📚 Solving Simultaneous Equations | 联立方程专题讲解
Simultaneous equations are a core topic in IGCSE mathematics. They test your ability to work with two or more unknown values at the same time. A solid understanding of this topic will make later problems in algebra, coordinate geometry and calculus much easier.
联立方程是 IGCSE 数学的核心考点。它考查你同时处理两个或多个未知数的能力。扎实掌握这一主题，会使后续代数、坐标几何和微积分等内容的学习轻松很多。
1. What Are Simultaneous Equations? | 什么是联立方程？
A linear equation with two variables, for example x + y = 10, has infinitely many solutions. If we add a second equation, for example x − y = 2, we are looking for values of x and y that satisfy both equations at the same time.
一个包含两个变量的线性方程，例如 x + y = 10，有无穷多组解。当我们再加入一个方程，例如 x − y = 2 时，就需要找到同时满足这两个方程的 x 和 y 的值。
x + y = 10
x − y = 2
Here the common solution is x = 6 and y = 4. The ordered pair (6, 4) is the point where the two straight lines cross on a graph.
这里公共解是 x = 6，y = 4。有序数对 (6, 4) 就是两条直线在图像上的交点。
2. The Graphical Method | 图像法
To solve simultaneous equations graphically, rearrange each equation into the form y = mx + c. Then plot both lines on the same axes.
用图像法解联立方程时，先把每个方程改写成 y = mx + c 的形式，然后在同一坐标系中画出两条直线。
For the equations above:
对于上面的方程组：
y = 10 − x
y = x − 2
The first line has slope −1 and y-intercept 10. The second line has slope 1 and y-intercept −2. Their intersection is at (6, 4).
第一条直线斜率为 −1，y 轴截距为 10；第二条直线斜率为 1，y 轴截距为 −2。两条直线相交于点 (6, 4)。
If the lines are parallel, there is no solution. If both equations represent the same line, there are infinitely many solutions.
如果两直线平行，则方程组无解；如果两个方程表示同一条直线，则方程组有无穷多组解。
3. Substitution Method | 代入消元法
The substitution method is especially useful when one variable has coefficient 1, or when one equation is already written as y = … or x = … .
代入消元法特别适合某个变量系数为 1，或者其中一个方程已经写成 y = … 或 x = … 的情形。
Example: solve
例如，解方程组：
y = 2x + 1
3x + y = 11
-
Substitute y = 2x + 1 into the second equation.
将 y = 2x + 1 代入第二个方程。
-
3x + (2x + 1) = 11
得到 3x + (2x + 1) = 11。
-
Simplify: 5x + 1 = 11, so 5x = 10 and x = 2.
化简：5x + 1 = 11，所以 5x = 10，x = 2。
-
Substitute back: y = 2 × 2 + 1 = 5.
回代：y = 2 × 2 + 1 = 5。
So the solution is x = 2, y = 5.
因此解为 x = 2，y = 5。
4. Elimination Method | 加减消元法
Elimination means adding or subtracting the equations to remove one variable. Use it when the coefficients of x or y match, or can easily be matched.
加减消元法是通过将两个方程相加或相减，消去其中一个变量。当 x 或 y 的系数相同或容易化成相同时，这种方法很有效。
Example: solve
例如，解方程组：
2x + 3y = 12
4x − 3y = 6
Add the two equations to eliminate y:
将两个方程相加，消去 y：
2x + 4x + 3y − 3y = 12 + 6
6x = 18
x = 3
Then substitute x = 3 into 2x + 3y = 12:
然后把 x = 3 代入 2x + 3y = 12：
2 × 3 + 3y = 12
6 + 3y = 12
3y = 6
y = 2
The solution is x = 3, y = 2.
解为 x = 3，y = 2。
5. Choosing the Best Method | 如何选择最佳方法
Different questions call for different methods. A quick comparison can help you decide.
不同题目适合不同方法。快速对比可以帮助你做出选择。
| Method 方法 |
When to use it 使用时机 |
Example 示例 |
| Graphical 图像法 |
You need to see the intersection, or the question asks you to draw graphs. 需要直观看到交点，或题目要求画图。 |
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