📚 Mastering Simultaneous Equations for Cambridge IGCSE Mathematics | 剑桥 IGCSE 数学联立方程核心技巧
Simultaneous equations are a fundamental algebra topic in the Cambridge IGCSE Mathematics syllabus (0580). They require you to find the values of unknown variables that satisfy two or more equations at the same time. This article will guide you through the key methods, common pitfalls and exam-style applications.
联立方程是剑桥 IGCSE 数学（0580）代数部分的核心内容。题目要求我们找到一组未知数的值，使两个或更多方程同时成立。本文带你系统掌握主要解法、常见陷阱以及考试中的实际应用。
1. What are Simultaneous Equations? | 什么是联立方程？
In Cambridge IGCSE Mathematics, a system of simultaneous equations usually consists of two equations with two unknown variables, such as x and y. The solution is an ordered pair (x, y) that satisfies both equations at the same time. Linear simultaneous equations can be solved algebraically or graphically.
在剑桥 IGCSE 数学中，联立方程组通常由两个含有两个未知数（如 x 和 y）的方程组成。它的解是一个有序数对 (x, y)，能够同时满足两个方程。线性联立方程可以通过代数方法或图像方法求解。
x + y = 8
x − y = 2
raight lines intersect. In this example, the pair x = 5, y = 3 makes both statements true.
联立方程的解就是两条直线交点的坐标。例如本例中，x = 5，y = 3 能使两个等式同时成立。
2. Why Simultaneous Equations Matter | 为什么联立方程很重要
Simultaneous equations allow you to model situations with two unknown quantities linked by two conditions. In IGCSE exams, they appear in algebra, coordinate geometry, and word problems. Strong skills here support later topics such as functions, sequences and quadratic systems.
联立方程可以用来描述含有两个未知量且由两个条件相互联系的实际情形。在 IGCSE 考试中，联立方程常出现在代数、坐标几何和文字应用题中。掌握扎实也能为后续的函数、数列和二次方程组等内容打好基础。
For example, if you buy 2 textbooks and 3 pens for £13, and another day 1 textbook and 1 pen for £6, you can form two equations and solve for the price of each item.
例如，如果买 2 本教材和 3 支笔花费 13 英镑，另一次买 1 本教材和 1 支笔花费 6 英镑，就可以建立两个方程，求出每件物品的价格。
3. The Substitution Method | 代入法
The substitution method involves three steps. First, rearrange one equation to make one variable the subject. Second, substitute this expression into the other equation. Third, solve the resulting single-variable equation and back-substitute to find the second variable. This method is especially useful when one coefficient is 1 or when a variable is already isolated.
代入法包括三步：第一步，把其中一个方程变形，使某个未知数成为主体；第二步，把这个表达式代入另一个方程；第三步，解得到的一元方程，再回代求出另一个未知数。当某个未知数的系数为 1 或已经被单独表示出来时，代入法特别方便。
Solve: y = 2x + 1 and 3x + y = 16
+ 1) = 16, so 5x = 15, giving x = 3. Then y = 2(3) + 1 = 7. The solution is x = 3, y = 7.
将 y = 2x + 1 代入 3x + y = 16：3x + (2x + 1) = 16，所以 5x = 15，x = 3。再得 y = 2(3) + 1 = 7。解为 x = 3，y = 7。
4. The Elimination Method | 消元法
ou add or subtract the equations to cancel one variable. If the coefficients are not already equal, multiply one or both equations by suitable constants first. Then solve the simple equation that remains.
当两个方程都具有 ax + by = c 的形式时，消元法通常更快。通过将两个方程相加或相减，抵消其中一个未知数。如果系数不相等，需要先给一个或两个方程乘以适当的常数。然后解得到的一元方程即可。
3x + 2y = 12
5x − 2y = 4
Adding the two equations gives
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