📚 Mastering Simultaneous Equations: Elimination and Substitution | 掌握联立方程：消元法与代入法
Simultaneous equations are a core topic in IGCSE Mathematics. They require you to find values that satisfy two or more equations at the same time, and they appear frequently in both Paper 2 and Paper 4-style questions.
联立方程是 IGCSE 数学的核心主题之一。它要求找出同时满足两个或多个方程的值，并且在 Paper 2 和 Paper 4 类型的题目中都经常出现。
This revision guide explains the two main algebraic methods, elimination and substitution, with worked examples, special cases, and exam tips to help you answer questions confidently.
本复习指南将讲解两种主要代数方法——消元法和代入法，并提供例题、特殊情况以及考试技巧，帮助你自信地解答相关题目。
1. What Are Simultaneous Equations? | 什么是联立方程？
A simultaneous equation system is a set of equations that share the same variables. In IGCSE, you will usually work with two linear equations in two unknowns, such as x and y, or with one linear and one quadratic equation.
联立方程组是由共享相同变量的一组方程组成。在 IGCSE 中，通常会遇到两个未知数（如 x 和 y）的两个线性方程，或者一个线性方程与一个二次方程组成的方程组。
The goal is to find the pair of values (x, y) that makes every equation true at the same time.
目标是找到使每个方程同时成立的一组值 (x, y)。
For example, the system below is a pair of simultaneous linear equations:
例如，下面这个方程组就是一组联立线性方程：
x + y = 7
2x − y = 2
Only one value of x and one value of y will satisfy both equations together.
只有一个 x 值和一个 y 值能够同时满足这两个方程。
2. The Graphical Meaning of a Solution | 解的几何意义
Each linear equation represents a straight line on the coordinate plane. The solution of a pair of simultaneous linear equations is the point where the two lines intersect.
每个线性方程在坐标平面上表示一条直线。一组联立线性方程的解就是两条直线的交点。
If the lines intersect at one point, the system has one unique solution. If the lines are parallel, there is no solution. If the two equations describe the same line, there are infinitely many solutions.
如果两条直线相交于一点，则方程组有唯一解；如果两条直线平行，则无解；如果两个方程描述的是同一条直线，则有无穷多解。
This geometric idea helps you check whether an algebraic answer is reasonable. A unique solution should give one clear intersection point.
这个几何概念可以帮助你判断代数答案是否合理。唯一解应该对应一个清晰的交点。
3. Method 1: Solving by Elimination | 方法一：消元法
Elimination is often the fastest method when both equations are linear. You add or subtract the equations to remove one variable.
当两个方程都是线性方程时，消元法通常是最快的方法。通过将方程相加或相减来消去一个变量。
To use elimination, make sure the coefficients of one variable are the same or opposite. Then add or subtract the left sides and right sides separately.
使用消元法时，需要使某个变量的系数相同或互为相反数，然后将等号左右两边分别相加或相减。
A useful set of steps is:
一组有用的步骤如下：
- Label the equations (1) and (2). 给方程编号为 (1) 和 (2)。
- Match the coefficient of x or y. 使 x 或 y 的系数相同。
- Add or subtract to eliminate that variable. 相加或相减以消去该变量。
- Solve the remaining one-variable equation. 解剩下的一元方程。
- Substitute back to find the other variable. 回代求出另一个变量。
Always check that you apply the same operation to both sides of the equation. This keeps the equation balanced.
务必对等号两边同时进行相同的运算。这样才能保持方程平衡。
4. Worked Example: Elimination with Positive Coefficients | 例题：正系数消元
Solve the system:
解方程组：
x + y = 7
2x − y = 2
Label the equations:
给方程编号：
(1) x + y = 7
(2) 2x − y = 2
Add equation (1) and equation (2) to eliminate y because +y and −y cancel.
将方程 (1) 与方程 (2) 相加，因为 +y 与 −y 可以相互抵消。
3x = 9
Divide both sides by 3:
两边同时除以 3：
x = 3
Substitute x = 3 into equation (1):
将 x = 3 代入方程 (1)：
3 + y = 7
Subtract 3 from both sides:
两边同时减去 3：
y = 4
Therefore, the solution is x = 3, y = 4.
因此，解为 x = 3, y = 4。
eck by substituting both values into equation (2): 2(3) − 4 = 6 − 4 = 2, which is correct.
你可以将两个值代入方程 (2) 检验：2(3) − 4 = 6 − 4 = 2，结果正确。
5. Worked Example: Elimination by Multiplying One Equation | 例题：一方程乘以常数后消元
Solve the system:
解方程组：
3x + 2y = 12
x − y = 1
Label the equations and multiply equation (2) by 2 so the y-coefficients become +2y and −2y.
给方程编号，并将方程 (2) 乘以 2，使 y 的系数变为 +2y 和 −2y。
(1) 3x + 2y = 12
(3) 2x − 2y = 2
Add equations (1) and (3):
将方程 (1) 与 (3) 相加：
5x = 14
Divide by 5:
两边除以 5：
x = 14/5 or 2.8
Substitute x = 14/5 into equation (2):
将 x = 14/5 代入方程 (2)：
