📚 Solving Other Types of Equations Using Graphs | 利用图像解其他类型方程
Graphs are not just pictures of equations — they are powerful problem-solving tools. In this tutorial, we learn how to use graphs to solve equations that are not simple linear or quadratic, including cubic, reciprocal, and equations that need rearranging.
图像不只是方程的图像——它们是强大的解题工具。在本教程中，我们学习如何利用图像解非简单线性或二次方程，包括三次方程、反比例方程，以及需要重排的方程。
1. Why Use Graphs for ‘Other’ Equations? | 为什么用图像解“其他”方程？
Some equations are hard to solve algebraically, especially cubic equations or equations mixing x with 1/x. Drawing graphs gives a quick visual answer, especially when you only need approximate solutions.
有些方程很难用代数方法解，尤其是三次方程或同时含 x 和 1/x 的方程。画图能快速给出直观答案，尤其是只需要近似解时。
| Equation Type | Number of Solutions (usually) | Example |
|---|---|---|
| Linear | 1 | 3x + 2 = 11 |
| Quadratic | 0, 1 or 2 | x² − 5x + 6 = 0 |
| Cubic | 1, 2 or 3 | x³ − 4x = 0 |
| Reciprocal | 0, 1 or 2 | 2/x = x − 1 |
2. Solving Equations When the Graph is Already Drawn | 当图像已经画好时解方程
If you are given y = x³ − 2x and asked to solve x³ − 2x = 1, just draw the horizontal line y = 1 on the same axes. The x-coordinates of the intersection points are the solutions.
若给你 y = x³ − 2x 并要你解 x³ − 2x = 1，只需在同一坐标系上画水平线 y = 1，交点的 x 坐标就是解。
Similarly, to solve x³ − 2x = −3, draw y = −3 and read off the x-values at the intersections.
类似地，解 x³ − 2x = −3 时画 y = −3，读出交点处的 x 值。
Worked Example | 例题
The graph of y = x³ − 2x is given. Solve x³ − 2x = 1.
已知 y = x³ − 2x 的图像，解 x³ − 2x = 1。
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Draw the line y = 1 on the same axes.
在同一坐标系上画直线 y = 1。
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Find the x-coordinates of the intersection points.
找出交点的 x 坐标。
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The solutions are approximately x ≈ −1.3, x ≈ 0.6, x ≈ 1.7.
解约为 x ≈ −1.3、x ≈ 0.6、x ≈ 1.7。
x ≈ −1.3, 0.6, 1.7
3. Rearranging the Equation to Match the Graph | 重排方程以匹配图像
Sometimes the equation to solve looks different from the given graph. For example, given y = x² − 2x + 1, to solve x² − 2x − 3 = 0, rewrite it as x² − 2x + 1 = 4, so the intersection with y = 4 gives the solutions.
有时要解的方程与已给图像不同。例如，已知 y = x² − 2x + 1，要解 x² − 2x − 3 = 0，可重写为 x² − 2x + 1 = 4，因此与 y = 4 的交点即为解。
The general idea: keep the LHS exactly equal to the given expression, and move everything else to the RHS. Then the RHS becomes the horizontal line to draw.
一般思路：保持左边与已给表达式完全一致，把其余项移到右边，右边就是你要画的水平线。
Worked Example | 例题
Given y = x² − 2x + 1, solve x² − 2x − 3 = 0 by drawing a suitable line.
已知 y = x² − 2x + 1，通过画合适的直线解 x² − 2x − 3 = 0。
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Rewrite: x² − 2x + 1 = 4.
改写为 x² − 2x + 1 = 4。
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Plot the horizontal line y = 4.
画水平线 y = 4。
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The x-coordinates of the intersections are the solutions.
交点的 x 坐标就是解。
x = −1 or x = 3
4. Solving Equations Using Two Graphs | 利用两个图像解方程
To solve f(x) = g(x), draw y = f(x) and y = g(x) on the same axes; the solutions are the x-coordinates of their intersection points.
要解 f(x) = g(x)，在同一坐标系上画 y = f(x) 和 y = g(x)，解就是它们交点的 x 坐标。
Worked Example | 例题
Using the graphs of y = x³ and y = 2x + 1, solve x³ = 2x + 1.
利用 y = x³ 和 y = 2x + 1 的图像，解 x³ = 2x + 1。
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Plot both curves on the same set of axes.
在同一坐标系中画出两条曲线。
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The x-coordinates of the intersection points are the solutions.
交点的 x 坐标即为解。
x ≈ −0.6, 1.5
5. Solving Cubic Equations Graphically | 用图像解三次方程
A cubic equation like x³ + 2x − 4 = 0 can be solved by drawing y = x³ and y = −2x + 4, or by drawing y = x³ + 2x − 4 and reading the x-axis crossings.
三次方程如 x³ + 2x − 4 = 0 可以通过画 y = x³ 和 y = −2x + 4，或画 y = x³ + 2x − 4 并读 x 轴交点来解。
The x-axis crossings of y = f(x) correspond to solving f(x) = 0.
y = f(x) 与 x 轴的交点对应解 f(x) = 0。
Worked Example | 例题
Solve x³ + 2x − 4 = 0 graphically.
用图像法解 x³ + 2x − 4 = 0。
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Draw y = x³ + 2x − 4.
画 y = x³ + 2x − 4。
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Read the x-coordinate where it crosses the x-axis.
读出与 x 轴交点的 x 坐标。
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The solution is approximately x ≈ 1.18.
解约为 x ≈ 1.18。
6. Solving Reciprocal and Fractional Equations | 解反比例和分式方程
Equations involving 1/x are hard to solve by hand. To solve 2/x = x − 1, plot y = 2/x and y = x − 1; the intersections give the solutions.
含 1/x 的方程很难手算。要解 2/x = x − 1，画 y = 2/x 和 y = x − 1，交点给出解。
Worked Example | 例题
Solve 2/x = x − 1 using graphs.
利用图像解 2/x = x − 1。
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Plot y = 2/x (a reciprocal graph).
画 y = 2/x（反比例图像）。
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Plot y = x − 1 (a straight line).
画 y = x − 1（直线）。
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The x-coordinates of the intersections are x ≈ −1 and x ≈ 2.
交点的 x 坐标约为 x ≈ −1 和 x ≈ 2。
7. Choosing a Suitable Line to Draw | 选择适合画的直线
When a graph y = f(x) is given and you must solve an equation, rewrite the equation so that the left side is f(x). The right side becomes the line y = m, or another curve you can handle.
当给定图像 y = f(x) 且需解某方程时，改写方程使左边为 f(x)，右边就成为直线 y = m 或另一条你能处理的曲线。
Golden Rules | 黄金法则
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Keep the given expression unchanged on one side.
保持给定表达式不变在一边。
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Move all other terms to the other side.
把所有其他项移到另一边。
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Read the intersection points carefully.
仔细读出交点。
8. Estimating and Reading Solutions Accurately | 准确估算并读出解
Reading solutions from a graph requires care. Use the grid lines, estimate halfway points, and give answers to one decimal place unless asked otherwise.
从图像上读解需要仔细。利用网格线，估算中间点，除非另有要求，否则答案给到一位小数。
| Reading Mistake | How to Avoid |
|---|---|
| Wrong point | Check the y-value matches |
| Wrong scale | Check each square’s value |
| Rough estimate | Use the nearest grid lines |
9. Checking Solutions by Substitution | 通过代入检验解
Always check a graphical solution by substituting back into the original equation. If the left and right sides are close, your answer is reliable.
务必把图像解代回原方程检验。如果左右两边很接近，你的答案就是可靠的。
Quick Check | 快速检验
Check x ≈ 1.5 for x³ = 2x + 1.
检验 x ≈ 1.5 是否满足 x³ = 2x + 1。
LHS: 1.5³ = 3.375; RHS: 2(1.5) + 1 = 4. They are not close, so 1.5 is not a good solution.
左边 1.5³ = 3.375；右边 2(1.5) + 1 = 4。两边不接近，所以 1.5 不是好的解。
10. Common Exam Question Types | 常见考试题型
Typical questions ask: solve f(x) = 0, solve f(x) = k, solve f(x) = g(x), or ask which line to draw. Practise all types.
典型问题包括：解 f(x) = 0、解 f(x) = k、解 f(x) = g(x)，或问应画哪条线。要练习所有类型。
Exam-Style Example | 考试风格例题
The graph y = x³ − 3x is shown. Which line should be drawn to solve x³ − 3x + 5 = 0?
图中给出 y = x³ − 3x。应画哪条线来解 x³ − 3x + 5 = 0？
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Rewrite: x³ − 3x = −5.
改写为 x³ − 3x = −5。
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So draw the horizontal line y = −5.
所以画水平线 y = −5。
11. Summary | 总结
Graphical methods turn algebra problems into geometry problems. They are useful for cubics, reciprocals, and any equation that is hard to solve exactly.
图像法把代数问题变成几何问题。它适用于三次、反比例以及任何难以精确求解的方程。
Master the skill of reading, rearranging, and checking — and graphical equation solving becomes a reliable exam technique.
掌握读取、重排和检验的技巧——利用图像解方程就会成为可靠的考试技能。
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