📚 Solving Simultaneous Equations | 解联立方程
Simultaneous equations are a core topic in IGCSE Mathematics. They appear in algebra papers, coordinate geometry and real-life problems. This guide explains the main methods clearly, with worked examples and exam-style tips.
联立方程是 IGCSE 数学的核心内容，常出现在代数试卷、坐标几何以及实际应用题中。本指南将清晰讲解主要方法，并附有例题和考试技巧。
1. What Are Simultaneous Equations? | 什么是联立方程？
A linear equation in two variables, such as ax + by = c, has infinitely many solutions. A simultaneous system asks us to find the single pair of values (x, y) that satisfies two or more equations at the same time.
含两个变量的线性方程如 ax + by = c 有无穷多组解。而联立方程组要求我们找出同时满足两个或两个以上方程的唯一一组值 (x, y)。
Key fact: For two straight lines, the solution is the intersection point. If the lines are parallel, there is no solution; if they are identical, there are infinitely many solutions.
关键事实：两条直线的交点就是方程组的解。若两直线平行，则无解；若两直线重合，则有无穷多组解。
2. The Graphical Method | 图解法
To solve a pair of simultaneous equations graphically, plot both lines on the same set of axes. Read the coordinates of the intersection point directly from the graph.
用图解法解联立方程时，需要把两条直线画在同一坐标系中，然后直接从图上读出交点的坐标。
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Rearrange each equation into the form y = mx + c.
把每个方程整理成 y = mx + c 的形式。
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Plot both lines accurately using a ruler.
用直尺准确画出两条直线。
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Write down the intersection point (x, y).
写下交点的坐标 (x, y)。
For example, the lines y = x + 1 and y = -2x + 7 intersect at (2, 3), so x = 2 and y = 3.
例如，直线 y = x + 1 与 y = -2x + 7 交于点 (2, 3)，因此 x = 2，y = 3。
3. The Substitution Method | 代入法
Substitution is the most reliable algebraic method. It works especially well when one equation already has a variable as its subject.
代入法是最可靠的代数方法之一，尤其适合某个方程中已有变量被单独表示的情况。
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Make one variable the subject of one equation.
在其中一个方程中，用另一个变量表示该变量。
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Substitute this expression into the other equation.
将这个表达式代入另一个方程。
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Solve the resulting linear equation.
解出得到的一元一次方程。
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Substitute back to find the second variable.
代回原式求出第二个变量。
Worked example: Solve y = x + 1 and 2x + y = 10.
例题：解方程组 y = x + 1 和 2x + y = 10。
2x + (x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3
Then y = 3 + 1 = 4, so the solution is x = 3, y = 4.
于是 y = 3 + 1 = 4，所以解为 x = 3，y = 4。
4. The Elimination Method | 消元法
In the elimination method, we multiply one or both equations so that the coefficients of one variable become equal or opposite. Then we add or subtract to eliminate that variable.
消元法的思路是：将一个方程或两个方程乘以适当数值，使某个变量的系数相同或互为相反数，然后通过相加或相减消去该变量。
Worked example: Solve 3x + 2y = 12 and 5x – 2y = 4.
例题：解方程组 3x + 2y = 12 和 5x – 2y = 4。
(3x + 2y) + (5x – 2y) = 12 + 4 → 8x = 16 → x = 2
Substitute x = 2 into 3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3.
将 x = 2 代入 3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3。
5. Solving Linear and Quadratic Systems | 线性与二次方程组
When one equation is linear and the other is quadratic, substitute the linear expression into the quadratic equation. This gives a quadratic equation in one variable, which you solve by factorisation, completing the square, or the quadratic formula.
当一个方程是线性方程、另一个是二次方程时，把线性表达式代入二次方程，得到一个关于单个变量的一元二次方程，再利用因式分解、配方法或二次公式求解。
Worked example: Solve y = x² – 3x + 1 and y = 2x – 5.
例题：解方程组 y = x² – 3x + 1 和 y = 2x – 5。
x² – 3x + 1 = 2x – 5 → x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0
Hence x = 2 or x = 3. When x = 2, y = -1; when x = 3, y = 1. The solutions are (2, -1) and (3, 1).
因此 x = 2 或 x = 3。当 x = 2 时，y = -1；当 x = 3 时，y = 1。解为 (2, -1) 和 (3, 1)。
6. Solving Word Problems | 应用题
Word problems require you to translate the given information into a pair of equations. Define your variables clearly before writing the equations.
应用题需要你把题目中的信息转化为两个方程。列方程之前，要清楚定义变量。
Worked example: The sum of two numbers is 15 and their difference is 3. Find the numbers.
例题：两个数的和是 15，差是 3。求这两个数。
x + y = 15 and x – y = 3
Adding the equations gives 2x = 18 → x = 9. Then 9 + y = 15 → y = 6. The numbers are 9 and 6.
两式相加得 2x = 18 → x = 9。再由 9 + y = 15 → y = 6。这两个数是 9 和 6。
7. Common Mistakes | 常见错误
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Sign errors when substituting negative expressions into another equation.
代入含负号的表达式时，出现符号错误。
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Forgetting to multiply every term of an equation during
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