📚 Solving Linear-Quadratic Simultaneous Equations | 求解线性与二次联立方程组
In this lesson we work through a typical AQA A-Level Mathematics problem that requires solving a pair of simultaneous equations where one equation is linear and the other is quadratic. The substitution method used here is not only an exam-ready skill but also the foundation for curve-sketching, intersection problems, and many real-world modelling tasks in the pure mathematics units.
本节课我们完成一个典型的 AQA A-Level 数学问题：求解由线性方程和二次方程组成的联立方程组。这里所用的代入法不仅是考场必备技巧，也是函数图象、曲线交点以及纯数学单元中许多实际建模问题的基础。
1. Understanding the Problem | 理解题目
The problem is to solve the simultaneous equations involving two unknown quantities, x and y:
y = 2x + 1 (linear)
y = x² + x − 1 (quadratic)
The first equation is linear because its highest power of x is 1; it describes a straight line. The second equation is quadratic because its highest power of x is 2; it describes a parabola. To 'solve' these simultaneous equations means to find every ordered pair (x, y) that makes both equations true at the same time.
题目要求我们解一个关于两个未知数 x 和 y 的联立方程组：第一个方程是线性的，因为 x 的最高次幂是 1，它表示一条直线；第二个方程是二次的，因为 x 的最高次幂是 2，它表示一条抛物线。"求解"这个联立方程组，就是找出所有同时满足这两个方程的坐标对 (x, y)。
2. Why Substitution Works | 为什么用代入法
There are several methods for solving simultaneous equations: elimination, substitution, and graphical methods. When one of the equations is quadratic, elimination is usually awkward because we cannot simply add or subtract the equations to remove a variable cleanly. Substitution, however, always works: because both equations give y in terms of x, we can replace the y in one equation with the expression for y from the other equation.
解联立方程组有几种常用方法：消元法、代入法和图解法。当其中一个方程是二次方程时，消元法往往不太方便，因为我们不能简单地通过加减两个方程来消去一个变量。而代入法总是可靠：既然两个方程都把 y 用含 x 的式子表示，我们就能把一个方程中的 y 用另一个方程的表达式整体替换掉。
3. Equating the Two Expressions for y | 令两个 y 的表达式相等
Since both equations are already solved for y, we can set their right-hand sides equal to each other:
2x + 1 = x² + x − 1
This is the critical step. At every common solution, the y-coordinate on the straight line must be exactly the same as the y-coordinate on the parabola, so the two expressions for y must be equal.
由于两个方程都已经把 y 单独放在等号左边，我们可以令它们等号右边的表达式相等。这是关键的一步：在每一个公共解处，直线上的 y 坐标必须与抛物线上的 y 坐标完全相同，因此两个关于 y 的表达式必然相等。
4. Rearranging into Standard Quadratic Form | 整理成标准二次方程形式
To use our quadratic-solving skills, we must rearrange the equation so that one side becomes zero. We subtract 2x and subtract 1 from both sides:
x² − x − 2 = 0
2x − 1 − 1 = x² − x − 2. Take special care with the signs: subtracting 1 from −1 gives −2
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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