📚 Simultaneous Equations in IGCSE Mathematics | IGCSE 数学联立方程精讲
In IGCSE Mathematics, simultaneous equations appear frequently in both Core and Extended papers. They require you to find values of unknowns that satisfy two or more equations at the same time. This article explains elimination, substitution, graphical methods, word problems, special cases, and common errors, with clear worked examples for revision.
在 IGCSE 数学中，联立方程在 Core 和 Extended 试卷中都经常出现。它要求你求出同时满足两个或更多方程的未知数的值。本文将讲解消元法、代入法、图像法、应用题、特殊情形和常见错误，并配有清晰的例题，方便复习。
1. What Are Simultaneous Equations? | 什么是联立方程？
uations are a set of two or more equations that share the same unknown variables. The word "simultaneous" 2x + y = 7 and x − y = 2 form a system with two unknowns. Their solution is x = 3, y = 1, because 2(3) + 1 = 7 and 3 − 1 = 2.
在 IGCSE 数学中，联立方程是一组共享相同未知数的两个或更多方程。"联立"意味着这些方程必须同时成立。解是使方程组中每个方程都成立的有序数对 (x, y)。例如，2x + y = 7 和 x − y = 2 组成一个含两个未知数的方程组。它们的解是 x = 3，y = 1，因为 2(3) + 1 = 7 且 3 − 1 = 2。
If there are two unknowns, you normally need two independent equations. One equation alone has infinitely many solutions, because you can choose any x and then find a matching y. In the same way, three unknowns generally need three independent equations before a unique solution can be found.
如果有两个未知数，通常需要两个独立方程。单独一个方程有无穷多个解，因为你可以任选 x，再求出对应的 y。同样地，三个未知数通常需要三个独立方程才能找到唯一解。
2x + y = 7, x − y = 2 → x = 3, y = 1
eet one linear equation and one non-linear equation, such as a quadratic equation. The basic aim is still the same: find values that work in both equations.
在 IGCSE 阶段，大多数联立方程是线性的，即每个方程都可以写成 ax + by = c 的形式。在 Extended 试卷中，你还可能遇到一个线性方程和一个非线性方程，例如二次方程。基本目标仍然相同：找出同时满足两个方程的值。
2. Core vs Extended Expectations | Core 与 Extended 的考查要求
Core candidates usually solve two linear equations with two unknowns by elimination or substitution. They may also read a solution from a given graph or plot two lines and find their intersection. The numbers are usually straightforward, and the working is often guided by the question.
Core 考生通常用消元法或代入法解两个二元一次方程。他们也可能从给定的图像上读取解，或画出两条直线并求交点。数字通常比较直接，题目也常常会引导解题过程。
Extended candidates are expected to handle more complex forms. These include equations with brackets, fractions, or negative coefficients, as well as word problems that must be translated into equations. Some Extended questions combine a linear equation with a quadratic, giving two possible solutions. Three-unknown systems may also appear as extension work.
Extended 考生需要处理更复杂的形式，包括含括号、分数或负系数的方程，以及需要转化为方程的应用题。有些 Extended 题目会把一个线性方程和一个二次方程结合起来，得到两个可能的解。三个未知数的方程组也可能作为拓展内容出现。
| Area | 领域 | Core | Core 要求 | Extended | Extended 要求 |
| Linear systems | 线性方程组 | Two equations, two unknowns | 两个方程、两个未知数 | Also fractions, brackets, negatives | 还包括分数、括号、负数 |
| Graphs | 图像 | Read intersection from a graph | 从图像读取交点 | Draw and interpret graphs | 画图并解释图像 |
| Word problems | 应用题 | Simple contexts | 简单情境 | Multi-step contexts | 多步骤情境 |
| Non-linear | 非线性 | Not usually required | Linear + quadratic | 线性 + 二次 |
The syllabus expects clear working, not just a final answer. Marks are often awarded for a correct method even if the final value has a small arithmetic slip. This is why you should always show the equations you form and the steps you take to solve them.
考纲要求写出清晰过程，而不只是最终答案。即使最后数值有小的算术失误，正确方法通常也能得分。因此，你应该始终写出所列方程以及解题步骤。
3. The Elimination Method | 消元法
oth equations so that the coefficients of x or y have the same magnitude. Add if the signs are opposite, and subtract if the signs are the same. This leaves a single equation with one unknown.
消元法通过相加或相减方程来消去一个变量。首先，把两个方程整理成 ax + by = c 的形式。然后通过乘一个或两个方程，使 x 或 y 的系数绝对值相同。符号相反时相加，符号相同时相减。这样便留下一个含一个未知数的一元方程。
Step 1: Write both equations neatly, one below the other, with like terms aligned. Step 2: Choose the variable that is easier to eliminate. Look for coefficients that are the same, opposite, or easy to match by multiplying. Step 3: Multiply one or both equations if necessary. Step 4: Add or subtract to remove that variable. Step 5: Solve for the remaining variable. Step 6: Substitute back into one original equation to find the other variable. Step 7: Check both values in both original equations.
第 1 步：把两个方程整齐写出，上下对齐同类项。第 2 步：选择更容易消去的变量。寻找相同、相反或容易通过乘法配平的系数。第 3 步：如有必要，乘一个或两个方程。第 4 步：相加或相减，消去该变量。第 5 步：解出剩下的变量。第 6 步：代回其中一个原方程，求出另一个变量。第 7 步：把两个值代入两个原方程检验。
3x + 2y = 16, 5x − 2y = 8 → 8x = 24 → x = 3 → y = 3.5
7 and y = 3.5. Always write the substitution step clearly, because examiners award method marks for it.
在这个例子中，y 项是 +2y 和 −2y，所以把两个方程相加可立即消去 y。得到 x = 3 后，代入 3x + 2y = 16：9 + 2y = 16，所以 2y = 7，y = 3.5。要清楚写出代入步骤，因为考官会为此给方法分。
= 3. Substitute into 3x − y = 7: 9 − y = 7, so y = 2. Check: 2(3) + 3(2) = 12 and 3(3) − 2 = 7.
第二个例子是 2x + 3y = 12 和 3x − y = 7。把第二个方程乘 3，得到 9x − 3y = 21。此时 y 项是 +3y 和 −3y，相加得 11x = 33，所以 x = 3。代入 3x − y = 7：9 − y = 7，所以 y = 2。检验：2(3) + 3(2) = 12 且 3(3) − 2 = 7。
4. The Substitution Method | 代入法
Substitution is useful when one equation is already solved for one variable, or when a coefficient is 1 or −1. Rearrange one equation to make x or y the subject, then replace that variable in the other equation. Solve the resulting equation, and substitute back to find the other variable. This method is especially important for non-linear systems in Extended papers.
当一个方程已经解出某个变量，或某个系数是 1 或 −1 时，代入法很方便。把一个方程变形，使 x 或 y 成为主元，然后代入另一个方程。解出所得方程，再代回求另一个变量。这个方法对 Extended 试卷中的非线性方程组尤其重要。
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