Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are sets of equations that share the same unknown variables. Solving them means finding values for the variables that satisfy all equations at the same time. This is a core skill for the Edexcel IGCSE Mathematics exam.

联立方程是包含相同未知数的多个方程。解联立方程就是找到同时满足所有方程的一组变量值。这是爱德思 IGCSE 数学考试的核心技能。

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

For two linear equations in two variables, such as x and y, a solution is a pair (x, y) that makes both equations true. Graphically, each equation represents a straight line, and the solution is the point where the two lines intersect.

对于含有两个未知数（如 x 和 y）的两个线性方程，解是一组 (x, y) 使得两个方程同时成立。从图像上看，每个方程代表一条直线，解就是两条直线的交点。

Example: 2x + y = 7 and x − y = 2. The values x = 3, y = 1 work because 2(3) + 1 = 7 and 3 − 1 = 2.

例如：2x + y = 7 和 x − y = 2。x = 3，y = 1 满足条件，因为 2(3) + 1 = 7 且 3 − 1 = 2。

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 opposites.

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

Solve: 2x + y = 7 and x − y = 2.

解方程组：2x + y = 7 和 x − y = 2。

Add the two equations: (2x + y) + (x − y) = 7 + 2.

把两个方程相加：(2x + y) + (x − y) = 7 + 2。

3x = 9 → x = 3

Substitute x = 3 into 2x + y = 7: 2(3) + y = 7 → 6 + y = 7 → y = 1.

将 x = 3 代入 2x + y = 7：2(3) + y = 7 → 6 + y = 7 → y = 1。

So the solution is x = 3, y = 1.

所以解为 x = 3，y = 1。

3. The Substitution Method | 代入法

In the substitution method, one equation is rearranged to express one variable in terms of the other. This expression is then substituted into the other equation.

代入法是将一个方程变形，用另一个变量表示某个变量，然后把这个表达式代入另一个方程。

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

例如：y = x + 2 和 3x + y = 10。

Since y = x + 2, substitute into the second equation: 3x + (x + 2) = 10.

因为 y = x + 2，代入第二个方程：3x + (x + 2) = 10。

4x + 2 = 10 → 4x = 8 → x = 2

Then y = 2 + 2 = 4. The solution is x = 2, y = 4.

于是 y = 2 + 2 = 4。解为 x = 2，y = 4。

4. Solving Linear-Quadratic Systems | 解线性-二次方程组

Edexcel IGCSE often includes one linear and one quadratic equation. The substitution method is usually the best approach.

爱德思 IGCSE 经常考一个线性方程和一个二次方程的组合。通常用代入法最合适。

Example: y = x² + 1 and y = 2x + 1.

例如：y = x² + 1 和 y = 2x + 1。

Since both equal y, set them equal: x² + 1 = 2x + 1.

因为两边都等于 y，所以令它们相等：x² + 1 = 2x + 1。

x² − 2x = 0 → x(x − 2) = 0

This gives x = 0 or x = 2. Substitute back to find y: when x = 0, y = 1; when x = 2, y = 5.

因此 x = 0 或 x = 2。代回求 y：当 x = 0 时，y = 1；当 x = 2 时，y = 5。

The solutions are (0, 1) and (2, 5).

解为 (0, 1) 和 (2, 5)。

5. Using Graphs to Find Solutions | 用图像求交

You can plot both equations on the same axes. The coordinates of the intersection point(s) are the solutions.

你可以在同一坐标系中画出两个方程的图像。交点的坐标就是方程组的解。

For a linear and a quadratic equation, the line may cut the curve twice, once, or not at all. This corresponds to two, one, or zero real solutions.

对于线性方程和二次方程，直线可能与曲线相交两次、一次或不相交，分别对应两个、一个或零个实数解。

Use a table of values for each equation, plot the points, and read off the intersection coordinates.

先为每个方程列值表，然后描点画图，读出交点的坐标。

6. Word Problems with Simultaneous Equations | 应用题

Real-world problems can be modelled with two variables and two equations. Look for key information that connects the quantities.

现实世界的问题可以用两个变量和两个方程来建模。注意寻找连接数量的关键信息。

Example: The sum of two numbers is 12, and their difference is 4. Let x and y be the numbers.

例如：两个数之和为 12，差为 4。设这两个数为 x 和 y。

x + y = 12, x − y = 4

Add the equations: 2x = 16 → x = 8. Then y = 12 − 8 = 4.

两式相加：2x = 16 → x = 8。于是 y = 12 − 8 = 4。

The numbers are 8 and 4. Always check the answer in the original conditions.

这两个数是 8 和 4。一定要把答案代回原条件检查。

7. Special Cases: No Solution or Infinite Solutions | 特殊情况：无解或无穷多解

When solving two linear equations, there are three possible outcomes:

解两个线性方程时，可能有三种结果：

One unique solution: the lines intersect at a single point.

唯一解：两条直线交于一点。
No solution: the lines are parallel. The coefficients of x and y are proportional, but the constants are not.

无解：两条直线平行。x 和 y 的系数成比例，但常数项不成比例。
Infinite solutions: the two lines are the same line. Both equations are multiples of each other.

无穷多解：两条直线重合。两个方程互为倍数。

Example: y = 2x + 1 and y = 2x − 3 have no solution because they are parallel.

例如：y = 2x + 1 和 y = 2x − 3 无解，因为两直线平行。

8. Common Mistakes to Avoid | 常见错误

Many students lose marks on simultaneous equations because of small errors. Here are the most common ones:

很多学生在联立方程上丢分是因为小错误。以下是最常见的问题：

Forgetting to multiply every term when eliminating a variable.

消元时忘记乘以每一项。
Making sign errors when subtracting equations, especially with negative coefficients.

相减方程时出现符号错误，尤其是负系数时。
Substituting the wrong value for a variable.

代入错误变量的值。
Not checking the final answer by substituting back into both original equations.

不把最终答案代回原方程检查。

Always write each step clearly to avoid confusion.

每一步都写清楚，可以避免混淆。

9. Practice Questions | 练习

Try these questions to test your understanding. Use the methods you have learned.

试试以下题目，检验你的理解。使用你学过的方法。

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

题目 1：解方程组 3x + 2y = 12 和 x − 2y = 4。

Question 2: Solve y = 3x − 5 and y = x² − 3.

题目 2：解方程组 y = 3x − 5 和 y = x² − 3。

Question 3: The cost of 2 apples and 3 bananas is $2.60. The cost of 4 apples and 1 banana is $3.20. Find the cost of one apple and one banana.

题目 3：2 个苹果和 3 根香蕉共 2.60 美元，4 个苹果和 1 根香蕉共 3.20 美元。求一个苹果和一根香蕉的单价。

Work through each one carefully, then check your answers by substitution.

仔细完成每一题，然后用代入法检查答案。

10. Summary | 总结

Simultaneous equations can be solved by elimination, substitution, or graphical methods. For linear-quadratic systems, substitution is almost always the simplest.

联立方程可以用消元法、代入法或图像法求解。对于线性-二次方程组，代入法几乎总是最简单的。

Always check your solution by substituting the values into the original equations. This guarantees accuracy and prevents careless mistakes.

始终把解代入原方程检查，这样能保证准确，避免粗心错误。

Master this topic with regular practice, and it will become a reliable source of marks in your IGCSE exam.

通过经常练习掌握这个主题，它将成为你在 IGCSE 考试中可靠的得分点。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Simultaneous Equations | 解联立方程