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 variables that must be solved together, because the solution must satisfy all equations at the same time. In IGCSE Mathematics, you will often deal with two linear equations in two unknowns, usually x and y.

联立方程组是一组包含多个变量的方程，需要同时求解，因为解必须同时满足所有方程。在IGCSE数学中，你通常会遇到含有两个未知数 x 和 y 的两个线性方程。

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

A single linear equation with two variables, such as x + y = 7, has infinitely many solutions. For example, x = 1, y = 6 and x = 2, y = 5 both satisfy it. When we have two such equations, we look for a pair (x, y) that satisfies both equations. This pair is called the simultaneous solution.

一个含有两个变量的线性方程，例如 x + y = 7，有无穷多组解。比如 x = 1，y = 6 和 x = 2，y = 5 都满足它。当我们有两个这样的方程时，我们寻找同时满足两个方程的一组 (x, y)，这组解称为联立解。

Example: Solve x + y = 7 and x − y = 3. The only solution is x = 5, y = 2, because 5 + 2 = 7 and 5 − 2 = 3.

例如：解 x + y = 7 和 x − y = 3。唯一解是 x = 5，y = 2，因为 5 + 2 = 7 且 5 − 2 = 3。

2. Graphical Method | 图像法

The graphical method is the most visual way to solve simultaneous equations. Each equation is drawn as a straight line on the same pair of axes. The point where the two lines intersect gives the values of x and y that solve both equations.

图像法是解决联立方程组最直观的方法。每个方程都被画成同一直角坐标系中的一条直线。两条直线的交点坐标就是同时满足两个方程的 x 和 y 值。

Steps: 1) Rearrange each equation into the form y = mx + c, if necessary. 2) Plot both lines accurately. 3) Read the coordinates of the intersection point.

步骤：1) 如有需要，将每个方程改写为 y = mx + c 的形式；2) 精确画出两条直线；3) 读出交点的坐标。

Example: y = x + 1 and y = −2x + 7

The lines intersect at (2, 3). Therefore x = 2, y = 3 is the simultaneous solution.

两条直线相交于点 (2, 3)。因此 x = 2，y = 3 就是联立解。

3. Substitution Method | 代入消元法

The substitution method works best when one equation is already written as y = … or x = … . You replace that variable in the other equation with the expression given by the first equation.

代入消元法在其中一个方程已经写成 y = … 或 x = … 的形式时最方便。你将第一个方程中的表达式代入另一个方程中的相应变量。

Example: Solve y = 3x − 2 and 2y + x = 10.

示例：解 y = 3x − 2 和 2y + x = 10。

Substitute y = 3x − 2 into the second equation:

将 y = 3x − 2 代入第二个方程：

2(3x − 2) + x = 10 → 6x − 4 + x = 10 → 7x = 14 → x = 2

Then y = 3(2) − 2 = 4. The solution is x = 2, y = 4.

再计算 y = 3(2) − 2 = 4。解为 x = 2，y = 4。

4. Elimination Method | 加减消元法

The elimination method is often the fastest for equations in the form ax + by = c. You multiply one or both equations so that the coefficients of one variable become the same or opposites. Then you add or subtract the equations to eliminate that variable.

加减消元法对于形如 ax + by = c 的方程通常是最快的。你可以将一个或两个方程乘以适当的数，使某个变量的系数相同或互为相反数，然后通过相加或相减消去该变量。

Example: Solve 3x + 2y = 12 and 2x − 2y = 8.

示例：解 3x + 2y = 12 和 2x − 2y = 8。

Add the two equations to eliminate y:

将两个方程相加以消去 y：

(3x + 2x) + (2y − 2y) = 12 + 8 → 5x = 20 → x = 4

Substitute x = 4 into 3x + 2y = 12:

将 x = 4 代入 3x + 2y = 12：

3(4) + 2y = 12 → 12 + 2y = 12 → y = 0

So the solution is x = 4, y = 0.

因此解为 x = 4，y = 0。

5. Solving by Equating | 等式相立法

If both equations are already in the form y = …, you can set the right-hand sides equal to each other. This creates one linear equation in x, which is easy to solve.

如果两个方程都已经写成 y = … 的形式，你可以令等号右边部分相等。这样就得到一个关于 x 的线性方程，很容易求解。

Example: y = 2x + 1 and y = 5 − x.

示例：y = 2x + 1 和 y = 5 − x。

Equate the right-hand sides:

令等号右边相等：

2x + 1 = 5 − x → 3x = 4 → x = 4/3

Then y = 2(4/3) + 1 = 8/3 + 1 = 11/3. The solution is x = 4/3, y = 11/3.

然后 y = 2(4/3) + 1 = 8/3 + 1 = 11/3。解为 x = 4/3，y = 11/3。

6. Forming Simultaneous Equations from Word Problems | 由应用题建立联立方程组

Many real-life problems can be translated into two linear equations. The key is to define the variables clearly and then find two independent pieces of information that connect them.

许多现实生活中的问题可以转化为两个线性方程。关键在于清晰定义变量，然后找出连接它们的两个独立信息。

Example: The total cost of 2 apples and 3 bananas is 13 yuan. The cost of 4 apples and 2 bananas is 18 yuan. Find the price of one apple and one banana.

示例：2个苹果和3根香蕉共13元，4个苹果和2根香蕉共18元。求1个苹果和1根香蕉的价格。

Let the price of an apple be x and a banana be y. Then:

设一个苹果的价格为 x，一根香蕉的价格为 y。则：

2x + 3y = 13 and 4x + 2y = 18

Using elimination, multiply the first equation by 2: 4x + 6y = 26. Subtract the second equation from it: (4x + 6y) − (4x + 2y) = 26 − 18 → 4y = 8 → y = 2. Then 2x + 3(2) = 13 → 2x = 7 → x = 3.5. So an apple costs 3.5 yuan and a banana costs 2 yuan.

使用加减消元法，将第一个方程乘以2：4x + 6y = 26。用它减去第二个方程：(4x + 6y) − (4x + 2y) = 26 − 18 → 4y = 8 → y = 2。然后 2x + 3(2) = 13 → 2x = 7 → x = 3.5。所以一个苹果3.5元，一根香蕉2元。

7. Special Cases: Unique, No Solution, or Infinite Solutions | 特殊情况：唯一解、无解与无穷多解

When solving simultaneous equations, the two lines may behave in three different ways. If the lines intersect at one point, there is exactly one solution. If the lines are parallel, there is no solution. If the lines are identical, there are infinitely many solutions.

解联立方程组时，两条直线可能有三种不同情况。如果两条直线相交于一点，那么有唯一解。如果两条直线平行，则无解。如果两条直线重合，则有无穷多组解。

For example, the equations x + y = 3 and x + y = 5 represent parallel lines. They have the same slope but different intercepts, so they never meet.

例如，方程 x + y = 3 和 x + y = 5 表示两条平行线。它们有相同的斜率但不同的截距，因此永远不会相交。

In contrast, x + y = 3 and 2x + 2y = 6 represent the same line. Every point on the line satisfies both equations.

相比之下，x + y = 3 和 2x + 2y = 6 表示同一条直线。直线上的每一个点都同时满足两个方程。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is forgetting to multiply every term when using elimination. Always check that the new equation is still equivalent to the original.

一个常见错误是使用加减消元时忘记乘每一项。始终检查新方程仍然与原方程等价。

Another mistake is making a sign error when subtracting equations. If you subtract a negative term, it becomes positive. For example, subtracting −2y is the same as adding 2y.

另一个错误是相减方程时符号出错。如果你减去一个负数项，它会变成正数。例如，减去 −2y 相当于加上 2y。

Tips: 1) Always write down each step clearly. 2) After finding x and y, substitute both values into both original equations to check. 3) If the question asks for an exact answer, avoid decimals unless stated. Use fractions when needed.

技巧：1) 每一步都清晰写下。2) 求得 x 和 y 后，将两个值同时代入两个原方程进行验证。3) 如果题目要求精确答案，除非特别说明，否则避免小数，需要时使用分数。

9. Summary and Further Practice | 总结与进阶练习

In this article you have learned four main methods: graphical, substitution, elimination, and equating. Each has its advantages: the graphical method gives a visual picture, substitution works well when one variable is already isolated, elimination is efficient for standard forms, and equating is useful when both equations are in y = form.

在本文中你学习了四种主要方法：图像法、代入消元法、加减消元法和等式相立法。每种方法各有优势：图像法提供直观的图形，代入法在变量已经被单独表示时很有效，加减法对于标准形式很高效，而等式相立法在两个方程都是 y = 形式时很方便。

To master simultaneous equations, practice with a variety of problems. Start with simple integer solutions, then move to fractional solutions and word problems. Always verify your answers by substitution. With regular practice, you will be able to choose the best method quickly and accurately in exams.

要想掌握联立方程组，需要练习多种类型的问题。从简单的整数解开始，然后过渡到分数解和应用题。始终通过代入验证答案。通过定期练习，你将在考试中快速而准确地选择最佳方法。

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Solving Simultaneous Equations | 解联立方程组