📚 Solving Simultaneous Equations | 解联立方程
Simultaneous equations are a fundamental topic in IGCSE Mathematics. They involve two or more equations that share the same variables, and you need to find values that satisfy all equations at the same time. Mastering this skill is essential for both Paper 1 and Paper 2 of the Edexcel IGCSE.
联立方程是 IGCSE 数学中的一个基础专题。它涉及两个或更多共享相同变量的方程，你需要找出同时满足所有方程的数值。掌握这一技能对 Edexcel IGCSE 的 Paper 1 和 Paper 2 都至关重要。
1. What Are Simultaneous Equations? | 什么是联立方程？
Simultaneous equations are a set of equations with multiple unknown variables, usually \(x\) and \(y\). For linear simultaneous equations, each equation represents a straight line on a coordinate grid. The solution is the point where the lines intersect.
联立方程是一组含多个未知变量（通常是 x 和 y）的方程。对于线性联立方程，每个方程在坐标网格上都表示一条直线。解就是这些直线的交点。
For example:
2x + y = 7
x − y = 2
The values \(x = 3\) and \(y = 1\) satisfy both equations, so the solution is (3, 1).
例如：
2x + y = 7
x − y = 2
数值 x = 3、y = 1 同时满足两个方程，因此解为 (3, 1)。
2. The Elimination Method | 消元法
The elimination method involves adding or subtracting the equations to remove one variable. To do this, the coefficients of one variable must be made the same (or equal and opposite) in both equations.
消元法通过相加或相减两个方程来消去一个变量。为此，需要使某一变量在两个方程中的系数相同（或互为相反数）。
Solve:
3x + 2y = 12
5x − 2y = 4
Here the coefficients of \(y\) are \(+2\) and \(−2\). Adding the equations eliminates \(y\):
(3x + 2y) + (5x − 2y) = 12 + 4
8x = 16
x = 2
Substitute \(x = 2\) into the first equation: \(3(2) + 2y = 12\), so \(6 + 2y = 12\), hence \(2y = 6\), \(y = 3\). The solution is (2, 3).
这里 y 的系数分别为 +2 和 −2。将两个方程相加可以消去 y：
(3x + 2y) + (5x − 2y) = 12 + 4
8x = 16
x = 2
将 x = 2 代入第一个方程：3(2) + 2y = 12，所以 6 + 2y = 12，因此 2y = 6，y = 3。解为 (2, 3)。
If the coefficients are not equal, multiply one or both equations by a suitable number before adding or subtracting.
如果系数不相等，需要先将一个或两个方程乘以适当的倍数，再进行相加或相减。
3. The Substitution Method | 代入法
The substitution method is useful when one variable has a coefficient of 1. Rearrange one equation to express one variable in terms of the other, then substitute into the second equation.
当某个变量的系数为 1 时，代入法非常有用。先整理其中一个方程，用一个变量表示另一个变量，然后代入第二个方程。
Solve:
y = 2x + 1
x + y = 10
Substitute \(y = 2x + 1\) into \(x + y = 10\):
x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3
Then \(y = 2(3) + 1 = 7\). The solution is (3, 7).
将 y = 2x + 1 代入 x + y = 10：
x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3
然后 y = 2(3) + 1 = 7。解为 (3, 7)。
Always check your answer by substituting both values into the original equations.
务必检查答案，将两个值代入原方程验证。
4. Graphical Solutions | 图解方法
Graphically, the solution of two linear simultaneous equations is the point of intersection of their graphs. To solve by drawing, rearrange each equation into the form \(y = mx + c\), plot the lines, and read the coordinates of the intersection.
从图像上看，两个线性联立方程的解就是它们图像的交点。要用画图法求解，先把每个方程整理成 y = mx + c 的形式，画出直线，然后读出交点的坐标。
Example: Solve graphically:
x + y = 5
y = x + 1
For the first equation, \(y = 5 − x\). Plot both lines on the same axes. The lines intersect at (2, 3), so \(x = 2\), \(y = 3\).
例如：用图像法解：
x + y = 5
y = x + 1
第一个方程即 y = 5 − x。在同一坐标系中画出两条直线。它们交于 (2, 3)，所以 x = 2，y = 3。
Graphical methods are often less precise than algebraic methods unless you use graph paper carefully.
除非你仔细使用坐标纸，否则图解方法通常不如代数方法精确。
5. Solving Word Problems | 解应用题
Many real-world problems can be modelled using simultaneous equations. The key is to define variables clearly and translate the given information into two equations.
许多实际问题可以用联立方程建模。关键在于清晰定义变量，并将题目信息转化为两个方程。
Example: The total cost of 3 apples and 2 bananas is 7 dollars. The cost of 1 apple and 4 bananas is 9 dollars. Find the cost of one apple and one banana.
设苹果单价为 a，香蕉单价为 b。可得：
3a + 2b = 7
a + 4b = 9
Using elimination: multiply the second equation by 3 to get \(3a + 12b = 27\). Subtract the first equation: \(10b = 20\), so \(b = 2\). Substitute \(b = 2\) into \(a + 4(2) = 9\), giving \(a = 1\). So one apple costs 1 dollar and one banana costs 2 dollars.
例如：3 个苹果和 2 个香蕉的总价是 7 美元。1 个苹果和 4 个香蕉的总价是 9 美元。求一个苹果和一个香蕉的单价。
设苹果单价为 a，香蕉单价为 b。可得：
3a + 2b = 7
a + 4b = 9
使用消元法：将第二个方程乘以 3，得到 3a + 12b = 27。减去第一个方程：10b = 20，所以 b = 2。将 b = 2 代入 a + 4(2) = 9，得到 a = 1。因此一个苹果 1 美元，一个香蕉 2 美元。
Always state the units in your final answer.
在最终答案中务必注明单位。
6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解
Not every pair of linear equations has a unique solution. There are two special cases you must recognise.
并非每一对线性方程都有唯一解。你必须识别两种特殊情况。
No solution occurs when the two lines are parallel. For example:
y = 2x + 3
y = 2x − 1
Both lines have the same gradient (2), but different intercepts, so they never meet. The equations are inconsistent.
无解发生在两条直线平行时。例如：
y = 2x + 3
y = 2x − 1
两条直线斜率相同（都是 2），但截距不同，因此它们永不相交。这组方程是不一致的。
Infinite solutions occur when the two equations represent the same line. For example:
2x + 4y = 8
x + 2y = 4
The second equation is simply the first equation divided by 2, so both lines are identical. Every point on the line is a solution.
无穷多解发生在两个方程表示同一条直线时。例如：
2x + 4y = 8
x + 2y = 4
第二个方程其实就是第一个方程除以 2，所以两条直线完全重合。直线上的每一个点都是解。
7. Simultaneous Equations with Quadratics | 含二次方程的联立方程组
In the higher tier of Edexcel IGCSE, you may need to solve a linear equation and a quadratic equation simultaneously. This often produces two pairs of solutions.
在 Edexcel IGCSE 的高阶（Higher）试卷中，你可能需要同时解一个线性方程和一个二次方程。这通常会产生两组解。
Example: Solve
y = x² − 3x + 2
y = 2x − 1
Since both expressions equal \(y\), set them equal to each other:
x² − 3x + 2 = 2x − 1
x² − 5x + 3 = 0
Use the quadratic formula:
x = [5 ± √(25 − 12)] / 2 = [5 ± √13] / 2
Then substitute each \(x\) value into \(y = 2x − 1\) to find the corresponding \(y\) values.
例如：解
y = x² − 3x + 2
y = 2x − 1
因为两个表达式都等于 y，将它们设为相等：
x² − 3x + 2 = 2x − 1
x² − 5x + 3 = 0
使用二次求根公式：
x = [5 ± √(25 − 12)] / 2 = [5 ± √13] / 2
然后将每个 x 值代入 y = 2x − 1 求出对应的 y 值。
Remember: when substituting into a quadratic, you may get two, one, or zero real solutions. If the discriminant is negative, there are no real solutions.
记住：代入二次方程后，可能有两个、一个或零个实数解。若判别式为负，则没有实数解。
8. Exam Tips and Common Mistakes | 考试技巧与常见错误
Here are some essential tips to avoid losing marks in your exam.
以下是一些避免失分的关键技巧。
-
Check your solution. Always substitute both values back into both original equations.
检查答案。始终将两个值代回两个原方程中验证。
-
Use consistent signs. When subtracting equations, be careful with negative signs: \(3x − (−2x) = 5x\), not \(x\).
注意符号。相减方程时，小心负号：3x − (−2x) = 5x，而不是 x。
-
Multiply every term. When multiplying an equation, multiply every term on both sides.
每一项都要乘。对方程进行乘法运算时，等式两边的每一项都要乘。
-
State the final answer as a coordinate pair. Write \(x = …\) and \(y = …\), or \((x, y)\).
最终答案写成坐标对。写出 x = … 和 y = …，或 (x, y)。
-
Choose the method wisely. Use elimination when coefficients are already equal or opposite; use substitution when one variable is already isolated.
合理选择方法。当系数已相等或互为相反数时用消元法；当某个变量已被单独表示时用代入法。
9. Practice Questions | 练习
Try these problems to test your understanding. Answers are provided after each question.
尝试以下题目来检验你的理解。每题后附有答案。
Question 1: Solve \(2x + 3y = 8\) and \(4x − y = 9\).
题目 1：解 2x + 3y = 8 和 4x − y = 9。
Answer: \(x = 2.5, y = 1\) (or \(x = 5/2, y = 1\)).
答案：x = 2.5，y = 1（即 x = 5/2，y = 1）。
Question 2: Solve \(y = 3x − 4\) and \(y = x^2 − 4x + 5\).
题目 2：解 y = 3x − 4 和 y = x² − 4x + 5。
Answer: \(x = 3, y = 5\) or \(x = 1, y = −1\).
答案：x = 3，y = 5 或 x = 1，y = −1。
Question 3: Find the point of intersection of \(x + 2y = 6\) and \(2x + 4y = 12\).
题目 3：求 x + 2y = 6 和 2x + 4y = 12 的交点。
Answer: Infinite solutions because the two equations are equivalent.
答案：无穷多解，因为两个方程等价。
10. Conclusion | 结论
Simultaneous equations are a powerful tool for solving problems with multiple unknowns. By mastering the elimination method, substitution method, and graphical interpretation, you can confidently tackle both linear and quadratic cases. Remember to check your work and understand the special cases. With regular practice, you will be fully prepared for the Edexcel IGCSE Mathematics examination.
联立方程是解决多未知数问题的强大工具。通过掌握消元法、代入法和图解方法，你可以自信地应对线性与二次情况。记得检查你的计算，并理解特殊情况。通过定期练习，你将为 Edexcel IGCSE 数学考试做好充分准备。
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