IGCSE Maths: Solving Simultaneous Equations | IGCSE 数学：解联立方程

📚 IGCSE Maths: Solving Simultaneous Equations | IGCSE 数学：解联立方程

Simultaneous equations appear throughout the IGCSE Mathematics syllabus, from linear systems in Paper 2 to more demanding quadratic-linear problems in Paper 4. They test your ability to handle two or more unknowns by combining equations logically and accurately.

联立方程在 IGCSE 数学大纲中出现频率很高，从 Paper 2 的线性方程组到 Paper 4 中更具挑战性的二次与线性方程组。这类题目考查学生能否通过合理、准确地组合方程来处理两个或多个未知数。

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

A simultaneous equation system is a set of two or more equations that share the same variables. A solution is a pair of values (x, y) that makes every equation true at the same time.

联立方程组是由两个或两个以上含有相同变量的方程组成的集合。一个解就是一组 (x, y) 值，能够同时使每一个方程成立。

For example, the system

例如，方程组

3x + 2y = 12 and 2x − y = 1

has the solution x = 2, y = 3 because both equations are satisfied.

的解为 x = 2，y = 3，因为这两个方程都成立。

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2. The Elimination Method | 消元法

The elimination method works by adding or subtracting equations to remove one variable. First make the coefficients of one unknown the same size, then add or subtract the equations.

消元法通过将方程相加或相减来消去一个变量。首先使某个未知数的系数大小相同，然后将两个方程相加或相减。

For the system above, multiply the second equation by 2:

对于上面的方程组，将第二个方程乘以 2：

3x + 2y = 12 and 4x − 2y = 2

Adding gives 7x = 14, so x = 2. Substitute x = 2 into 2x − y = 1 to get y = 3.

两式相加得到 7x = 14，所以 x = 2。将 x = 2 代入 2x − y = 1，得到 y = 3。

Always write the new equation clearly after each step. This reduces sign errors and helps the examiner follow your method.

每一步后都要清晰地写下新的方程。这样可以减少符号错误，也有助于考官理解你的解题过程。

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3. The Substitution Method | 代入法

... or x = ... . Rearrange one equation, then replace that variable in the other equation.

当其中一个方程已经是 y = ... 或 x = ... 的形式时，代入法非常有用。先将一个方程变形，再把该变量代入另一个方程。

From 2x − y = 1 we can write y = 2x − 1. Substituting into 3x + 2y = 12 gives:

由 2x − y = 1 可得 y = 2x − 1。代入 3x + 2y = 12，得到：

3x + 2(2x − 1) = 12

This simplifies to 7x − 2 = 12, so x = 2. Then y = 2(2) − 1 = 3.

化简得到 7x − 2 = 12，因此 x = 2。再由 y = 2(2) − 1 得到 y = 3。

Choose substitution when coefficients are awkward for elimination, such as when one variable has coefficient 1.

当系数不适合消元时，比如某个变量的系数为 1，优先选择代入法。

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4. Graphical Solution and Intersection Points | 图解与交点

Each linear equation represents a straight line on the coordinate plane. The solution of the system is the point where the two lines intersect.

每个线性方程在坐标平面上表示一条直线。方程组的解就是两条直线相交的点。

For our system, drawing y = (12 − 3x) ÷ 2 and y = 2x − 1 shows the intersection at (2, 3).

对于我们的方程组，画出 y = (12 − 3x) ÷ 2 和 y = 2x − 1，可以看到两条直线在 (2, 3) 处相交。

In IGCSE questions, you may be asked to solve graphically. Plot at least three points for each line, draw them accurately, and read the intersection to 1 decimal place if needed.

在 IGCSE 题目中，可能会要求用图解求解。每条直线至少描三个点，准确画出图像，并在需要时以一位小数的精度读出交点坐标。

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5. Setting Up Equations from Word Problems | 由文字题建立方程

Many real-life IGCSE problems ask you to form equations before solving. Define the unknowns clearly using letters, then translate each sentence into an equation.

许多 IGCSE 实际应用题要求先建立方程再求解。要清楚地用字母定义未知数，然后把每句话转化为一个方程。

Example: Two apples and one banana cost 3 dollars, while one apple and two bananas cost 2.70 dollars. Let a be the apple cost and b be the banana cost.

例如：两个苹果和一根香蕉共 3 美元，一个苹果和两根香蕉共 2.70 美元。设 a 为苹果单价，b 为香蕉单价。

2a + b = 3 and a + 2b = 2.70

gives a = 1.10 and b = 0.80. Always state the final answer in the context of the question.

解得 a = 1.10，b = 0.80。最后一定要结合题意写出答案。

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6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无穷多解

Two linear equations may be parallel lines, so there is no solution. This happens when the left-hand sides are proportional but the constants are different.

两条直线可能平行，此时方程组无解。这种情况发生在等号左边成比例但常数项不同的时候。

= 5 and 4x + 2y = 9 are inconsistent because doubling the first gives 4x + 2y = 10, not 9.

例如，2x + y = 5 与 4x + 2y = 9 是矛盾的，因为第一个方程乘以 2 后得到 4x + 2y = 10，而不是 9。

If one equation is just a multiple of the other, there are infinitely many solutions because the two lines coincide.

如果一个方程只是另一个方程的倍数，那么两条直线重合，方程组有无穷多解。

In IGCSE, you may be asked to explain why a system has no solution, so mention parallel lines and equal gradients but different y-intercepts.

在 IGCSE 中，可能会要求解释为什么方程组无解，此时要提到两条直线平行、斜率相等但截距不同。

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7. Quadratic and Linear Systems | 二次与线性方程组

When one equation is quadratic and the other is linear, use substitution. Replace y in the quadratic equation with the linear expression, then solve the resulting quadratic.

当一个方程是二次方程、另一个是线性方程时，使用代入法。将二次方程中的 y 替换为线性表达式，然后求解得到的二次方程。

Example: y = x² − 2x − 3 and y = 2x − 3. Equate the right-hand sides:

例如：y = x² − 2x − 3 与 y = 2x − 3。令两个等号右边相等：

x² − 2x − 3 = 2x − 3

o x² − 4x = 0, so x(x − 4) = 0. Thus x = 0 or x = 4. The solutions are (0, −3) and (4, 5).

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