Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a pair of equations that share two unknown variables. Students need to find values of x and y that satisfy both equations at the same time. This is a core topic in the IGCSE Edexcel Mathematics syllabus and appears in both Paper 1 and Paper 2.

联立方程组是包含两个未知数的方程组。学生需要找到同时满足两个方程的 x 和 y 的值。这是 IGCSE Edexcel 数学大纲中的核心考点，在 Paper 1 和 Paper 2 中都会出现。

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

Simultaneous equations consist of two or more equations involving two or more variables. For IGCSE, we focus on two linear equations with two unknowns, typically x and y. The solution is an ordered pair (x, y) that makes both equations true.

联立方程组由两个或多个包含两个或更多变量的方程组成。在 IGCSE 中，我们重点研究含有两个未知数（通常为 x 和 y）的两个线性方程。解是一组有序数对 (x, y)，使两个方程同时成立。

For example:

例如：

2x + y = 7
x − y = 2

The first equation has infinitely many solutions alone, but when paired with the second, only one solution satisfies both: x = 3 and y = 1.

单独的第一个方程有无数解，但当与第二个方程结合时，只有一组解同时满足两者：x = 3，y = 1。

2. The Elimination Method | 消元法

The elimination method involves adding or subtracting the two equations to remove one variable. This is usually the fastest method when the coefficients are simple.

消元法通过两个方程相加或相减来消去一个变量。当系数比较简单时，这通常是最快的方法。

Step 1: Arrange both equations in the form ax + by = c.

步骤1：将两个方程整理为 ax + by = c 的形式。

Step 2: If the coefficients of one variable are equal in magnitude, add or subtract to eliminate it.

步骤2：如果某个变量的系数大小相等，则通过相加或相减将其消去。

Step 3: Solve the resulting equation for the remaining variable.

步骤3：解出剩余变量的一次方程。

Step 4: Substitute the value back into either original equation to find the other variable.

步骤4：将解出的值代回任一原方程，求出另一个变量。

Example: Solve

例题：求解

3x + 2y = 12 … (1)
x − 2y = 4 … (2)

Add the equations to eliminate y: (3x + 2y) + (x − 2y) = 12 + 4, so 4x = 16, hence x = 4.

将两式相加消去 y：(3x + 2y) + (x − 2y) = 12 + 4，得 4x = 16，所以 x = 4。

Substitute x = 4 into (2): 4 − 2y = 4, so −2y = 0, hence y = 0.

将 x = 4 代入 (2)：4 − 2y = 4，得 −2y = 0，所以 y = 0。

Solution: x = 4, y = 0.

解：x = 4，y = 0。

If the coefficients are not equal, multiply one or both equations by suitable numbers before adding or subtracting.

如果系数不相等，先将一个或两个方程乘以适当的数，再进行加减。

3. The Substitution Method | 代入法

The substitution method is ideal when one equation has a variable with coefficient 1, or when the equations are not linear (in Higher tier).

当某个变量的系数为 1 时，或对于高级试卷中的非线性方程组，代入法是理想的选择。

Step 1: Rearrange one equation to make one variable the subject.

步骤1：将其中一个方程变形，使一个变量成为主项。

Step 2: Substitute this expression into the other equation.

步骤2：将表达式代入另一个方程。

Step 3: Solve the resulting equation.

步骤3：解出得到的新方程。

Step 4: Substitute back to find the other variable.

步骤4：代回求出另一个变量。

Example: Solve

例题：求解

y = 2x + 1 … (1)
5x + y = 15 … (2)

From (1), y = 2x + 1. Substitute into (2): 5x + (2x + 1) = 15, so 7x = 14, hence x = 2.

由 (1) 得 y = 2x + 1。代入 (2)：5x + (2x + 1) = 15，得 7x = 14，所以 x = 2。

Substitute x = 2 into (1): y = 2(2) + 1 = 5.

将 x = 2 代入 (1)：y = 2(2) + 1 = 5。

Solution: x = 2, y = 5.

解：x = 2，y = 5。

4. The Graphical Method | 图像法

The graphical method involves drawing both lines on the same set of axes. The intersection point gives the solution.

图像法是将两条直线画在同一坐标轴上，交点即为方程组的解。

Steps:

步骤：

Rearrange each equation into the form y = mx + c.
Draw both lines accurately on graph paper.
Read the coordinates of the intersection point.

将每个方程整理为 y = mx + c 的形式。
在坐标纸上准确画出两条直线。
读出交点的坐标。

The graphical method is rarely required in the exam for exact answers, but it is useful for checking your work and for understanding the meaning of a solution.

考试中很少要求用图像法求出精确答案，但它适合用于检查计算结果，并帮助理解解的含义。

Note: If the lines are parallel, there is no solution. If the lines are identical, there are infinitely many solutions.

注意：如果两条直线平行，则方程组无解；如果两条直线重合，则有无穷多组解。

5. Word Problems | 应用题

Examiners love to set simultaneous equations in real-life contexts. The key is to translate the words into two equations.

命题者喜欢将联立方程组放在实际生活情境中。关键在于把文字转化为两个方程。

Example: Two apples and three bananas cost £1.70. Three apples and two bananas cost £1.80. Find the cost of one apple and one banana.

例题：两个苹果和三个香蕉共 1.70 英镑，三个苹果和两个香蕉共 1.80 英镑。求一个苹果和一个香蕉的价格。

Let a = cost of one apple, b = cost of one banana.

设 a 为一个苹果的价格，b 为一个香蕉的价格。

2a + 3b = 1.70 … (1)
3a + 2b = 1.80 … (2)

Multiply (1) by 2 and (2) by 3 to make the b coefficients equal: 4a + 6b = 3.40 and 9a + 6b = 5.40.

将 (1) 乘以 2，(2) 乘以 3，使 b 的系数相等：4a + 6b = 3.40 和 9a + 6b = 5.40。

Subtract: 5a = 2.00, so a = 0.40. Substitute a = 0.40 into (1): 0.80 + 3b = 1.70, so 3b = 0.90, hence b = 0.30.

相减得 5a = 2.00，所以 a = 0.40。将 a = 0.40 代入 (1)：0.80 + 3b = 1.70，得 3b = 0.90，所以 b = 0.30。

One apple costs 40p and one banana costs 30p.

一个苹果 40 便士，一个香蕉 30 便士。

6. Common Mistakes to Avoid | 常见错误

Here are the most frequent errors students make in this topic.

以下是学生在这个主题中最常犯的错误。

Forgetting to multiply every term when scaling an equation.
Sign errors when subtracting equations.
Failing to substitute the found value into both original equations as a check.
Mixing up x and y coordinates in the final answer.

方程整体缩放时忘记乘以每一项。
方程相减时出现符号错误。
没有将求出的值代回两个原方程进行验证。
在最终答案中混淆 x 和 y 的坐标。

Example of a sign error: when subtracting (3x + 2y = 12) minus (x − 2y = 4), be careful: 2y − (−2y) = 4y, not 0. Only add when the signs are opposite, subtract when they are the same.

符号错误示例：当 (3x + 2y = 12) 减去 (x − 2y = 4) 时，注意：2y − (−2y) = 4y，而不是 0。只有当符号相反时才相加，符号相同时才相减。

7. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧

In the Edexcel IGCSE exam, simultaneous equations appear in Foundation and Higher tiers. Higher-tier students must also solve a linear equation paired with a quadratic equation.

在 Edexcel IGCSE 考试中，联立方程组出现在基础卷和高级卷中。高级卷学生还需要解一个线性方程与一个二次方程联立的方程组。

For linear-quadratic systems:

对于线性-二次方程组：

y = x² + 2x − 1
y = 3x + 1

Set the two equations equal: x² + 2x − 1 = 3x + 1, so x² − x − 2 = 0, which factors to (x − 2)(x + 1) = 0. Hence x = 2 or x = −1. Then find the corresponding y values.

令两个方程相等：x² + 2x − 1 = 3x + 1，得 x² − x − 2 = 0，因式分解为 (x − 2)(x + 1) = 0。所以 x = 2 或 x = −1，再求出对应的 y 值。

Always check your final answer by substituting both values into both original equations. This takes 10 seconds and prevents careless marks being lost.

始终将两个值同时代回两个原方程进行检验。这只需 10 秒，但能避免因粗心丢分。

8. Practice Questions | 练习题

Try these questions to test your understanding.

尝试以下题目来检验你的理解。

Question 1: Solve 3x + 4y = 10 and 2x − 4y = 0.

题目1：解方程组 3x + 4y = 10 和 2x − 4y = 0。

Question 2: Solve y = 3x − 5 and 2x + y = 10.

题目2：解方程组 y = 3x − 5 和 2x + y = 10。

Question 3: The sum of two numbers is 25 and their difference is 7. Find the two numbers.

题目3：两个数之和为 25，差为 7。求这两个数。

Answers: 1) x = 2, y = 1 2) x = 3, y = 4 3) The numbers are 16 and 9.

答案：1) x = 2，y = 1 2) x

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Solving Simultaneous Equations | 解联立方程组