Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a set of equations with multiple unknown variables that are solved together. In IGCSE Mathematics, you most often meet two linear equations with two unknowns, x and y. The solution is the pair of values that satisfies both equations at the same time.

联立方程组是一组含有多个未知数的方程，需要放在一起求解。在 IGCSE 数学中，最常见的是含有两个未知数 x 和 y 的两个线性方程。解就是同时满足这两个方程的一组 x 和 y 的数值。

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

A linear equation in two variables can be written in the form ax + by = c, where a, b and c are constants. For example, 2x + 3y = 8 is a linear equation. One equation alone has infinitely many solutions: for every value of x, you can find a corresponding y.

含有两个变量的线性方程可以写成 ax + by = c 的形式，其中 a、b、c 是常数。例如，2x + 3y = 8 就是一个线性方程。单独一个方程有无数多组解：对于每一个 x 值，都能找到一个对应的 y 值。

When you have two such equations, they usually share exactly one common solution. This common solution represents the point where the two straight lines intersect on a graph.

当你有两个这样的方程时，它们通常只有一个公共解。这个公共解代表两条直线在图像上的交点。

Key terminology: the word “simultaneous” means “at the same time”. A solution must make both equations true at once. If the point (1, 2) is substituted into both equations and both become true statements, then (1, 2) is the solution.

关键术语：”simultaneous” 意为”同时的”。一个解必须同时使两个方程成立。如果把点 (1, 2) 代入两个方程，并且两个方程都成为真命题，那么 (1, 2) 就是解。

There are three main methods to solve simultaneous equations: elimination, substitution, and graphical. You should be confident with all three because the exam may ask you to use a specific method.

解联立方程组主要有三种方法：消元法、代入法和图像法。你需要熟练掌握这三种方法，因为考试可能要求你用特定方法解题。

2. The Elimination Method | 消元法

The elimination method works by adding or subtracting the two equations to remove one variable. This leaves a single equation with only one unknown, which you can solve directly.

消元法的核心是通过两式相加或相减来消去一个变量。这样只剩下一个只含一个未知数的方程，可以直接求解。

Consider this pair of equations:

考虑如下方程组：

2x + 3y = 8 (1)
3x − 3y = 3 (2)

Notice that the y coefficients are +3 and −3. If we add the two equations, the y terms cancel out:

注意 y 的系数分别是 +3 和 −3。如果我们把两个方程相加，y 项就会互相抵消：

(2x + 3x) + (3y − 3y) = 8 + 3
5x = 11

This gives x = 2.2 or x = 11/5. Then substitute back into equation (1): 2(2.2) + 3y = 8, so 4.4 + 3y = 8, thus 3y = 3.6, so y = 1.2. The solution is x = 2.2, y = 1.2.

于是得到 x = 2.2 或 x = 11/5。然后代回方程 (1)：2(2.2) + 3y = 8，即 4.4 + 3y = 8，所以 3y = 3.6，y = 1.2。解为 x = 2.2，y = 1.2。

When the coefficients do not match, you must multiply one or both equations by a constant first. For example, to solve:

当系数不一致时，你需要先将一个或两个方程乘以某个常数。例如，解方程组：

2x + y = 5 (1)
3x + 2y = 8 (2)

Multiply equation (1) by 2 to get 4x + 2y = 10. Now subtract equation (2) from this new equation: (4x − 3x) + (2y − 2y) = 10 − 8, so x = 2. Substitute back to find y = 1.

将方程 (1) 乘以 2，得到 4x + 2y = 10。现在用这个新方程减去方程 (2)：(4x − 3x) + (2y − 2y) = 10 − 8，所以 x = 2。代回原方程得 y = 1。

General strategy: decide which variable to eliminate, make their coefficients equal in magnitude, then add if the signs are opposite or subtract if the signs are the same.

一般策略：决定要消去哪个变量，使它的系数绝对值相等；如果符号相反就用加法，如果符号相同就用减法。

Always check your answer by substituting both values into the original equations. This prevents careless arithmetic errors.

务必把两个值代回原方程进行检验。这样可以避免粗心的计算错误。

3. The Substitution Method | 代入法

The substitution method involves rearranging one equation to make one variable the subject, then substituting that expression into the other equation. This method is especially useful when one equation has a coefficient of 1 for x or y.

代入法的步骤是：先重新整理其中一个方程，把某个变量表示成另一个变量的表达式，再把这个表达式代入另一个方程。当一个方程中 x 或 y 的系数为 1 时，这种方法特别方便。

Solve these equations using substitution:

用代入法解方程组：

y = 2x − 1 (1)
3x + 2y = 12 (2)

Equation (1) already gives y in terms of x. Substitute y = 2x − 1 into equation (2):

方程 (1) 已经给出了 y 关于 x 的表达式。将 y = 2x − 1 代入方程 (2)：

3x + 2(2x − 1) = 12
3x + 4x − 2 = 12
7x = 14
x = 2

Then substitute x = 2 into y = 2x − 1 to get y = 3. Check: 3(2) + 2(3) = 6 + 6 = 12, correct.

然后将 x = 2 代入 y = 2x − 1，得到 y = 3。检验：3(2) + 2(3) = 6 + 6 = 12，正确。

If neither equation has a variable with coefficient 1, choose the simpler equation and rearrange it. For example, from 2x + 3y = 8 you can write x = (8 − 3y)/2.

如果两个方程中都没有系数为 1 的变量，就选择较简单的方程进行变形。例如，从 2x + 3y = 8 可以写成 x = (8 − 3y)/2。

Substitution is also the natural method for solving a linear equation together with a quadratic equation, which appears in Higher Tier IGCSE papers.

代入法也是解线性方程与二次方程联立的自然选择，这在 IGCSE 高级试卷中会出现。

4. The Graphical Method | 图像法

To solve simultaneous equations graphically, plot both straight lines on the same coordinate axes. The coordinates of the intersection point give the solution.

用图像法解联立方程组时，需要在同一坐标平面内画出两条直线。交点的坐标就是方程组的解。

For the equations y = 2x − 1 and 3x + 2y = 12, you would plot both lines. The first line has gradient 2 and y-intercept −1. The second line can be rearranged to y = −1.5x + 6. Draw them carefully; they intersect at (2, 3).

对于 y = 2x − 1 和 3x + 2y = 12，你需要画出两条直线。第一条直线的斜率为 2，y 截距为 −1。第二条直线可以改写为 y = −1.5x + 6。仔细画出这两条直线，它们相交于点 (2, 3)。

To plot a line from an equation in the form ax + by = c, you can:

要从 ax + by = c 形式的方程画直线，你可以：

Find the x-intercept by setting y = 0. This gives the point (c/a, 0).

令 y = 0 求 x 截距，得到点 (c/a, 0)。
Find the y-intercept by setting x = 0. This gives the point (0, c/b).

令 x = 0 求 y 截距，得到点 (0, c/b)。
Draw a straight line through these two points.

通过这两点画一条直线。

If the intersection point has non-integer coordinates, your graphical solution will be approximate. In exams, the mark scheme usually accepts answers within a small tolerance range.

如果交点坐标不是整数，图像法得到的解只是近似值。在考试评分标准中，通常允许一定误差范围内的答案。

5. Word Problems | 应用题

Many exam questions describe a real-life situation and ask you to form and solve simultaneous equations. The key skill is translating words into algebraic equations.

许多考试题目会描述一个实际情境，要求你列出并联立求解方程组。关键能力是把文字转换成代数方程。

Example: A shop sells apples and bananas. Two apples and three bananas cost $2.60. Three apples and two bananas cost $2.90. Find the price of one apple and one banana.

例题：一家商店出售苹果和香蕉。2 个苹果和 3 根香蕉共 2.60 美元。3 个苹果和 2 根香蕉共 2.90 美元。求一个苹果和一根香蕉的价格。

Let x be the price of one apple and y be the price of one banana. The equations are:

设 x 为一个苹果的价格，y 为一根香蕉的价格。可得方程组：

2x + 3y = 2.60
3x + 2y = 2.90

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

第一个方程乘以 3，第二个方程乘以 2：

6x + 9y = 7.80
6x + 4y = 5.80

Subtract: 5y = 2.00, so y = 0.40. Substitute to find x = 0.70. One apple costs $0.70 and one banana costs $0.40.

两式相减：5y = 2.00，所以 y = 0.40。代入求得 x = 0.70。一个苹果 0.70 美元，一根香蕉 0.40 美元。

Tips for word problems:

应用题小贴士：

Clearly define your variables using “let x be…” before writing any equation.

在列方程之前，先明确写出”设 x 为……”来定义变量。
Use units consistently. Do not mix dollars and cents without converting.

保持单位一致。不要在不换算的情况下混用美元和美分。
Check that your final answer makes sense in the context of the problem.

检查最终答案是否符合题目的实际情境。

6. Special Cases: No Solution and Infinite Solutions | 特殊情况：无解与无数解

Not every pair of simultaneous equations has exactly one solution. There are two special cases you must recognise.

并非所有联立方程组都有唯一解。有两种特殊情况你必须能够识别。

Case 1: No solution. If the two lines are parallel, they never intersect, so there is no common solution. This happens when the coefficients of x and y are proportional but the constant terms are not.

情形一：无解。如果两条直线平行，它们永远不会相交，因此没有公共解。当 x 和 y 的系数成比例，但常数项不成比例时，就会出现这种情况。

For example, x + 2y = 4 and 2x + 4y = 10 have no solution. Notice that 2x + 4y = 10 is exactly double the left side of the first equation, but double 4 is 8, not 10. The lines are parallel.

例如，x + 2y = 4 和 2x + 4y = 10 无解。注意 2x + 4y = 10 的左边恰好是第一个方程左边的两倍，但 4 的两倍是 8，不是 10。所以这两条直线平行。

Case 2: Infinite solutions. If one equation is a multiple of the other, the two equations represent the same line. Every point on the line is a solution.

情形二：无数解。如果一个方程是另一个方程的倍数，那么两个方程表示同一条直线。直线上每一个点都是解。

For example, x + 2y = 4 and 3x + 6y = 12 are the same line because the second is the first multiplied by 3. When you attempt to solve, you will find 0 = 0 after elimination.

例如，x + 2y = 4 和 3x + 6y = 12 表示同一条直线，因为第二个方程是第一个方程乘以 3。当你尝试求解时，消元后会发现 0 = 0。

In exams, if elimination leads to a false statement like 0 = 5, write “no solution”. If it leads to 0 = 0, write “infinite solutions” or “the equations represent the same line”.

在考试中，如果消元后得到 0 = 5 这样的假命题，就写”无解”。如果得到 0 = 0，就写”无数解”或”两个方程表示同一条直线”。

7. Exam Tips and Common Mistakes | 应试技巧与常见错误

Here are the most important tips to maximise your marks on simultaneous equation questions.

以下是在联立方程组题目中拿高分的最重要技巧。

Align like terms. Write both equations in the same form, usually ax + by = c, before comparing coefficients.

对齐同类项。在比较系数之前，先把两个方程写成相同的形式，通常是 ax + by = c。
Label your equations (1) and (2). This makes your working clear and helps you refer to them correctly.

给方程标号 (1) 和 (2)。这样可以让步骤清晰，并且帮助你正确引用它们。
Always show substitution of your answer. A final check earns you method marks even if a minor slip occurred earlier.

始终代入检验。即使前面有小错误，最后的检验也能为你赢得方法分。
Beware of sign errors. When subtracting equations, subtract every term carefully. A common mistake is forgetting to subtract negative terms correctly.

当心符号错误。两式相减时，要小心地对每一项进行减法。常见的错误是忘记正确处理负数项的相减。
Read the question carefully. Some questions ask only for the value of x+y, or for the point of intersection. Answer exactly what is asked.

仔细审题。有些题目只要求 x+y 的值，或只要求交点。要准确回答题目所问。

If you have time, solve the system using a second method as a check. If elimination gives x = 2 and y = 3, substitution should give the same result.

如果时间充裕，可以用第二种方法再解一遍作为检验。如果消元法得到 x = 2 和 y = 3，代入法也应该得到相同结果。

8. Practice Questions | 练习题

Try these questions before checking the answers below.

先尝试做下面的题目，再对照答案。

Question 1: Solve 2x + 5y = 11 and 4x − 3y = 9 using elimination.

第 1 题：用消元法解 2x + 5y = 11 和 4x − 3y = 9。

Question 2: Solve y = 3x + 2 and 2x + y = 22 using substitution.

第 2 题：用代入法解 y = 3x + 2 和 2x + y = 22。

Question 3: The sum of two numbers is 15. Twice the larger number minus the smaller number is 24. Find the two numbers.

第 3 题：两个数之和是 15。较大数的 2 倍减去较小数等于 24。求这两个数。

Question 题号	Answer 答案
1	x = 3, y = 1
2	x = 4, y = 14
3	larger = 13, smaller = 2

9. Summary | 总结

Simultaneous equations are a core topic in IGCSE Mathematics. You must know how to solve them by elimination, substitution, and graphing, and be able to interpret the results in context.

联立方程组是 IGCSE 数学的核心内容。你必须掌握消元法、代入法和图像法，并能在实际情境中解释结果。

Remember the three key steps: form the equations, solve using an appropriate method, and verify your solution. Special cases with no solution or infinite solutions must be recognised and explained clearly.

记住三个关键步骤：列出方程、选择合适的方法求解、验证你的解。必须能够识别并清楚地解释无解和无数解的特殊情况。

With regular practice, simultaneous equation questions become routine and are an excellent source of marks in the exam.

通过定期练习，联立方程组题目会变得非常常规，是考试中获取分数的重要来源。

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Solving Simultaneous Equations | 解联立方程组