📚 Graphical Solution of Simultaneous Equations: Intersections and Solutions | 联立方程组的图像解法：交点与解
In this revision article, we explore how to solve simultaneous equations graphically. Every equation in two variables can be plotted as a curve on an x-y coordinate plane. A solution of the system is any point that satisfies all equations at once, and on the graph that point appears as an intersection of the curves.
在本篇复习文章中，我们探讨如何用图像法解联立方程组。每个含两个变量的方程都可以在 x-y 坐标平面上画成一条曲线。方程组的解是同时满足所有方程的点，在图上表现为曲线的交点。
1. What are Simultaneous Equations? | 什么是联立方程组
x + 1 and y = -x + 4. The goal is to find values of x and y that make both equations true.
联立方程组是一组共享相同变量的两个或更多方程。例如：y = 2x + 1 和 y = -x + 4。目标就是找到同时使两个方程成立的 x 与 y 值。
Such problems appear frequently in mathematics, physics and economics. Solving them algebraically often uses substitution or elimination, but the graphical approach offers a direct visual understanding.
这类问题在数学、物理和经济学中经常出现。代数上通常用代入法或消元法求解，而图像法能提供直观的视觉理解。
2. The Graphical Method | 图像解法
The fundamental idea is simple: each equation is drawn as a graph on the same axes. The coordinates of any intersection point satisfy both equations, so those coordinates form a solution.
基本思想很简明：把每个方程画在同一坐标轴上。任何交点的坐标都同时满足两个方程，因此这些坐标就是一组解。
Steps: (i) rearrange each equation to make y the subject; (ii) produce a table of values for each graph; (iii) plot the points and draw smooth curves; (iv) identify the intersection(s) and read off the coordinates.
步骤：(i) 将每个方程变形为 y 的表达式；(ii) 为每个图像制作取值表；(iii) 描点并画出平滑曲线；(iv) 确定交点并读出坐标。
Solution = coordinates of the intersection point
解 = 交点的坐标
3. Plotting Linear Graphs | 绘制线性图像
as slope m and y-intercept c. To plot it, choose at least three x-values, compute y, and connect the points with a straight edge.
形如 y = mx + c 的线性方程中，m 是斜率，c 是 y 轴截距。画图时至少选三个 x 值，计算对应的 y，然后用直尺连线。
For example, for y = 2x + 1: when x = -1, 0, 1, the y-values are -1, 1, 3.
例如，对于 y = 2x + 1：当 x = -1, 0, 1 时，y 值分别为 -1, 1, 3。
Use a table to organise values:
用表格整理取值：
| x | -1 | 0 | 1 |
| y = 2x+1 | -1 | 1 | 3 |
For y = -x + 4: when x = -1, 0, 1, y = 5, 4, 3.
对于 y = -x + 4：当 x = -1, 0, 1 时，y = 5, 4, 3。
Plot both sets of points on the same set of axes. Choose a scale that allows all calculated points to fit comfortably.
在同一坐标轴上描出两组点。选择能让所有计算点都合适放置的比例尺。
4. Intersection of Two Lines | 两条直线的交点
Using the two lines from the previous section, we see that y = 2x
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