Using Graphs to Solve Non-Linear Simultaneous Equations | 利用图像求解非线性联立方程

📚 Using Graphs to Solve Non-Linear Simultaneous Equations | 利用图像求解非线性联立方程

When two equations are solved simultaneously, we are finding values of x and y that satisfy both equations at once. If at least one equation is non-linear – for example, it contains x², x³, xy, or circles – then an algebraic solution can be difficult. However, by drawing both graphs on the same set of axes, the solutions appear visually as the points where the curves intersect.

当我们联立求解两个方程时，实际上是在寻找同时满足两个方程的 x 和 y 的值。如果至少有一个方程是非线性的——例如包含 x²、x³、xy 或圆的形式——那么代数求解可能会比较困难。不过，将两个图像绘制在同一坐标系中，解就会以曲线交点的方式直观地呈现出来。

1. What Are Non-Linear Simultaneous Equations? | 什么是非线性联立方程

A simultaneous equation system normally contains two equations in two unknowns. In IGCSE Edexcel Mathematics, the linear case is straightforward: both equations are straight lines, and the solution is their single intersection. A non-linear system includes at least one curve, such as a parabola, a circle, a cubic, or a reciprocal graph.

联立方程组通常包含两个方程和两个未知数。在 IGCSE Edexcel 数学中，线性情形比较简单：两个方程都是直线，解就是它们的唯一交点。非线性方程组则至少包含一条曲线，例如抛物线、圆、三次函数或反比例图像。

Common non-linear forms you will encounter include:

你常会遇到的非线性形式包括：

Quadratic: y = ax² + bx + c, whose graph is a parabola.

二次函数：y = ax² + bx + c，其图像为抛物线。
Circle: x² + y² = r², centred at the origin with radius r.

圆：x² + y² = r²，以原点为圆心，半径为 r。
Cubic: y = ax³ + bx² + cx + d, with one or more turning points.

三次函数：y = ax³ + bx² + cx + d，具有一个或多个转向点。
Reciprocal: y = k/x, giving a hyperbola with two branches.

反比例函数：y = k/x，图像为具有两支的双曲线。

2. Why Use Graphs? | 为什么要使用图像

Algebraic substitution works well for simple cases, for example solving y = x² with y = 2x + 3. But when the equations are more complicated, or when the question explicitly asks you to use a graph, the graphical method becomes essential. Graphs also show the number of solutions at a glance: a line and a parabola can intersect at zero, one, or two points.

代数代入法适用于简单情形，例如联立 y = x² 与 y = 2x + 3。但当方程更加复杂，或题目明确要求使用图像时，图像法就变得至关重要。图像还能一眼看出解的个数：直线与抛物线可能相交于零个、一个或两个点。

The graphical approach also helps you check algebraic answers. If your algebra gives an intersection that does not appear on the graph, you know a mistake has been made.

图像法还能帮助你检验代数结果。如果代数求得的交点没有出现在图像上，就说明计算中出现了错误。

3. The Intersection Principle | 交点的基本原理

Every point on the graph of y = f(x) satisfies that equation. Every point on the graph of y = g(x) satisfies the other equation. Therefore, any point that lies on both graphs automatically satisfies both equations simultaneously. This is why the solution set of a pair of simultaneous equations is exactly the set of intersection points of their graphs.

y = f(x) 图像上的每一个点都满足该方程；y = g(x) 图像上的每一个点也都满足另一个方程。因此，同时位于两个图像上的任何点，必定同时满足两个方程。这就是为什么一对联立方程组的解集恰好是两个图像交点的集合。

Solutions of simultaneous equations = coordinates of intersection points of their graphs

联立方程的解 = 两个图像交点的坐标

You should write the answer as coordinate pairs, for example (2, 5) and (−1, 1). Each coordinate pair is one complete solution.

作答时应写成坐标对的形式，例如 (2, 5) 和 (−1, 1)。每一个坐标对就是一个完整的解。

4. Key Curves You Must Recognise | 你必须认识的关键曲线

Before using graphs to solve equations, you need to recognise the shapes of common curves and know how to plot them accurately from a table of values.

在利用图像解方程之前，你需要认识常见曲线的形状，并知道如何根据数值表准确地绘制它们。

Equation 方程	Shape 形状	Key Features 关键特征
y = x²	Parabola 抛物线	Vertex at (0,0), opens upwards 顶点在 (0,0)，开口向上
y = −x²	Parabola 抛物线	Vertex at (0,0), opens downwards 顶点在 (0,0)，开口向下
x² + y² = r²	Circle 圆	Centre (0,0), radius r 圆心 (0,0)，半径 r
y = x³	Cubic 三次曲线	Passes through origin, one inflection 过原点，一个拐点
y = 1/x	Hyperbola 双曲线	Two branches, asymptotes at x = 0 and y = 0 两支，渐近线为 x = 0 和 y = 0

In an exam, you may be given a table of x-values and asked to complete the y-values. Take care with negative values of x: for y = x², (−3)² = 9, but for y = −x², −(3)² = −9. For y = x³, (−2)³ = −8.

在考试中，你可能拿到一组 x 值并要求补全对应的 y 值。处理负 x 值时要格外小心：对于 y = x²，(−3)² = 9；但对于 y = −x²，−(3)² = −9。对于 y = x³，(−2)³ = −8。

5. Step-by-Step Plotting Method | 逐步绘图方法

To solve a non-linear pair of simultaneous equations graphically, follow these steps carefully.

要用图像法求解非线性联立方程组，请仔细遵循以下步骤。

Step 1: Construct a table of values for each equation over the given range of x.

第 1 步：在给定的 x 范围内，为每个方程构建数值表。
Step 2: Plot the points for the first equation on a suitable grid, then join them with a smooth curve.

第 2 步：在合适的网格上绘出第一个方程的点，然后用平滑曲线连接。
Step 3: On the same axes, plot the points for the second equation and draw its curve or straight line.

第 3 步：在相同的坐标系上，绘出第二个方程的点并画出它的曲线或直线。
Step 4: Mark all points where the two graphs cross. Read off the x-coordinate and the y-coordinate of each intersection.

第 4 步：标出两个图像相交的所有点，读出每个交点的 x 坐标和 y 坐标。
Step 5: Write each solution as an ordered pair (x, y). If the question asks for estimates, round to the nearest sensible degree of accuracy.

第 5 步：将每个解写成有序数对 (x, y)。如果题目要求估算值，请四舍五入到合理的精度。

Use a sharp pencil and plot points accurately. A small plotting error can shift an intersection significantly, causing you to lose marks on the final answer.

请使用削尖的铅笔并精确描点。一个微小的绘图误差都可能明显移动交点位置，导致最终答案失分。

6. Worked Example 1: Parabola and Straight Line | 示例 1：抛物线与直线

Solve the simultaneous equations y = x² and y = x + 2 using graphs, for −2 ≤ x ≤ 3.

用图像法求解联立方程 y = x² 和 y = x + 2，其中 −2 ≤ x ≤ 3。

First, complete the table for y = x².

首先，补全 y = x² 的数值表。

x	−2	−1	0	1	2	3
y = x²	4	1	0	1	4	9
y = x + 2	0	1	2	3	4	5

Plot both sets of points on the same axes. The parabola y = x² passes through (0, 0) and turns upwards. The line y = x + 2 crosses the y-axis at 2 and has a gradient of 1.

在同一坐标系中绘出两组点。抛物线 y = x² 经过 (0, 0) 并向上开口。直线 y = x + 2 与 y 轴交于 2，斜率为 1。

Reading from the graph, the two curves intersect at approximately (−1, 1) and (2, 4). These are the simultaneous solutions.

从图像上读得，两条曲线大约相交于 (−1, 1) 和 (2, 4)。这就是联立方程的解。

x = −1, y = 1 and x = 2, y = 4

Algebraic check: substitute x = 2 into both equations. y = 2² = 4 and y = 2 + 2 = 4. Both give the same y-value, confirming the intersection.

代数验证：将 x = 2 代入两个方程。y = 2² = 4，y = 2 + 2 = 4。两者得到相同的 y 值，确认了交点正确。

7. Worked Example 2: Circle and Line | 示例 2：圆与直线

Solve x² + y² = 25 and y = 2x + 1 graphically for −6 ≤ x ≤ 6.

用图像法求解 x² + y² = 25 和 y = 2x + 1，其中 −6 ≤ x ≤ 6。

The first equation represents a circle centred at the origin with radius 5. To plot it, use the key points: (5, 0), (−5, 0), (0, 5), (0, −5), and draw a smooth circle through them.

第一个方程表示以原点为圆心、半径为 5 的圆。绘制时可使用关键点：(5, 0)、(−5, 0)、(0, 5)、(0, −5)，并画出经过这些点的平滑圆。

The second equation is a straight line with gradient 2 and y-intercept 1. Draw it by plotting (0, 1) and, say, (2, 5).

第二个方程是斜率为 2、y 截距为 1 的直线。通过 (0, 1) 和，例如，(2, 5) 两点来画出该直线。

From the graph, the line cuts the circle at two points. Suppose the coordinates read approximately (1.8, 4.6) and (−2.6, −4.2).

从图像上看，直线与圆相交于两点。假设读得的坐标约为 (1.8, 4.6) 和 (−2.6, −4.2)。

Solutions: x ≈ 1.8, y ≈ 4.6 and x ≈ −2.6, y ≈ −4.2

Notice that every point on the circle satisfies only the first equation, while every point on the line satisfies only the second. Only the two crossing points satisfy both, which is exactly what ‘simultaneous’ means.

请注意，圆上的每个点只满足第一个方程，而直线上的每个点只满足第二个方程。只有两个交点同时满足两个方程，这正是“联立”的含义。

8. Estimating Accuracy and Reading Scales | 估算精度与刻度读数

Graphical solutions are approximations, not exact answers. Your accuracy depends on the scale of your axes and the care you take in reading coordinates. Always estimate to the nearest half a small square, and do not give more decimal places than the graph reasonably supports.

图像解是近似值而非精确答案。你的精度取决于坐标轴的比例以及读取坐标时的仔细程度。始终估读到最小格的一半，不要给出超出图像合理支持范围的小数位数。

Use a ruler to read coordinates where possible, and draw thin construction lines from the intersection down to the x-axis and across to the y-axis. This avoids parallax errors and makes your method clear to the examiner.

尽可能使用直尺读取坐标，并从交点向下画细辅助线到 x 轴、水平画到 y 轴。这样可以避免视差误差，也让阅卷者看清你的方法。

If an intersection appears between two grid lines, record your reading as a decimal, such as 1.6 or 2.3, rather than a fraction like 8/5 unless the grid makes the fraction obvious.

如果交点位于两条网格线之间，请将读数记为小数，例如 1.6 或 2.3，除非网格使分数十分明显，否则不要写成 8/5 这样的分数。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Several errors appear repeatedly when students solve non-linear simultaneous equations graphically. Being aware of them will help you avoid losing easy marks.

学生在用图像法求解非线性联立方程时，有几个反复出现的错误。了解它们可以帮助你避免丢掉容易得到的分数。

Mistake 1: Plotting negative y-values incorrectly for curves like y = −x². Remember the negative sign applies to x², not to x.

错误 1：对 y = −x² 等曲线，负 y 值绘制错误。记住负号作用于 x²，而不是作用于 x。
Mistake 2: Drawing the parabola as V-shaped straight lines. A quadratic graph must be a smooth U-shaped curve.

错误 2：将抛物线画成 V 形直线。二次函数图像必须是平滑的 U 形曲线。
Mistake 3: Forgetting to check whether the line actually touches the circle or parabola. If they do not intersect in your drawing, re-check your plotting.

错误 3：忘记确认直线是否真的与圆或抛物线相交。如果图中没有相交，请重新检查你的描点。
Mistake 4: Giving only the x-values without the y-values. A coordinate pair is required for each solution.

错误 4：只给出 x 值而忽略 y 值。每个解都需要一个坐标对。
Mistake 5: Using a tiny grid that does not show the intersections clearly. Choose a scale that fits the whole range and shows every crossing point.

错误 5：使用过小的网格导致交点显示不清。选择能覆盖整个范围并显示所有交点的比例。

Exam tip: If the question provides a grid with pre-drawn axes, pay attention to the labelled scale before plotting. If the question says ‘estimate’, it accepts a range of values, so do not waste time trying to be exact.

考试技巧：如果题目提供了带坐标轴的网格，请先注意标注的比例再描点。如果题目说“estimate（估算）”，它接受一个数值范围，因此不必浪费时间追求精确。

10. When the Number of Solutions Varies | 解的个数何时变化

A useful insight when solving graphically is that the number of intersections tells you how many real solutions exist. A straight line and a circle may have two intersections, one tangent point, or none at all.

用图像法求解时有一个有用的视角：交点的个数告诉你存在多少个实数解。一条直线和一个圆可能有两个交点、一个切点，或完全没有交点。

For a tangent, the line touches the curve at exactly one point. In that case, the simultaneous equations have one repeated solution. For example, the line y = 5 is tangent to the circle x² + y² = 25 at the point (0, 5).

对于切线情形，直线与曲线恰好相切于一个点。此时，联立方程有一个重解。例如，直线 y = 5 与圆 x² + y² = 25 在点 (0, 5) 处相切。

If the line is too far from the curve, there is no intersection and therefore no real solution. The graph shows this immediately, whereas algebraic methods may involve a negative discriminant that is less intuitive.

如果直线离曲线太远，则没有交点，因此没有实数解。图像能立即显示这一点，而代数方法可能涉及负判别式，不够直观。

11. Linking Graphs and Algebra | 图像与代数的联系

You can check a graphical solution by solving algebraically. For y = x² and y = x + 2, set the expressions equal:

你可以通过代数求解来检验图像解。对于 y = x² 和 y = x + 2，令两个表达式相等：

x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0

x = 2 or x = −1

Substituting back: when x = 2, y = 4; when x = −1, y = 1. These match the graph intersections exactly. This confirms that the graphical reading of (−1, 1) and (2, 4) is correct.

代回原式：当 x = 2 时，y = 4；当 x = −1 时，y = 1。这与图像交点完全吻合，确认了从图像读出的 (−1, 1) 和 (2, 4) 是正确的。

In fact, solving the two equations algebraically is equivalent to finding the x-values where the two graphs cross. The graph is a visual representation of the same mathematics.

事实上，用代数方法解这两个方程，等价于找出两个图像交叉处的 x 值。图像是同一数学问题的直观表示。

12. Practice Questions for You | 请你练习的题目

Try the following questions on your own graph paper, then check your answers by substitution.

请在自己的方格纸上尝试以下题目，然后用代入法检验答案。

Q1: Draw y = x² − 2 and y = x on the axes −3 ≤ x ≤ 3. State the coordinates of the intersections.

第 1 题：在 −3 ≤ x ≤ 3 的坐标系中画出 y = x² − 2 和 y = x，写出交点的坐标。
Q2: On the axes −5 ≤ x ≤ 5, draw x² + y² = 16 and y = 2x − 1. Estimate the two intersection points.

第 2 题：在 −5 ≤ x ≤ 5 的坐标系中画出 x² + y² = 16 和 y = 2x − 1，估算两个交点。
Q3: How many real solutions does y = x² + 2 have with y = x? Sketch both graphs and justify your answer.

第 3 题：y = x² + 2 与 y = x 有多少个实数解？画出两个图像并证明你的答案。

For Q1, expect intersections near (−1, −1) and (2, 2). For Q3, the parabola sits entirely above the line, so there are no intersection points and hence no real solutions.

对于第 1 题，预计交点接近 (−1, −1) 和 (2, 2)。对于第 3 题，抛物线完全位于直线上方，因此没有交点，也没有实数解。

Graphical methods give you a powerful tool for solving non-linear simultaneous equations. By drawing each equation on the same axes and reading off the intersection points, you can find approximate solutions quickly, check your algebra, and understand the number of real solutions at a glance. Master this skill, and you will be well prepared for the Edexcel IGCSE examination.

图像法为你提供了解非线性联立方程的强大工具。通过在同一坐标系中绘制每个方程并读取交点，你可以快速找到近似解、检验代数结果，并一目了然地理解实数解的个数。掌握这项技能，你将为 Edexcel IGCSE 考试做好充分准备。

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Using Graphs to Solve Non-Linear Simultaneous Equations | 利用图像求解非线性联立方程