Mastering Simultaneous Equations | 掌握联立方程

📚 Mastering Simultaneous Equations | 掌握联立方程

Simultaneous equations are a cornerstone of IGCSE Mathematics. They test your ability to reason algebraically and interpret relationships between variables, appearing in both Paper 2 and Paper 4.

联立方程是IGCSE数学的核心内容之一，它考查你进行代数推理和理解变量之间关系的能力，在Paper 2和Paper 4中都会出现。

In this revision guide, we will break down the three main solution methods — elimination, substitution, and graphical — and show you how to solve systems involving one linear and one quadratic equation.

在本复习指南中，我们将详细讲解三种主要解法——消元法、代入法和图解法——并教你如何求解含有一个线性方程和一个二次方程的方程组。

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

A simultaneous equation system consists of two or more equations that share the same unknown variables. The solution is the set of values that satisfy all equations at the same time.

联立方程组由两个或多个含有相同未知数的方程组成。方程组的解就是同时满足所有方程的未知数值。

For example, consider the system below. We need to find values of x and y that make both equations true.

例如，考虑下面的方程组，我们需要找到同时使两个方程成立的x和y值。

2x + 3y = 8
x − y = −1

In IGCSE, you will usually deal with two equations in two unknowns. The most common types are linear–linear systems and linear–quadratic systems.

在IGCSE中，你通常需要处理含有两个未知数的两个方程。最常见的类型是线性-线性方程组和线性-二次方程组。

2. The Elimination Method | 消元法

The elimination method involves adding or subtracting the equations to cancel out one variable. This is best used when the coefficients of one variable are already equal or opposites.

消元法通过将两个方程相加或相减来消去一个变量。当一个变量的系数已经相等或互为相反数时，这个方法最为适用。

Consider the system below.

看下面的方程组。

2x + y = 7
x − y = 2

Adding the two equations eliminates y, because +y and −y cancel out.

将两个方程相加可以消去y，因为+y和−y相互抵消。

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

Substitute x = 3 back into the first equation: 2(3) + y = 7 → 6 + y = 7 → y = 1. The solution is x = 3, y = 1.

将x = 3代回第一个方程：2(3) + y = 7 → 6 + y = 7 → y = 1。解为x = 3，y = 1。

If the coefficients are not aligned, multiply one equation by a suitable constant first. For example:

如果系数没有对齐，需要先给其中一个方程乘以适当的常数。例如：

3x + 2y = 12
x + y = 5

Multiply the second equation by 2 to get 2x + 2y = 10. Then subtract: (3x + 2y) − (2x + 2y) = 12 − 10 → x = 2. Then y = 3.

将第二个方程乘以2，得到2x + 2y = 10。然后相减：(3x + 2y) − (2x + 2y) = 12 − 10 → x = 2。于是y = 3。

3. The Substitution Method | 代入法

The substitution method is ideal when one equation is written with a variable already isolated, such as y = … or x = … . Replace that variable in the other equation.

当一个方程中的某个变量已经被单独表示（如y = … 或 x = …）时，代入法是最理想的选择。将这个变量代入另一个方程即可。

Solve the system:

解方程组：

y = 2x + 1
x + y = 10

Substitute 2x + 1 for y in the second equation: x + (2x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3.

将第二个方程中的y替换为2x + 1：x + (2x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3。

Now find y: y = 2(3) + 1 = 7. Solution: x = 3, y = 7.

现在求y：y = 2(3) + 1 = 7。解为x = 3，y = 7。

Substitution is also the standard method for linear–quadratic systems, as we will see in Section 5.

代入法也是求解线性-二次方程组的标准方法，我们将在第5节中看到。

4. The Graphical Method | 图解法

Graphically, the solution to a pair of simultaneous linear equations is the coordinates of the point where the two lines intersect.

从图形上看，一对线性联立方程的解就是两条直线交点的坐标。

To use this method, rearrange each equation into the form y = mx + c, plot both lines, and read off the intersection point.

使用这个方法时，先将每个方程改写为y = mx + c的形式，画出两条直线，然后读出交点坐标。

For example, sketch y = 2x − 1 and y = −x + 5 on the same axes. The lines intersect at approximately (2, 3), which is the solution.

例如，在同一坐标系中画出y = 2x − 1和y = −x + 5。两条直线大约相交于(2, 3)，这就是方程组的解。

Note: graphical methods may give approximate answers, so always check by substituting into the original equations.

注意：图形法可能只给出近似答案，因此一定要通过代入原方程进行检验。

5. Linear–Quadratic Systems | 线性与二次方程联立

IGCSE also requires you to solve a system containing one linear and one quadratic equation, usually y = ax² + bx + c combined with a linear equation.

IGCSE还要求你求解含有一个线性方程和一个二次方程的方程组，通常形式为y = ax² + bx + c与一个线性方程的组合。

The substitution method is always used: replace y in the quadratic with the linear expression, then solve the resulting quadratic equation.

这里总是使用代入法：将线性表达式代入二次方程中的y，然后解出所得的二次方程。

Solve the system:

解方程组：

y = x²
y = x + 2

Substitute: x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = 2 or x = −1.

代入得：x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = 2 或 x = −1。

Now find the corresponding y-values: when x = 2, y = 4; when x = −1, y = 1. The solutions are (2, 4) and (−1, 1).

然后求对应的y值：当x = 2时，y = 4；当x = −1时，y = 1。解为(2, 4)和(−1, 1)。

Graphically, these are the two intersection points between the parabola and the straight line.

在图形上，这两个解就是抛物线与直线的两个交点。

6. Word Problems | 实际应用题

Many exam questions frame simultaneous equations in real-world contexts. The key is to define your variables clearly and translate the words into equations.

许多考试题目会将联立方程放在现实情境中。关键在于明确定义变量，并将文字转化为方程。

Example: The sum of two numbers is 12, and their difference is 4. Find the numbers.

例如：两个数之和是12，之差是4。求这两个数。

Let the numbers be a and b. Then write:

设这两个数为a和b，则可得：

a + b = 12
a − b = 4

Add the equations: 2a = 16 → a = 8. Then b = 12 − 8 = 4. The numbers are 8 and 4.

两式相加：2a = 16 → a = 8。于是b = 12 − 8 = 4。这两个数分别是8和4。

Always check your answer against the original wording of the problem.

务必根据题目的原始描述检验你的答案。

7. Common Mistakes to Avoid | 常见错误与避坑指南

Below are the most frequent errors students make and how to avoid them.

以下是学生最常犯的错误以及如何避免它们。

Sign errors when subtracting: subtract both terms of the second equation, not just the first term.
相减时发生符号错误：要减去第二个方程的每一项，而不只是第一项。
Forgetting to substitute back to find the second variable.
忘记代回原方程求第二个变量。
Mixing up the variables when writing the final answer — always write (x, y) clearly.
写最终答案时混淆变量——始终清晰地写成(x, y)。
When solving linear–quadratic systems, losing one of the two solutions.
求解线性-二次方程组时，遗漏两个解中的一个。

8. Practice

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Mastering Simultaneous Equations | 掌握联立方程