Mastering Simultaneous Equations | 方程组解法精讲

📚 Mastering Simultaneous Equations | 方程组解法精讲

Simultaneous equations are a core topic in IGCSE Mathematics. This article will guide you through the key methods, common pitfalls, and exam strategies to solve them confidently.

联立方程（方程组）是 IGCSE 数学的核心内容。本文将带你掌握主要解法、常见误区以及考试中的应试策略，帮助你从容应对这类题目。

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

Simultaneous equations are two or more algebraic equations that share the same variables. A solution must satisfy all equations at the same time.

联立方程是由两个或更多代数方程组成，它们包含相同的变量。一个解必须同时满足所有方程。

For example, in the pair:

例如，对于下面这一组方程：

2x + 3y = 7
x − y = 1

The only pair of values that works is x = 2 and y = 1. Check: 2(2) + 3(1) = 7 and 2 − 1 = 1.

唯一同时成立的解是 x = 2，y = 1。验证：2(2) + 3(1) = 7，且 2 − 1 = 1。

In the Edexcel IGCSE syllabus, you need to solve linear simultaneous equations using algebraic and graphical methods. You may also meet one linear and one quadratic equation.

在 Edexcel IGCSE 考纲中，你需要会用代数和图像方法解线性联立方程，也可能遇到一个线性方程和一个二次方程组成的方程组。

2. The Elimination Method | 消元法

Elimination means removing one variable by adding or subtracting the equations. This works best when the coefficients of one variable are the same or easy to make the same.

消元法指的是通过加法或减法去掉一个变量。当一个变量的系数相同或容易化成相同时，这种方法最有效。

Example:

例如：

3x + 2y = 12
5x + 2y = 16

Subtract the first equation from the second to eliminate y:

用第二个方程减去第一个方程，消去 y：

(5x − 3x) + (2y − 2y) = 16 − 12
2x = 4
x = 2

Then substitute x = 2 into one of the original equations:

然后把 x = 2 代入其中一个原方程：

3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3

Always check your answer in both equations.

一定要把答案代入两个方程进行检验。

3. The Substitution Method | 代入法

Substitution means rearranging one equation to make a variable the subject, then replacing that variable in the other equation.

代入法是把一个方程变形，使一个变量成为“主项”，然后用这个表达式替换另一个方程中的该变量。

Example:

例如：

y = 2x + 1
3x + 2y = 17

From the first equation we already have y in terms of x. Substitute into the second:

第一个方程已经给出了 y 关于 x 的表达式，把它代入第二个方程：

3x + 2(2x + 1) = 17
3x + 4x + 2 = 17
7x = 15
x = 15⁄7

Then substitute back to find y:

再代回去求 y：

y = 2(15⁄7) + 1 = 30⁄7 + 7⁄7 = 37⁄7

Substitution is especially useful when one equation already has x or y alone on one side.

特别地，当一个方程已经单独给出 x 或 y 时，代入法尤为方便。

4. The Graphical Method | 图像法

To solve simultaneous equations graphically, draw both lines on the same set of axes and read the coordinates of their intersection point.

用图像法解联立方程时，需要把两条直线画在同一坐标轴上，并读取它们交点的坐标。

For the equations y = 2x − 1 and y = −x + 5, plotting both lines gives an intersection at (2, 3).

例如对于 y = 2x − 1 和 y = −x + 5，画出两条直线后会得到交点 (2, 3)。

The graphical method is often slower, but it shows the meaning of the solution clearly.

图像法通常较慢，但能清楚地展示解的意义。

Remember to label axes, plot accurate points, and use a sharp pencil for the exam.

在考试中记得标出坐标轴、精确描点，并使用削尖的铅笔。

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

Sometimes a pair of linear equations has no common solution. This happens when the lines are parallel.

有时一对线性方程没有公共解，这种情况发生在两条直线平行时。

For example:

例如：

y = 2x + 3
y = 2x − 4

These lines have the same gradient 2 but different intercepts, so they never meet.

这两条直线斜率相同，都是 2，但截距不同，因此它们永远不会相交。

When you solve by elimination, you may arrive at an impossible statement like 0 = 7. This means the system is inconsistent.

如果使用消元法时得到一个不可能的等式，例如 0 = 7，说明该方程组无解。

If both equations are actually the same line, every point on the line is a solution, giving infinitely many solutions.

如果两个方程实际上表示同一条直线，那么直线上的每一个点都是解，此时有无穷多个解。

6. Setting Up Word Problems | 列方程解应用题

Many exam questions test your ability to convert words into algebraic equations.

许多考题考查你将文字描述转化为代数方程的能力。

Example: The sum of two numbers is 10 and their difference is 4. Find the numbers.

例如：两个数之和是 10，差是 4。求这两个数。

Let the numbers be x and y. Then:

设这两个数为 x 和 y，则：

x + y = 10
x − y = 4

Add the equations: 2x = 14, so x = 7. Then y = 3.

两式相加：2x = 14，所以 x = 7。于是 y = 3。

Always define your variables clearly and check that your answers match the original wording.

做题时一定要清楚定义变量，并检查答案是否符合原题的文字条件。

7. Simultaneous Equations with Fractions | 含分数的联立方程

Fractions can be removed by multiplying every term by the common denominator.

分数可以通过每一项都乘以公分母来消除。

Example:

例如：

x⁄2 + y⁄3 = 4
x⁄4 − y⁄2 = 0

Multiply the first equation by 6 and the second by 4:

第一个方程两边乘以 6，第二个方程两边乘以 4：

3x + 2y = 24
x − 2y = 0

Now use elimination. Add the two equations:

现在用消元法，将两个方程相加：

4x = 24
x = 6

From x − 2y = 0, we get 6 = 2y, so y = 3.

由 x − 2y = 0 得 6 = 2y，所以 y = 3。

8. One Linear and One Quadratic Equation | 一个线性方程与一个二次方程

For higher tier IGCSE, you may need to solve a linear equation together with a quadratic equation. Use substitution.

在 IGCSE 扩展级别中，你可能会遇到一个线性方程和一个二次方程组成的方程组。这时使用代入法。

Example:

例如：

y = x + 2
x² + y² = 20

Substitute y = x + 2 into the quadratic equation:

把 y = x + 2 代入二次方程：

x² + (x + 2)² = 20
x² + x² + 4x + 4 = 20
2x² + 4x − 16 = 0

Divide by 2: x² + 2x − 8 = 0. Factorise:

两边除以 2：x² + 2x − 8 = 0。因式分解：

(x + 4)(x − 2) = 0

So x = −4 or x = 2. Use y = x + 2 to find the corresponding y-values.

所以 x = −4 或 x = 2。再利用 y = x + 2 求出对应的 y 值。

This gives two solutions: (−4, −2) and (2, 4).

这样得到两组解：(−4, −2) 和 (2, 4)。

9. Common Mistakes to Avoid | 常见易错点

Forgetting to multiply the entire equation when dealing with fractions or brackets.

处理分数或括号时，忘记把整个方程都乘以相同的数。
Making a sign error when subtracting equations. For example, subtracting a negative term incorrectly.

在方程相减时出现符号错误，例如错误地处理含负号的项。
Only substituting into the rearranged equation instead of an original equation, causing errors.

只代回到变形后的方程，而不是原方程，导致错误。
Not checking answers in both original equations.

没有将答案代入两个原方程去检验。
Misreading the graph for the graphical method, especially with non-integer coordinates.

在图像法中读错交点坐标，尤其是非整数坐标时。

10. Exam Technique Tips | 考试技巧

In the exam, always show clear steps. Even if your final answer is wrong, you can gain method marks.

在考试中，要一步一步清晰地展示过程。即使最终答案算错，你仍然可以获得方法分。

Use the elimination method when coefficients are easy to match; use substitution when one variable is already isolated.

当系数容易配平时用消元法；当某个变量已经被单独表示时用代入法。

If the answer comes out as a fraction, keep it as a fraction unless the question asks for a decimal.

如果答案出现分数，除非题目要求小数，否则保留分数形式。

Check your solution mentally in less than five seconds: plug the numbers into both equations.

用不到五秒钟心算检查一下答案：把数值代入两个方程即可。

11. Practice Questions | 练习题目

Try these problems on your own.

请独立尝试以下题目。

1. 4x + 5y = 23
2x − y = 1

2. 3a + 2b = 8
5a − 3b = 7

3. y = x² − 3x + 2
y = 2x − 4

Answers: 1. x = 2, y = 3; 2. a = 2, b = 1; 3. Solutions are (2, 0) and (3, 2).

答案：1. x = 2, y = 3；2. a = 2, b = 1；3. 解为 (2, 0) 和 (3, 2)。

12. Summary | 总结

Simultaneous equations are solved by elimination, substitution, or graphing. Always define variables in word problems, remove fractions early, and check your answers.

联立方程可以用消元法、代入法或图像法求解。解应用题时要先定义变量，尽早去分母，并检查答案。

For one linear and one quadratic equation, substitution is usually the most reliable method. Be careful with signs and factorisation.

当一个线性方程搭配一个二次方程时，代入法通常最可靠。注意符号和因式分解。

With regular practice, you can turn this topic into easy marks in the IGCSE exam.

通过有规律的练习，你完全可以把这部分内容变成 IGCSE 考试中的送分题。

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Mastering Simultaneous Equations | 方程组解法精讲