Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are a core topic in IGCSE Mathematics. This guide explains the three essential methods – graphical, substitution and elimination – as well as how to handle systems that include a quadratic equation. Every method is illustrated with step-by-step examples and exam-style reasoning.

联立方程是IGCSE数学的核心考点。本指南系统讲解三种基本解法：图像法、代入法和消元法，并深入介绍含二次方程的联立方程组。每个方法均配有分步例题与考试型推理过程。

1. What Are Simultaneous Equations? | 什么是联立方程

A system of simultaneous equations consists of two or more equations that share the same unknown variables. In IGCSE Mathematics, you will normally meet two equations in two unknowns, usually x and y. The solution is an ordered pair (x, y) that satisfies both equations at the same time.

联立方程组由两个或更多共享未知变量的方程组成。在IGCSE数学中，最常见的形式是含有两个未知数（通常为x和y）的两个方程。方程组的解是一个有序数对 (x, y)，它必须同时满足两个方程。

For example, consider the system below. Only x = 3 and y = 2 make both equations true at once.

例如，对于下面的方程组，只有 x = 3, y = 2 才能同时使两个方程成立。

x + y = 5 与 x − y = 1

Depending on the equations, a system may have exactly one solution, infinitely many solutions when the two lines coincide, or no solution when the lines are parallel.

根据方程的不同，方程组可能只有一个解；当两条直线重合时有无穷多解；当两条直线平行时则无解。

2. The Graphical Method | 图像法

To solve by the graphical method, draw both lines on the same set of axes. The coordinates of their intersection point give the solution of the system.

图像法的步骤是把两条直线画在同一坐标系中，交点的坐标就是方程组的解。

Rearrange each equation into the form y = mx + c.

将每个方程整理为 y = mx + c 的形式。
Plot both lines on the same axes as accurately as possible.

在同一坐标系中尽可能精确地画出两条直线。
Read the coordinates of the intersection point carefully.

仔细读出交点的坐标。

Take the system x + y = 5 and x − y = 1. Rearranging gives y = −x + 5 and y = x − 1.

以方程组 x + y = 5 与 x − y = 1 为例。整理后得到 y = −x + 5 与 y = x − 1。

The two lines intersect at (3, 2), so the solution is x = 3, y = 2. Check: 3 + 2 = 5 and 3 − 2 = 1. Both equations are satisfied.

两条直线相交于点 (3, 2)，所以解为 x = 3, y = 2。检验：3 + 2 = 5，3 − 2 = 1，两个方程均成立。

The graphical method is rarely used for exact solutions in the exam, but it helps you visualise what a solution means and is often tested in paper-based questions.

图像法在考试中很少用于求精确解，但它有助于直观理解解的含义，也常在笔试试卷中出现。

3. The Substitution Method | 代入法

Substitution is the most straightforward approach when one equation is already solved for a variable, or when one variable has a coefficient of 1 and can be isolated easily.

当一个方程已经解出某个变量，或者某个变量的系数为1、便于单独分离时，代入法是最直接的方法。

Rearrange one equation so that x or y becomes the subject.

将其中一个方程变形，使x或y成为主项。
Substitute this expression into the other equation.

把这个表达式代入另一个方程。
Solve the resulting linear equation for the remaining variable.

解所得的一元一次方程，求出剩余变量。
Substitute the value back to find the second variable.

代回原方程求出第二个变量。

Example. Solve the system y = 2x + 1 and 3x + 2y = 16.

例：解方程组 y = 2x + 1 与 3x + 2y = 16。

Substitute y = 2x + 1 into the second equation:

将 y = 2x + 1 代入第二个方程：

3x + 2(2x + 1) = 16

Expanding gives 3x + 4x + 2 = 16, so 7x = 14 and x = 2. Then y = 2(2) + 1 = 5. The solution is x = 2, y = 5.

展开得 3x + 4x + 2 = 16，即 7x = 14，所以 x = 2。于是 y = 2(2) + 1 = 5。解为 x = 2, y = 5。

4. The Elimination Method | 消元法

Elimination removes one variable by making its coefficients equal and then adding or subtracting the equations. It is often the fastest method for linear systems.

消元法通过使某一变量的系数相等，然后相加或相减来消去该变量。对于线性方程组，这通常是最快的方法。

Decide which variable to eliminate.

确定要消去哪个变量。
Multiply one or both equations so that the chosen coefficients match.

将其中一个或两个方程乘以适当倍数，使选定变量的系数相等。
Add or subtract the equations to eliminate that variable.

将两个方程相加或相减，消去该变量。
Solve for the remaining variable, then substitute back.

解出剩余变量，再代回求另一个变量。

Example. Solve the following system.

例：解下面的方程组。

2x + 3y = 12 与 x − 2y = −1

To eliminate x, multiply the second equation by 2 to obtain 2x − 4y = −2. Now subtract the second result from the first equation:

为了消去x，将第二个方程乘以2，得到 2x − 4y = −2。然后用第一个方程减去这个新方程：

(2x + 3y) − (2x − 4y) = 12 − (−2)

This simplifies to 7y = 14, so y = 2. Substituting y = 2 into x − 2y = −1 gives x − 4 = −1, hence x = 3. The solution is x = 3, y = 2.

化简得 7y = 14，所以 y = 2。将 y = 2 代入 x − 2y = −1，得 x − 4 = −1，故 x = 3。解为 x = 3, y = 2。

5. Solving Linear-Quadratic Systems | 线性与二次联立方程

In IGCSE, you may be asked to solve one linear equation together with one quadratic equation. This situation always uses substitution, because the linear equation allows you to express one variable in terms of the other.

在IGCSE中，你可能会遇到一个线性方程与一个二次方程组成的方程组。这类题目必须使用代入法，因为线性方程可以将一个变量用另一个变量表示出来。

Example. Solve the system y = x² − 3x + 2 and y = 2x − 2.

例：解方程组 y = x² − 3x + 2 与 y = 2x − 2。

Step 1. Equate the two expressions for y:

第一步：令两个y的表达式相等：

x² − 3x + 2 = 2x − 2

Step 2. Rearrange to bring all terms to one side:

第二步：将所有项移到一边并整理：

x² − 5x + 4 = 0

Step 3. Factorise the quadratic:

第三步：对二次方程进行因式分解：

(x − 1)(x − 4) = 0

So x = 1 or x = 4. Step 4. Substitute back into the linear equation y = 2x − 2. When x = 1, y = 0. When x = 4, y = 6. The solutions are (1, 0) and (4, 6).

因此 x = 1 或 x = 4。第四步：代回线性方程 y = 2x − 2。当 x = 1 时，y = 0；当 x = 4 时，y = 6。解为 (1, 0) 和 (4, 6)。

A linear-quadratic system can have two, one or zero real solutions. If the quadratic obtained has a negative discriminant b² − 4ac, then the line does not intersect the curve and there are no real solutions.

线性与二次方程组可能有两个、一个或零个实数解。如果得到的二次方程判别式 b² − 4ac 为负数，则说明直线与曲线没有交点，不存在实数解。

6. Word Problems | 应用题

Many exam questions present simultaneous equations in written form. The key skill is to translate the worded conditions into a pair of algebraic equations.

许多考试题以文字描述的形式考查联立方程。关键在于将文字条件准确转化为代数方程组。

Example. The sum of two numbers is 15 and their difference is 3. Find the numbers.

例：两个数之和为15，差为3，求这两个数。

Let x be the larger number and y be the smaller number. The conditions give:

设x为较大的数，y为较小的数。题目条件可转化为：

x + y = 15 与 x − y = 3

Adding the two equations gives 2x = 18, so x = 9. Then 9 + y = 15, so y = 6. The two numbers are 9 and 6.

两式相加得 2x = 18，所以 x = 9。再由 9 + y = 15 得 y = 6。这两个数为9和6。

Always define your variables clearly and check your final answer against the original worded conditions.

解题时务必清晰定义变量，并对照原题的文字条件检查最终答案。

7. Common Mistakes | 常见错误

Forgetting to multiply every term when scaling an equation. If you multiply one equation by 2, every term on both sides must be multiplied.

对方程进行倍乘时忘记乘以所有项。如果将某个方程乘以2，则等式两边的每一项都必须乘以2。
Subtracting equations incorrectly, especially when dealing with negative numbers. Write the subtraction in brackets and handle signs carefully.

相减方程时出错，特别是涉及负数的时候。先用括号把方程写清楚，再小心处理符号。
Substituting only into a rearranged equation without checking the original system. Your solution must satisfy both original equations.

只代入变形后的方程而不检查原方程组。最终解必须同时满足两个原方程。
Mixing up signs when expanding brackets such as −2(y − 3). Remember that −2 × (−3) = +6.

展开括号时弄错符号，例如 −2(y − 3) 中，−2 × (−3) = +6，不要漏掉正号。
In linear-quadratic systems, forgetting to give both coordinate pairs. Exams expect two ordered pairs when there are two intersections.

解线性与二次方程组时，漏写两个坐标答案。当有两个交点时，考试要求写出两组有序数对。

8. Exam Tips | 应考技巧

Choose the most efficient method. Use elimination when coefficients align neatly; use substitution when a variable is already isolated.

选择最高效的方法。当系数整齐时用消元法；当某个变量已经单独表示时用代入法。
Show all working clearly. IGCSE mark schemes award method marks even if the final answer is wrong.

清晰展示全部解题过程。IGCSE评分标准即使最终答案错误，也会给方法分。
Always check your answer by substituting the values into both original equations.

务必通过将数值代回两个原方程来检验答案。
In linear-quadratic systems, use the discriminant b² − 4ac to confirm whether real solutions exist before doing full working.

在解线性与二次方程组时，先用判别式 b² − 4ac 判断实数解是否存在，再进行完整计算。
In word problems, write down the variables you define. This earns marks and prevents confusion.

在应用题中，写出你所定义的变量。这既得分又能避免混淆。

Mastering simultaneous equations gives you a powerful tool for solving many real-world and exam problems. Practise all three methods until you can select the best one quickly and accurately.

掌握联立方程，你就拥有了解决许多实际问题和考试题的有力工具。请反复练习三种方法，直到你能快速准确地选择最合适的解法。

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Solving Simultaneous Equations | 解联立方程