📚 Solving Simultaneous Equations | 解联立方程
Simultaneous equations are a set of two or more equations that share the same variables. Solving them means finding the values of the variables that satisfy all equations at the same time. This is one of the most useful tools in IGCSE Mathematics, appearing in algebra, geometry, and real-life problems.
联立方程是一组含有相同变量的两个或更多方程。解联立方程就是找到同时满足所有方程的变量值。这是 IGCSE 数学中最有用的工具之一，出现在代数、几何和现实生活问题中。
1. What Are Simultaneous Equations? | 什么是联立方程？
A linear simultaneous equation system looks like this:
线性联立方程组如下所示：
3x + 2y = 12
x − y = 1
Each equation represents a straight line on a graph. The solution is the point where the two lines intersect, which is the pair (x, y) that works for both equations.
每个方程在图象上都表示一条直线。解就是两条直线的交点，即同时满足两个方程的一对 (x, y) 值。
There are three main methods for solving simultaneous linear equations: elimination, substitution, and graphical. You should be comfortable with all three, because the best method depends on the question.
解线性联立方程主要有三种方法：消元法、代入法和图象法。你应该熟练掌握这三种方法，因为最佳方法取决于题目类型。
2. The Elimination Method | 消元法
The elimination method involves adding or subtracting the equations to cancel one variable. This is usually the fastest method when the coefficients are simple.
消元法是通过将两个方程相加或相减来消去一个变量。当系数比较简单时，这通常是最快的方法。
Example: Solve 3x + 2y = 12 and x − y = 1.
例题：解 3x + 2y = 12 和 x − y = 1。
Step 1: Make the coefficients of one variable the same. Multiply the second equation by 2:
第一步：使某个变量的系数相同。将第二个方程乘以 2：
2x − 2y = 2
Step 2: Add the two equations to eliminate y:
第二步：将两个方程相加，消去 y：
(3x + 2y) + (2x − 2y) = 12 + 2
5x = 14
x = 2.8
Step 3: Substitute x = 2.8 into x − y = 1:
第三步：将 x = 2.8 代入 x − y = 1：
2.8 − y = 1
y = 1.8
Step 4: Write the answer as a coordinate pair: (2.8, 1.8). Always check by substituting both values into the original equations.
第四步：将答案写成坐标对：(2.8, 1.8)。务必把两个值代入原方程进行检验。
When the coefficients of the same variable are the same sign, subtract; when they are opposite signs, add. This idea is key to solving quickly.
当同一个变量的系数符号相同时用减法；符号相反时用加法。这是快速解题的关键。
- If coefficients are equal and same sign → subtract the equations.
- 如果系数相等且符号相同 → 两式相减。
- If coefficients are equal and opposite sign → add the equations.
- 如果系数相等且符号相反 → 两式相加。
- If coefficients differ → multiply one or both equations first.
- 如果系数不同 → 先将一个或两个方程乘以适当的数。
3. The Substitution Method | 代入法
The substitution method works by rearranging one equation to make one variable the subject, then substituting that expression into the other equation.
代入法的步骤是：先将一个方程变形，把一个变量用另一个变量表示出来，然后将该表达式代入另一个方程。
Example: Solve y = 2x + 3 and 3x + 2y = 13.
例题：解 y = 2x + 3 和 3x + 2y = 13。
Step 1: The first equation already has y as the subject. Substitute y = 2x + 3 into 3x + 2y = 13:
第一步：第一个方程已经将 y 表示为 x 的形式。将 y = 2x + 3 代入 3x + 2y = 13：
3x + 2(2x + 3) = 13
3x + 4x + 6 = 13
7x = 7
x = 1
Step 2: Substitute x = 1 back into y = 2x + 3:
第二步：将 x = 1 代回 y = 2x + 3：
y = 2(1) + 3 = 5
The solution is (1, 5). The substitution method is especially useful when one equation is already solved for a variable, or when the system is non-linear.
解为 (1, 5)。当其中一个方程已经用某个变量表示时，或者方程组是非线性时，代入法尤其有用。
4. The Graphical Method | 图象法
To solve simultaneous equations graphically, draw both lines on the same axes and read off the coordinates of the intersection point.
用图象法解联立方程时，需要在同一坐标系中画出两条直线，然后读出交点的坐标。
Example: Solve x + y = 5 and 2x − y = 1 graphically.
例题：用图象法解 x + y = 5 和 2x − y = 1。
First, rearrange each equation into the form y = mx + c:
首先，将每个方程变形为 y = mx + c 的形式：
y = −x + 5
y = 2x − 1
| x | y = −x + 5 | y = 2x − 1 |
| 0 | 5 | −1 |
| 2 | 3 | 3 |
Both lines pass through (2, 3), so the solution is x = 2, y = 3. The graphical method is slower and less precise but very useful for visualising problems.
两条直线都经过 (2, 3)，所以解为 x = 2, y = 3。图象法较慢且精度较低，但非常有助于直观理解问题。
5. Solving Word Problems | 解应用题
Many IGCSE questions require you to set up equations from a word problem. The key steps are:
许多 IGCSE 题目要求你根据文字叙述建立方程。关键步骤如下：
- Define your variables clearly in terms of the question.
- 根据题意明确设定变量。
- Form one equation for each piece
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