Student’s Edition: IGCSE Mathematics Lesson 16-6 Simultaneous Equations | 学生版：IGCSE 数学第16-6课 联立方程

📚 Student’s Edition: IGCSE Mathematics Lesson 16-6 Simultaneous Equations | 学生版：IGCSE 数学第16-6课 联立方程

In IGCSE Mathematics, simultaneous equations are two or more equations that share the same variables. Solving them means finding the one pair of x and y values that makes every equation true at the same time. This topic is a central part of the algebra syllabus and regularly appears in Paper 2 and Paper 4.

在 IGCSE 数学中，联立方程是指含有相同未知数的两个或更多方程。解联立方程就是要找出能同时使每个方程都成立的一组 x 和 y 的值。这个主题是代数大纲的核心内容，经常在 Paper 2 和 Paper 4 中出现。

The three main methods you need to master are the graphical method, the elimination method, and the substitution method. You also need to be able to form simultaneous equations from real-life word problems and to check whether a given pair of values is correct.

你需要掌握的三种主要方法是图像法、消元法和代入法。你还需要学会从实际应用题中建立联立方程，并能够检验一组给定的值是否正确。

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

A simultaneous equation system usually contains two linear equations with two unknown values, often written as x and y. For example, the pair below must be satisfied by the same values of x and y:

联立方程组通常包含两个含有两个未知数 x 和 y 的线性方程。例如下面这一对方程必须由相同的 x 和 y 值同时满足：

2x + y = 7
x − y = 2

Each equation on its own has infinitely many solutions. The equation 2x + y = 7 is a straight line, and every point on that line is a solution. The equation x − y = 2 is another straight line. The solution to the system is the single point where the two lines cross.

每个方程单独来看都有无穷多个解。方程 2x + y = 7 是一条直线，这条直线上的每一个点都是它的解。方程 x − y = 2 是另一条直线。方程组的解就是这两条直线相交的唯一一个点。

In IGCSE questions, the unknowns are usually x and y, but they can also be other letters such as a and b, or p and q. The method stays the same regardless of the letters used.

在 IGCSE 考题中，未知数通常是 x 和 y，但也可以是其他字母，如 a 和 b，或 p 和 q。无论使用什么字母，解题方法都是一样的。

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2. Checking a Solution | 检验一个解

To check whether a pair of values is the solution to a system, substitute the values into both equations. If both equations give true statements, the pair is correct. Consider the system above and test x = 3 and y = 1.

要检验一组值是否是方程组的解，就把这组值代入两个方程。如果两个方程都成立，这组值就是正确的。以刚才的方程组为例，检验 x = 3 和 y = 1。

2(3) + 1 = 7
3 − 1 = 2

The first equation gives 6 + 1 = 7, which is true. The second equation gives 3 − 1 = 2, which is also true. Therefore, x = 3, y = 1 is the solution. This checking skill is very useful after solving a question, especially in the exam when you want to avoid careless mistakes.

第一个方程得到 6 + 1 = 7，成立。第二个方程得到 3 − 1 = 2，也成立。因此 x = 3, y = 1 是方程组的解。这种检验技巧在解题后非常有用，尤其是在考试中想避免粗心错误的时候。

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3. The Graphical Method | 图像法

The graphical method involves drawing both equations on the same coordinate grid and finding the coordinates of their intersection point. To draw each line, rearrange the equation into the form y = mx + c, where m is the gradient and c is the y-intercept.

图像法是在同一个坐标系中画出两个方程的图像，并找出它们交点的坐标。要画出每条直线，需要先把方程整理成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。

For the system above, rearrange the equations as follows:

对于上面的方程组，先把方程整理如下：

y = −2x + 7
y = x − 2

The first line has gradient −2 and y-intercept 7. The second line has gradient 1 and y-intercept −2. When you plot both lines accurately, they cross at the point (3, 1). This gives the solution x = 3, y = 1.

第一条直线的斜率是 −2，y 轴截距是 7。第二条直线的斜率是 1，y 轴截距是 −2。当你准确地画出这两条直线时，它们会在点 (3, 1) 处相交。由此得到解 x = 3, y = 1。

The graphical method is excellent for understanding what a solution means, but it can be slow and less accurate when the solution involves fractions or decimals. In exams, you should use algebra methods for exact answers unless the question specifically asks for a graph.

图像法有助于理解解的含义，但当解含有分数或小数时，它可能比较慢而且不够精确。在考试中，除非题目明确要求画图，否则应该使用代数方法求精确答案。

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4. The Elimination Method | 消元法

The elimination method works by adding or subtracting the two equations to remove one unknown. The goal is to make the coefficients of one variable equal in size, then eliminate it so you are left with a simple equation in one variable.

消元法通过将两个方程相加或相减来消去一个未知数。目标是使其中一个未知数的系数大小相等，然后消去它，从而得到一个只含一个未知数的简单方程。

For the system 2x + y = 7 and x − y = 2, the coefficients of y are +1 and −1. Adding the two equations removes y immediately:

对于方程组 2x + y = 7 和 x − y = 2，y 的系数分别是 +1 和 −1。将两个方程相加可以立刻消去 y：

(2x + y) + (x − y) = 7 + 2
3x = 9
x = 3

Once x = 3 is found, substitute it back into either original equation. Using x − y = 2 gives 3 − y = 2, so y = 1. This is quick and gives the exact answer.

求出 x = 3 后，把它代回任意一个原方程。使用 x − y = 2 可得 3 − y = 2，所以 y = 1。这种方法快速且能给出精确答案。

Sometimes the coefficients do not match. For example, in 3x + 2y = 12 and 2x + y = 7, you need to multiply one or both equations before adding or subtracting. Multiply the second equation by 2 to get 4x + 2y = 14. Then subtract the first equation from this new equation:

有时系数并不匹配。例如在 3x + 2y = 12 和 2x + y = 7 中，需要先将一个或两个方程乘以适当的数，然后再相加或相减。将第二个方程乘以 2，得到 4x + 2y = 14。然后用这个新方程减去第一个方程：

(4x + 2y) − (3x + 2y) = 14 − 12
x = 2

Substitute x = 2 into 2x + y = 7 to get 4 + y = 7, so y = 3. Always choose the multiplication that keeps the numbers small and positive where possible.

将 x = 2 代入 2x + y = 7，得到 4 + y = 7，所以 y = 3。选择乘数时应尽量使数字较小且为正数。

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5. The Substitution Method | 代入法

The substitution method is especially useful when one equation already has one variable isolated, or when it can be isolated very easily. You rearrange one equation to make one variable the subject, then replace that variable in the other equation.

代入法在其中一个方程已经有一个变量被分离出来，或者可以很容易地分离出来时特别有用。你把一个方程变形，使其中一个变量成为主项，然后将这个变量代入另一个方程。

For example, if the system is y = 2x + 1 and 3x + y = 11, the first equation already gives y as the subject. Substitute y = 2x + 1 into the second equation:

例如，如果方程组是 y = 2x + 1 和 3x + y = 11，第一个方程已经将 y 表示出来。将 y = 2x + 1 代入第二个方程：

3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2

Then use y = 2x + 1 to find y = 2(2) + 1 = 5. The solution is x = 2, y = 5. Substitution is also the best method when one coefficient is 1, because rearranging is simple.

然后使用 y = 2x + 1 求出 y = 2(2) + 1 = 5。解为 x = 2, y = 5。当其中一个系数是 1 时，代入法也是最好的方法，因为变形很简单。

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6. Special Cases: Parallel Lines and Coincident Lines | 特殊情况：平行线与重合线

Not every system of linear equations has exactly one solution. If the two lines are parallel, they never meet, so the system has no solution. If the two equations represent the same line, there are infinitely many solutions.

并不是每一个线性方程组都恰好有一个解。如果两条直线平行，它们永远不会相交，因此方程组无解。如果两个方程表示同一条直线，那么方程组有无穷多个解。

Consider the system x + y = 3 and x + y = 5. Subtracting one equation from the other gives 0 = 2, which is impossible. This tells you the lines are parallel and there is no solution.

考虑方程组 x + y = 3 和 x + y = 5。用一个方程减去另一个方程会得到 0 = 2，这是不可能的。这说明两条直线平行，方程组无解。

Now consider x + y = 3 and 2x + 2y = 6. The second equation is just double the first, so both describe the same line. Any point on that line satisfies both equations. In an exam, you would describe this as infinitely many solutions.

现在考虑 x + y = 3 和 2x + 2y = 6。第二个方程只是第一个方程的两倍，所以两者描述的是同一条直线。这条直线上的任何点都满足两个方程。在考试中，你应该把它描述为无穷多个解。

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7. Forming Equations from Word Problems | 由应用题建立方程

Many IGCSE questions describe a situation in words, and you must first write down two equations before solving them. Start by defining the unknowns clearly, for example let x be the cost of one book and let y be the cost of one pen.

许多 IGCSE 题目会用文字描述一个情境，你必须先写出两个方程，然后再求解。首先要清楚地定义未知数，例如设 x 为一本书的价格，设 y 为一支笔的价格。

Example: Three books and two pens cost 19 dollars, while five books and two pens cost 27 dollars. Form two equations and solve them.

例题：三本书和两支笔共花费 19 美元，五本书和两支笔共花费 27 美元。建立两个方程并求解。

3x + 2y = 19
5x + 2y = 27

Subtract the first equation from the second to eliminate y: 2x = 8, so x = 4. Substitute into the first equation: 3(4) + 2y = 19, giving 12 + 2y = 19, so 2y = 7 and y = 3.5. State the answer clearly: a book costs 4 dollars and a pen costs 3.5 dollars.

用第二个方程减去第一个方程以消去 y：2x = 8，所以 x = 4。代入第一个方程：3(4) + 2y = 19，得到 12 + 2y = 19，所以 2y = 7，y = 3.5。清楚地写出答案：一本书 4 美元，一支笔 3.5 美元。

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8. Common Mistakes to Avoid | 常见错误

One common error is forgetting to multiply every term in an equation before using elimination. If you multiply 2x + y = 7 by 3, the result must be 6x + 3y = 21, not 6x + y = 21. Check every coefficient and constant carefully.

一个常见错误是在使用消元法前忘记乘以方程中的每一项。如果把 2x + y = 7 乘以 3，结果必须是 6x + 3y = 21，而不是 6x + y = 21。要仔细检查每一个系数和常数。

Another mistake is substituting the found value into the wrong equation or mixing up the values of x and y at the final answer stage. Always write the final answer as x = … and y = …, then check both original equations.

另一个错误是把求出的值代入错误的方程，或者在最后答案阶段混淆 x 和 y 的值。一定要把最终答案写成 x = … 和 y = …，然后检验两个原方程。

• Use direct quotes and clear notation: do not confuse y = −2x + 7 with y = 2x − 7.
• 使用清晰的书写：不要混淆 y = −2x + 7 和 y = 2x − 7。
• If the solution is a fraction, leave it as an improper fraction unless the question asks for a decimal.
• 如果解是分数，保留假分数形式，除非题目要求写成小数。

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9. IGCSE Exam-Style Practice | IGCSE 考试风格练习

Try this exam-style question: Solve the simultaneous equations 4x + y = 14 and 2x − y = 4.

尝试这道考试风格的问题：解联立方程 4x + y = 14 和 2x − y = 4。

Adding the equations gives 6x = 18, so x = 3. Substitute into 2x − y = 4: 6 − y = 4, so y = 2. The solution is x = 3, y = 2.

将两个方程相加得到 6x = 18，所以 x = 3。代入 2x − y = 4：6 − y = 4，所以 y = 2。解为 x = 3, y = 2。

Another useful practice type is: The sum of two numbers is 25 and their difference is 7. Find the two numbers. Let the numbers be x and y. Then x + y = 25 and x − y = 7. Adding gives 2x = 32, so x = 16. Substituting gives y = 9.

另一种常见练习类型是：两个数的和是 25，差是 7。求这两个数。设这两个数为 x 和 y。则 x + y = 25 且 x − y = 7。相加得 2x = 32，所以 x = 16。代入得 y = 9。

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10. Summary and Key Takeaways | 总结与关键要点

Simultaneous equations are solved by finding the values that satisfy all equations at once. The algebraic methods of elimination and substitution are precise and faster than drawing graphs in most exam situations.

联立方程通过求出能同时满足所有方程的值来解答。在大多数考试情境下，消元法和代入法这两种代数方法比画图更精确、更快。

Method	Best used when
Elimination | 消元法	Coefficients of one variable match or can be matched easily | 一个变量的系数相同或容易化为相同
Substitution | 代入法	One variable is already the subject or has coefficient 1 | 一个变量已经是主项或系数为 1
Graphical | 图像法	Question asks for a graph or approximate solution | 题目要求画图或近似解

Always define unknowns in word problems, write two clear equations, solve carefully, and check your answers in both original equations. This routine will earn full marks on both straightforward and applied IGCSE simultaneous equation questions.

在应用题中要始终先定义未知数，写出两个清晰的方程，认真求解，并在两个原方程中检验答案。这个流程能帮助你在简单和应用型的 IGCSE 联立方程题目中获得满分。

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