Solving Simultaneous Equations | 联立方程的解法

📚 Solving Simultaneous Equations | 联立方程的解法

Simultaneous equations are one of the most frequently tested topics in IGCSE Mathematics. Whether you are sitting the Core or Extended paper, you will encounter systems of linear equations, and sometimes a linear equation paired with a quadratic. This article breaks down every method you need, with step-by-step worked examples, examiner-style tips, and practice questions.

联立方程是 IGCSE 数学中最高频的考点之一。无论是 Core 还是 Extended 试卷，你都会遇到线性方程组，有时还会遇到线性方程与二次方程的组合。本文将系统讲解所有需要掌握的方法，包含逐步演算的例题、考官风格的技巧提示以及练习题。

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

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

联立方程组由两个或两个以上含有相同未知数的方程组成。方程组的解是同时满足其中每一个方程的一组数值。

Consider the system below:

2x + y = 7
x − y = 2

The only pair that works is x = 3 and y = 1, because 2(3) + 1 = 7 and 3 − 1 = 2. We say the solution is x = 3, y = 1, often written as (3, 1).

唯一同时成立的解是 x = 3，y = 1，因为 2(3) + 1 = 7，且 3 − 1 = 2。我们称方程组的解为 x = 3，y = 1，通常记作 (3, 1)。

Equation	Check with x = 3, y = 1	Result
2x + y = 7	2(3) + 1 = 6 + 1	7 ✓
x − y = 2	3 − 1 = 2	2 ✓

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2. The Substitution Method | 代入法

The substitution method is ideal when one equation is already written in the form ‘y = …’ or ‘x = …’. You replace that variable in the other equation with its equivalent expression.

代入法适用于其中一个方程已经写成 ‘y = …’ 或 ‘x = …’ 的形式。此时，把这个表达式代入另一个方程中的对应变量即可。

Worked Example: Solve the system y = 2x + 1 and 3x + 2y = 23.

例题：解方程组 y = 2x + 1 与 3x + 2y = 23。

Substitute 2x + 1 for y in the second equation:

把 y = 2x + 1 代入第二个方程：

3x + 2(2x + 1) = 23
3x + 4x + 2 = 23
7x = 21
x = 3

Now substitute x = 3 back into y = 2x + 1:

再将 x = 3 代回 y = 2x + 1：

y = 2(3) + 1 = 7

Therefore, the solution is x = 3, y = 7.

因此，方程组的解为 x = 3，y = 7。

• Rearrange one equation to make one variable the subject, if needed. | 如有需要，先将一个方程变形，使某个变量成为主项。
• Substitute the expression into the other equation. | 将得到的表达式代入另一个方程。
• Solve the resulting single-variable equation. | 解出这个只含一个变量的方程。
• Substitute the value back to find the second variable. | 代回原式求出另一个变量。

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3. The Elimination Method | 消元法

The elimination method works by adding or subtracting the equations to remove one variable. You may first need to multiply one or both equations so that the coefficients of a variable match.

消元法的核心是通过将两个方程相加或相减，消去其中一个变量。必要时，需要先将一个或两个方程乘以适当的数，使某个变量的系数相等。

Worked Example: Solve the system 3x + 2y = 18 and 2x − y = 5.

例题：解方程组 3x + 2y = 18 与 2x − y = 5。

Multiply the second equation by 2 so that the y coefficients are opposite:

将第二个方程乘以 2，使 y 的系数互为相反数：

4x − 2y = 10

Add the two equations to eliminate y:

将两个方程相加以消去 y：

3x + 2y + 4x − 2y = 18 + 10
7x = 28
x = 4

Substitute x = 4 into 3x + 2y = 18:

将 x = 4 代入 3x + 2y = 18：

3(4) + 2y = 18
12 + 2y = 18
2y = 6
y = 3

The solution is x = 4, y = 3. Always check in the other equation: 2(4) − 3 = 5 ✓.

方程组的解为 x = 4，y = 3。务必代入另一个方程检验：2(4) − 3 = 5 ✓。

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4. Which Method Should You Choose? | 如何选择方法？

Both methods always give the same answer, but one is often quicker. Use the table below to decide.

两种方法得到的答案一定相同，但其中一种往往更快。请参考下表进行选择。

Situation | 情况	Recommended Method | 推荐方法
One variable is already the subject (y = …)	Substitution | 代入法
Coefficients of one variable match or can match easily	Elimination | 消元法
One linear and one quadratic equation	Substitution | 代入法
You need an estimate or a visual check	Graphical method | 图像法

If an equation is given in the form y = mx + c, substitution is usually the most direct route. If both equations are written in the form ax + by = c, elimination is often neater.

如果某个方程已写成 y = mx + c 的形式，代入法通常最直接。如果两个方程都写成 ax + by = c 的形式，消元法通常更简洁。

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5. Solving Word Problems | 应用题解法

Exam questions often wrap simultaneous equations in a real-life context. The key is to translate words into algebra: define two variables, write two equations, then solve using either method.

考试题常常把联立方程包装在实际生活情境中。关键是把文字转译为代数：先设定两个变量，列出两个方程，再用任意一种方法求解。

Worked Example: A theatre sells adult tickets at $8 and child tickets at $5. A group buys 12 tickets for a total of $84. How many adult and child tickets were bought?

例题：剧院成人票每张 8 美元，儿童票每张 5 美元。一个团体共买了 12 张票，总价为 84 美元。问成人票和儿童票各买了多少张？

Let a = number of adult tickets and c = number of child tickets.

设 a 为成人票数量，c 为儿童票数量。

a + c = 12
8a + 5c = 84

From the first equation, c = 12 − a. Substitute into the second:

由第一个方程得 c = 12 − a，代入第二个方程：

8a + 5(12 − a) = 84
8a + 60 − 5a = 84
3a = 24
a = 8

Then c = 12 − 8 = 4. So the group bought 8 adult tickets and 4 child tickets. Check: 8(8) + 5(4) = 64 + 20 = 84 ✓.

因此 c = 12 − 8 = 4。该团体买了 8 张成人票和 4 张儿童票。检验：8(8) + 5(4) = 64 + 20 = 84 ✓。

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6. One Linear and One Quadratic | 线性与二次联立方程

On the Extended paper, you may be asked to solve a system where one equation is linear and the other is quadratic. Substitution always works: substitute the linear expression into the quadratic, then solve the resulting quadratic equation by factorising or using the quadratic formula.

在 Extended 试卷中，你可能会遇到一个线性方程与一个二次方程组成的方程组。代入法始终有效：把线性表达式代入二次方程，然后通过因式分解或求根公式解出所得的一元二次方程。

Worked Example: Solve y = x + 1 and x² + y² = 25.

例题：解方程组 y = x + 1 与 x² + y² = 25。

Substitute y = x + 1 into the quadratic equation:

将 y = x + 1 代入二次方程：

x² + (x + 1)² = 25
x² + x² + 2x + 1 = 25
2x² + 2x − 24 = 0

Divide both sides by 2 and factorise:

两边同时除以 2 并进行因式分解：

x² + x − 12 = 0
(x + 4)(x − 3) = 0

So x = −4 or x = 3. Now find the corresponding y values:

因此 x = −4 或 x = 3。再求对应的 y 值：

When x = −4, y = −4 + 1 = −3
When x = 3, y = 3 + 1 = 4

The solutions are (−4, −3) and (3, 4). A linear-quadratic system can have up to two solutions.

方程组的解为 (−4, −3) 和 (3, 4)。线性与二次联立方程组最多可以有两个解。

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7. The Graphical Method | 图像法

Each linear equation can be drawn as a straight line on a coordinate grid. The solution of the system is the coordinates of the point where the two lines intersect.

每个线性方程都可以在坐标平面上画成一条直线。方程组的解就是两条直线交点的坐标。

For example, the lines y = 2x + 1 and y = −x + 4 intersect at the point (1, 3). Therefore, x = 1, y = 3 is the solution.

例如，直线 y = 2x + 1 与 y = −x + 4 交于点 (1, 3)。因此 x = 1，y = 3 就是方程组的解。

The graphical method is useful for estimating solutions and for understanding the meaning of ‘no solution’ or ‘infinitely many solutions’. However, IGCSE examiners expect algebraic working for exact marks; graphs alone are rarely enough for full credit.

图像法有助于估算解，也能帮助你理解“无解”和“无穷多解”的含义。不过，IGCSE 考官期望考生写出代数步骤以获得精确分数；仅靠画图通常不足以拿到满分。

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

Not every system has a unique solution. If the two lines are parallel, the system has no solution. If the two equations represent the same line, every point on that line is a solution, giving infinitely many solutions.

并非每个方程组都有唯一解。如果两条直线平行，方程组无解；如果两个方程表示同一条直线，则直线上每一点都是解，此时有无穷多组解。

No solution: y = 3x + 2 and y = 3x − 1 have the same slope (3) but different intercepts, so the lines are parallel and never meet.

无解：y = 3x + 2 与 y = 3x − 1 的斜率相同（均为 3），但截距不同，因此两条直线平行，永不相交。

Infinite solutions: 2x + y = 5 and 4x + 2y = 10. The second equation is exactly double the first, so they represent the same straight line

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