Solving Simultaneous Equations | 联立方程解题指南

📚 Solving Simultaneous Equations | 联立方程解题指南

In IGCSE Mathematics, a linear equation in two variables, such as x + y = 10, has infinitely many solutions. When we are given two equations together, we want to find the ordered pair (x, y) that satisfies both equations at the same time. This is called solving simultaneous equations.

在 IGCSE 数学中，一个包含两个变量的线性方程，例如 x + y = 10，本身有无数多组解。当我们同时给出两个方程时，我们需要找到能同时满足这两个方程的有序对 (x, y)，这个过程称为解联立方程。

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

Simultaneous equations are two or more equations that share the same variables. For IGCSE, the most common case is a pair of linear equations in two variables, usually x and y.

联立方程是指两个或多个含有相同变量的方程。在 IGCSE 中，最常见的情况是含有 x 和 y 两个变量的两个线性方程。

For example:

例如：

x + y = 10
x − y = 2

The solution is x = 6 and y = 4, because 6 + 4 = 10 and 6 − 4 = 2.

这组方程的解是 x = 6，y = 4，因为 6 + 4 = 10，并且 6 − 4 = 2。

2. Setting Up Equations | 建立方程

Before solving, you must be able to translate a word problem into two equations. Look for key relationships such as sums, differences, totals, and comparisons.

在解方程之前，你需要能把文字题目转化为两个方程。注意题目中的关键关系，例如和、差、总值以及比较关系。

Example: The sum of two numbers is 12 and their difference is 2. If the numbers are a and b, then:

例如：两个数的和是 12，差是 2。若这两个数为 a 和 b，则：

a + b = 12
a − b = 2

Adding these equations gives 2a = 14, so a = 7. Then b = 5. Always define your variables first.

将两个方程相加得到 2a = 14，所以 a = 7，因此 b = 5。解题前一定要先定义变量。

3. The Elimination Method | 消元法

Elimination means removing one variable by adding or subtracting the equations. This works especially well when the coefficients of one variable are the same or opposite.

消元法是指通过两个方程相加或相减去掉其中一个变量。当一个变量的系数相同或互为相反数时，这种方法尤其有效。

Worked example:

示例：

2x + y = 11
x − y = 1

Add the two equations. The y terms cancel because y + (−y) = 0.

将两个方程相加，y 项相互抵消，因为 y + (−y) = 0。

3x = 12
x = 4

Substitute x = 4 into the first equation: 2(4) + y = 11, so y = 3. The solution is (4, 3).

将 x = 4 代入第一个方程：2(4) + y = 11，所以 y = 3。解为 (4, 3)。

Sometimes you need to multiply one or both equations before eliminating.

有时需要先对一个或两个方程进行倍数处理，然后才能消元。

2x + 3y = 12
5x − 2y = 11

Multiply the first equation by 2 and the second equation by 3:

将第一个方程乘以 2，第二个方程乘以 3：

4x + 6y = 24
15x − 6y = 33

Now add to eliminate y: 19x = 57, so x = 3. Substitute back to find y = 2.

现在相加消去 y：19x = 57，所以 x = 3。代回方程求得 y = 2。

4. The Substitution Method | 代入法

Substitution means solving one equation for one variable and replacing that expression in the other equation. It is very useful when one variable is already isolated.

代入法是指先从一个方程中解出一个变量，然后把该表达式代入另一个方程。当一个变量已经被单独表示出来时，这种方法非常方便。

Worked example:

示例：

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

Since y = 3x + 2, replace y in the second equation:

因为 y = 3x + 2，所以把第二个方程中的 y 替换掉：

2x + (3x + 2) = 12

Simplify: 5x + 2 = 12, so 5x = 10 and x = 2. Then y = 3(2) + 2 = 8.

化简：5x + 2 = 12，所以 5x = 10，x = 2。于是 y = 3(2) + 2 = 8。

Always substitute your answer into one of the original equations to check it works.

一定要把答案代回原方程之一进行验证。

5. Choosing a Method | 方法选择

Both elimination and substitution give the same correct answer when used carefully. Your choice depends on the form of the equations.

只要计算仔细，消元法和代入法都会得到相同的正确答案。如何选择取决于方程的形式。

Method	Best Used When
Elimination	Coefficients of x or y are the same, or can be made the same with simple multiplication.
Substitution	One equation has y = … or x = … already.
Graphical method	You need a visual check or are told to draw graphs.

For IGCSE, you should be comfortable with all three methods.

对于 IGCSE，你应该熟练掌握这三种方法。

6. Solving with Graphs | 图像法

Each linear equation can be drawn as a straight line. The solution to a pair of simultaneous equations is the point where the two lines intersect.

每个线性方程都可以画成一条直线。两个联立方程的解就是两条直线的交点。

Example: solve y = x + 1 and y = −x + 3 by drawing graphs.

例如：通过画图像求解 y = x + 1 和 y = −x + 3。

x	0	1	2
y = x + 1	1	2	3
y = −x + 3	3	2	1

Both lines pass through the point (1, 2), so the solution is x = 1 and y = 2.

两条直线都经过点 (1, 2)，所以解为 x = 1，y = 2。

When drawing graphs, always use a sharp pencil and plot at least three points for each line.

画图时务必使用削尖的铅笔，并且每条直线至少取三个点。

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

Not every pair of equations has exactly one solution. There are two important special cases.

不是每一组方程都有唯一解。有两种重要的特殊情况。

No solution: The lines are parallel and never meet.

无解：两条直线平行，永不相交。

x + y = 3
2x + 2y = 7

Dividing the second equation by 2 gives x + y = 3.5. This contradicts x + y = 3, so no pair (x, y) can satisfy both equations.

将第二个方程除以 2 得到 x + y = 3.5，这与 x + y = 3 矛盾，因此不存在能同时满足两个方程的 (x, y)。

Infinite solutions: The two equations represent the same line.

无穷多解：两个方程表示同一条直线。

x + y = 3
2x + 2y = 6

Here the second equation is exactly twice the first, so every point on the line x + y = 3 is a solution.

这里第二个方程正好是第一个方程的两倍，所以直线 x + y = 3 上的每一个点都是解。

8. Word Problems | 应用题

Simultaneous equations are often tested in word-problem form. The key is to define variables clearly and write two separate equations from the given information.

应用题常常用文字方式考查联立方程。关键是把变量定义清楚，并根据题目信息写出两个独立的方程。

Example: Three pens and two pencils cost $8.40. One pen and one pencil cost $3.30. Find the cost of one pen and one pencil.

例如：3 支钢笔和 2 支铅笔共需 8.40 美元。1 支钢笔和 1 支铅笔共需 3.30 美元。求 1 支钢笔和 1 支铅笔的价格。

Let p be the cost of one pen and q be the cost of one pencil.

设 p 为钢笔单价，q 为铅笔单价。

3p + 2q = 8.40
p + q = 3.30

From the second equation, q = 3.30 − p. Substitute into the first equation:

由第二个方程得 q = 3.30 − p。代入第一个方程：

3p + 2(3.30 − p) = 8.40

Simplify: 3p + 6.60 − 2p = 8.40, so p = 1.80. Then q = 1.50.

化简：3p + 6.60 − 2p = 8.40，所以 p = 1.80，q = 1.50。

Always finish with a sentence: “One pen costs $1.80 and one pencil costs $1.50.”

最后要写完整答句：“一支钢笔 1.80 美元，一支铅笔 1.50 美元。”

9. Simultaneous Equations with Quadratics | 含二次方程的联立方程

IGCSE Extended students may need to solve a pair of equations where one is linear and the other is quadratic. Substitution is usually the most reliable method.

IGCSE 扩展课程学生可能需要解一个线性方程与一个二次方程组成的联立方程。代入法通常是最可靠的方法。

Example: solve

例如：解方程组

y = x² + 2x − 1
y = 2x + 3

Since both expressions equal y, set them equal:

因为两个表达式都等于 y，所以令它们相等：

x² + 2x − 1 = 2x + 3

Simplify: x² − 4 = 0. Factorise: (x − 2)(x + 2) = 0, so x = 2 or x = −2.

化简得 x² − 4 = 0。因式分解得 (x − 2)(x + 2) = 0，所以 x = 2 或 x = −2。

Substitute back into the linear equation y = 2x + 3. When x = 2, y = 7. When x = −2, y = −1.

代回线性方程 y = 2x + 3。当 x = 2 时 y = 7；当 x = −2 时 y = −1。

The solutions are (2, 7) and (−2, −1).

解为 (2, 7) 和 (−2, −1)。

If the quadratic equation has no real roots, then the line and curve do not intersect.

如果二次方程没有实数根，说明直线与曲线不相交。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always check your final values in both original equations. One equation is not enough.

一定要把最终结果代入两个原方程验证。只检查一个方程是不够的。
When using elimination, write down each step. Sign errors are the most common mistake.

使用消元法时，写出每一步。正负号错误是最常见的错误。
When using substitution, put brackets around the expression you substitute.

使用代入法时，在代入的表达式中加上括号。
In graphical methods, use an appropriate scale and plot points accurately.

使用图像法时，要选择合适的比例尺，并准确描点。
If the equations contain fractions, multiply through by the common denominator first.

如果方程中含有分数，先乘以公分母去掉分母。
In word problems, never forget to define your variables and write your answer in context.

在应用题中，不要忘记定义变量，并在最后结合题目写出答案。

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Solving Simultaneous Equations | 联立方程解题指南