Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are one of the most important topics in IGCSE Mathematics. They appear in Paper 2 and Paper 4, both as direct questions and as part of longer problems involving coordinate geometry, sequences and real-life applications. This article guides you through every method you need, with worked examples, common pitfalls and exam-style practice.

联立方程是 IGCSE 数学中最重要的话题之一，在 Paper 2 和 Paper 4 中都会出现，既会单独出题，也会作为坐标几何、数列及实际应用问题的一部分。本文将带你掌握所有必备方法，包含例题、常见错误与考试风格练习。

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 unknown variables. The solution is the set of values that satisfy all equations at the same time. For linear equations, the solution corresponds to the point where the two lines intersect on a graph.

联立方程组是包含两个或两个以上、拥有相同未知数的方程的集合。解就是同时满足所有方程的未知数值。对于线性方程而言，解对应图像上两条直线的交点。

For example, consider the system below. We are looking for one pair (x, y) that makes both equations true.

例如，考虑下面的方程组。我们寻找一对 (x, y)，使两个方程同时成立。

3x + y = 11

2x − y = 4

There are three main methods to solve such systems: elimination, substitution and graphical. Each has advantages depending on the form of the equations.

解此类方程组主要有三种方法：消元法、代入法和图像法。每种方法各有优势，取决于方程的形式。

2. The Elimination Method | 消元法

The elimination method removes one variable by adding or subtracting the equations. This works best when the coefficients of one variable are the same or opposite.

消元法通过两个方程相加或相减来消去一个变量。当一个变量的系数相同或互为相反数时，这种方法最有效。

Take the system above. Notice that the y terms are +y and −y. Adding the two equations eliminates y completely.

以上面的方程组为例。注意两个方程中 y 的系数分别为 +y 和 −y。将两个方程相加可完全消去 y。

(3x + y) + (2x − y) = 11 + 4

5x = 15

x = 3

Substitute x = 3 back into the first equation:

将 x = 3 代回第一个方程：

3(3) + y = 11 → 9 + y = 11 → y = 2

The solution is x = 3, y = 2. Always check by substituting both values into the other equation: 2(3) − 2 = 6 − 2 = 4 ✓.

解为 x = 3，y = 2。一定要把两个值代入另一个方程验算：2(3) − 2 = 6 − 2 = 4 ✓。

If neither coefficient matches, multiply one or both equations first. For example, to solve 2x + 3y = 7 and 5x + y = 11, multiply the second equation by 3 to make the y coefficients +3 and +3, then subtract.

如果系数都不相同，需要先将一个或两个方程整体乘以某个数。例如解 2x + 3y = 7 和 5x + y = 11 时，将第二个方程乘以 3，使两个 y 系数都变成 +3，再相减。

2x + 3y = 7

15x + 3y = 33

(15x − 2x) + (3y − 3y) = 33 − 7 → 13x = 26 → x = 2

Then 2(2) + 3y = 7 → 3y = 3 → y = 1. The solution is x = 2, y = 1.

然后 2(2) + 3y = 7 → 3y = 3 → y = 1。解为 x = 2，y = 1。

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. It is especially useful when one equation is already in the form y = … or x = …

代入法是将其中一个方程变形，把某个变量表示成含另一个变量的代数式，再把这个代数式代入另一个方程。当一个方程已经是 y = … 或 x = … 的形式时尤其好用。

Example: Solve

例：解方程组

y = 2x + 1

x + y = 7

Since y is already the subject, substitute 2x + 1 for y in the second equation:

由于 y 已经是研究主题，将第二个方程中的 y 替换为 2x + 1：

x + (2x + 1) = 7

3x + 1 = 7 → 3x = 6 → x = 2

Now substitute x = 2 back into y = 2x + 1: y = 2(2) + 1 = 5. The solution is x = 2, y = 5.

再将 x = 2 代回 y = 2x + 1：y = 2(2) + 1 = 5。解为 x = 2，y = 5。

When the equations are given in the form ax + by = c, rearrange one equation first. For example, from x + y = 7 you could write y = 7 − x and substitute that into the other equation.

如果方程以 ax + by = c 的形式给出，先变形其中一个方程。例如从 x + y = 7 可得 y = 7 − x，再代入另一个方程。

4. Choosing Between Elimination and Substitution | 在消元法与代入法之间选择

The table below summarises when each method is most convenient. In an exam, you may use either method, but choosing the right one saves time and reduces errors.

下表总结了每种方法最适用的场景。考试中两种方法都可以用，但选择正确的方法能节省时间并减少错误。

Method \| 方法	Best used when \| 适用情况	Example \| 例子
Elimination 消元法	Coefficients of a variable match or can easily match	2x + y = 5; x + y = 3
Substitution 代入法	One variable is already the subject or easy to make the subject	y = 3x − 2; 2x + y = 8

In general, if one equation contains a single term in x or y, substitution is often cleaner. If both equations are in the form ax + by = c, elimination is usually quicker.

一般来说，如果一个方程中某个变量的系数为 1，代入法更简洁；如果两个方程都是 ax + by = c 的形式，消元法通常更快。

5. The Graphical Method | 图像法

Graphical methods are used when you need to estimate a solution or when the question specifically asks you to draw the lines. Each equation is rearranged into the form y = mx + c, plotted on the same axes, and the point of intersection gives the solution.

图像法用于需要估算解，或题目明确要求画图时。将每个方程化为 y = mx + c 的形式，画在同一坐标系中，交点就是解。

Example: Solve graphically x + y = 5 and 2x − y = 1.

例：用图像法解 x + y = 5 和 2x − y = 1。

First rearrange:

先变形：

y = −x + 5

y = 2x − 1

Plot both lines. The first passes through (0, 5) and (5, 0); the second passes through (0, −1) and (1, 1). The lines intersect at approximately x = 2, y = 3.

画出两条直线。第一条经过 (0, 5) 和 (5, 0)；第二条经过 (0, −1) 和 (1, 1)。两线交点约为 x = 2，y = 3。

Checking algebraically: 2 + 3 = 5 ✓ and 2(2) − 3 = 1 ✓.

代数验证：2 + 3 = 5 ✓，2(2) − 3 = 1 ✓。

Remember that graphical solutions are only as accurate as your drawing. In an exam, plotting these points carefully is essential — use a sharp pencil and a ruler.

注意图像解的精度取决于画图是否准确。考试中一定要用削尖的铅笔和直尺仔细描点。

6. Solving Linear and Quadratic Simultaneous Equations | 解线性与二次联立方程

In extended IGCSE papers, you will also meet a system where one equation is linear and the other is quadratic. The solution set may contain two pairs of values.

在 IGCSE 扩展卷中，你还会遇到一个方程为线性、另一个方程为二次的方程组。解可能包含两对数值。

Example: Solve

例：解方程组

y = x²

y = x + 2

Since both equations have y as the subject, set the right-hand sides equal to each other:

因为两个方程都给出了 y 的表达式，令右边相等：

x² = x + 2

x² − x − 2 = 0

(x − 2)(x + 1) = 0

So x = 2 or x = −1. Substitute each into y = x + 2:

因此 x = 2 或 x = −1。分别代入 y = x + 2：

x = 2 → y = 4

x = −1 → y = 1

The solutions are x = 2, y = 4 and x = −1, y = 1. These correspond to the two intersection points between a parabola and a straight line.

解为 x = 2，y = 4 以及 x = −1，y = 1。它们对应抛物线与直线的两个交点。

For systems like 2x + y = 5 and x² + y² = 13, start by making y = 5 − 2x from the linear equation, then substitute into the quadratic. This produces a quadratic equation in x that you solve by factorising or the quadratic formula.

对于像 2x + y = 5 和 x² + y² = 13 这样的方程组，先从线性方程得到 y = 5 − 2x，再代入二次方程。这会得到一个关于 x 的二次方程，通过因式分解或求根公式求解。

x² + (5 − 2x)² = 13

x² + 25 − 20x + 4x² = 13

5x² − 20x + 12 = 0

Using the quadratic formula gives x = 2 ± (2√10)/5 approximately, and each x produces a corresponding y value.

用求根公式可得 x = 2 ± (2√10)/5（近似值），每个 x 对应一个 y 值。

7. Forming Simultaneous Equations from Word Problems | 从应用题建立联立方程

Many exam questions do not give equations directly; you must translate the words into algebra. Read the problem carefully, define two unknowns, and look for two independent pieces of information.

许多考试题目并不直接给方程，需要你把文字转成代数表达式。仔细读题，设两个未知数，找出两条独立的信息。

Example: The sum of two numbers is 20 and their difference is 4. Let the larger number be x and the smaller be y.

例：两个数之和为 20，差为 4。设较大的数为 x，较小的数为 y。

x + y = 20

x − y = 4

Add the equations: 2x = 24, so x = 12. Then y = 20 − 12 = 8. Check: 12 + 8 = 20 ✓ and 12 − 8 = 4 ✓.

两式相加：2x = 24，所以 x = 12。则 y = 20 − 12 = 8。检验：12 + 8 = 20 ✓，12 − 8 = 4 ✓。

A common type involves costs. For example: 2 pens and 3 books cost $11; 1 pen and 2 books cost $7. Let p = price of a pen and b = price of a book. Write:

常见类型涉及价格。例如：2 支笔和 3 本书共 11 美元；1 支笔和 2 本书共 7 美元。设 p 为笔的单价，b 为书的单价：

2p + 3b = 11

p + 2b = 7

From the second equation, p = 7 − 2b. Substitute into the first: 2(7 − 2b) + 3b = 11 → 14 − 4b + 3b = 11 → −b = −3 → b = 3. Then p = 7 − 2(3) = 1. A book costs $3 and a pen costs $1.

由第二个方程得 p = 7 − 2b。代入第一个方程：2(7 − 2b) + 3b = 11 → 14 − 4b + 3b = 11 → −b = −3 → b = 3。则 p = 7 − 2(3) = 1。书每本 3 美元，笔每支 1 美元。

8. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Even strong students lose marks on simultaneous equations through careless errors. Here are the most frequent mistakes and the strategies to prevent them.

即使是成绩不错的学生也会在联立方程上因粗心而失分。以下是最常见的错误及预防策略。

Not multiplying every term when scaling an equation. For example, if you multiply x + y = 7 by 2, you must write 2x + 2y = 14, not 2x + y = 14. Always check each term.
Sign errors when subtracting equations. If you subtract (3x − 2y) from (5x + y), remember that subtracting a negative gives a positive: y − (−2y) = 3y.
Forgetting to substitute back into the other equation to justify the answer. Always verify both equations.
In quadratic systems, losing one solution. Remember that a quadratic equation usually gives two x-values, so there may be two pairs of solutions.
Mixing up coordinates when giving the answer. Write the final answer as x = …, y = … clearly, and when writing in coordinate form, put x first, e.g. (2, 5).

未在方程整体缩放时乘以每一项。例如将 x + y = 7 乘以 2 时，必须写成 2x + 2y = 14，不能写成 2x + y = 14。务必逐项检查。

相减时出现符号错误。从 (5x + y) 中减去 (3x − 2y) 时，减去负数等于加正数：y − (−2y) = 3y。

忘记代回另一个方程验算。一定要验证两个方程都成立。

在二次方程组中漏掉一个解。记住二次方程通常给出两个 x 值，因此可能有两对解。

写答案时混淆坐标顺序。最后清晰写成 x = …，y = …；写成坐标形式时，x 必须在前，如 (2, 5)。

9. Exam Tips and Practice Questions | 考试技巧与练习

In the IGCSE exam, show full working for every step. Partial marks are awarded for correct substitution, rearranging and factorising even if the final answer is wrong. Write well-structured steps in a logical order.

在 IGCSE 考试中，每一步都必须写出完整过程。即使最终答案错误，正确的代入、变形和因式分解仍能获得步骤分。按逻辑顺序书写清晰步骤。

Always check your answer by substituting into the original equations.
For graphs, plot at least three points for each line to avoid mistakes.
For quadratic systems, expect two solutions unless the line is tangent to the curve.
When a question says “estimate”, your graphical answer needs only to be close, but a clear intersection point is required.

始终把答案代回原方程验算。
画图时每条直线至少描三个点，避免出错。
对于二次方程组，除非直线与曲线相切，否则通常有两组解。
题目要求 estimate 时，图像答案只需接近，但必须清晰标出交点。

Try these exam-style questions. Answers are given below.

试试以下考试风格题目，答案在下方。

Question 1 | 题目 1: Solve 3x + 2y = 12 and x − 2y = −4.

Question 2 | 题目 2: Solve y = 3x + 1 and y = x² − 1.

Question 3 | 题目 3: A ticket office sells adult tickets for $5 and child tickets for $2. On one day, 120 tickets are sold for a total of $450. How many adult tickets were sold?

Answers | 答案:

Q1: Add equations: 4x = 8 → x = 2. Then 3(2) + 2y = 12 → 2y = 6 → y = 3. Solution: x = 2, y = 3.

Q1 答案: 两式相加：4x = 8 → x = 2。则 3(2) + 2y = 12 → 2y = 6 → y = 3。解：x = 2，y = 3。

Q2: Set 3x + 1 = x² − 1 → x² − 3x − 2 = 0 → x = (3 ± √17)/2. The exact solutions are x = (3 + √17)/2, y = (11 + 3√17)/2 and x = (3 − √17)/2, y = (11 − 3√17)/2.

Q2 答案: 令 3x + 1 = x² − 1 → x² − 3x − 2 = 0 → x = (3 ± √17)/2。精确解为 x = (3 + √17)/2，y = (11 + 3√17)/2 和 x = (3 − √17)/2，y = (11 − 3√17)/2。

Q3: Let a be adult tickets and c be child tickets. Then a + c = 120 and 5a + 2c = 450. From the first equation, c = 120 − a. Substitute: 5a + 2(120 − a) = 450 → 5a + 240 − 2a = 450 → 3a = 210 → a = 70. So 70 adult tickets were sold.

Q3 答案: 设成人票数量为 a，儿童票数量为 c。则 a + c = 120，5a + 2c = 450。由第一个方程得 c = 120 − a。代入：5a + 2(120 − a) = 450 → 5a + 240 − 2a = 450 → 3a = 210 → a = 70。所以售出 70 张成人票。

Simultaneous equations reward careful organisation and systematic working. Master the elimination and substitution methods, practise forming equations from word problems, and always verify your answers. With consistent practice, this topic becomes one of the most reliable sources of marks in the IGCSE examination.

联立方程考验的是细致的整理能力和系统的解题步骤。掌握消元法与代入法，练习从应用题中建立方程，并始终验算答案。坚持练习后，这一话题将成为 IGCSE 考试中最稳定的得分点之一。

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Solving Simultaneous Equations | 解联立方程