📚 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
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)
(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。
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。
(−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
actors 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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