one
0 = 5 (contradiction)	Parallel distinct lines	None
0 = 0 (identity)	Coincident lines	Infinitely many

Being able to link algebraic outcomes to geometric pictures is a key skill in Edexcel Pure Mathematics and helps with questions on linear systems with unknown parameters.

能够将代数结果与几何图像联系起来，是 Edexcel 纯数学中的一项关键技能，也有助于解答含未知参数的线性方程组问题。

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6. One Linear and One Quadratic Equation | 一次与二次方程联立

Edexcel Pure Mathematics also tests systems where one equation is linear and the other is quadratic, such as y = 2x + 1 and y = x² − 3x + 5. To solve, substitute the linear expression for y into the quadratic equation. This yields a quadratic in x, which can be solved by factorising, completing the square, or using the quadratic formula.

Edexcel 纯数学也考查一个是一次方程、另一个是二次方程的方程组，例如 y = 2x + 1 和 y = x² − 3x + 5。求解时，将一次方程中 y 的表达式代入二次方程，得到一个关于 x 的二次方程，可以通过因式分解、配方法或求根公式求解。

For the example above, setting x² − 3x + 5 = 2x + 1 gives x

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