Mastering Simultaneous Equations | 解方程组完全指南

📚 Mastering Simultaneous Equations | 解方程组完全指南

Simultaneous equations are a fundamental topic in IGCSE Mathematics. You will learn to solve systems of linear equations using algebraic and graphical methods, and apply them to real-world problems.

解方程组是 IGCSE 数学中的基础重要内容。你将学习用代数方法和图像法求解线性方程组，并将其应用于现实问题。

1. What Are Simultaneous Equations? | 什么是方程组？

Simultaneous equations are two or more equations that share the same variables. The solution is the set of values that satisfies all equations at the same time. For example:
2x + y = 7 and x − y = 2.

方程组是指两个或多个含有相同变量的方程。解就是同时满足所有方程的取值。例如：2x + y = 7 和 x − y = 2。

2x + y = 7
x − y = 2

The unique solution here is x = 3 and y = 1, because both equations become true when these values are substituted.

此处的唯一解是 x = 3, y = 1，因为代入这两个值时两个方程都成立。

2. The Substitution Method | 代入消元法

The substitution method is useful when one variable is already isolated, or can be easily isolated. Solve one equation for one variable, then substitute that expression into the other equation.

代入消元法适用于某个变量已经单独表示出来，或者很容易单独表示出来的情况。先从一个方程中解出一个变量，再将表达式代入另一个方程。

Example / 示例：

y = 2x + 1
x + y = 10

Substitute y = 2x + 1 into the second equation: x + (2x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3.

将 y = 2x + 1 代入第二个方程：x + (2x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3。

Then use y = 2(3) + 1 = 7. Solution: (3, 7). Always check by substituting both values into the original equations.

再利用 y = 2(3) + 1 = 7。解为 (3, 7)。务必把两个值代入原方程进行检验。

3. The Elimination Method | 加减消元法

The elimination method involves adding or subtracting the equations to cancel one variable. This works best when the coefficients of a variable are the same or opposite.

加减消元法通过两个方程相加或相减来消去一个变量。当某个变量的系数相同或互为相反数时最适合使用。

Example / 示例：

3x + 2y = 12
3x − 2y = 6

Add the equations: (3x + 2y) + (3x − 2y) = 12 + 6 → 6x = 18 → x = 3.

两式相加：(3x + 2y) + (3x − 2y) = 12 + 6 → 6x = 18 → x = 3。

Now substitute x = 3 into 3x + 2y = 12: 9 + 2y = 12 → 2y = 3 → y = 1.5. Solution: (3, 1.5).

将 x = 3 代入 3x + 2y = 12：9 + 2y = 12 → 2y = 3 → y = 1.5。解为 (3, 1.5)。

If coefficients are not the same, multiply one or both equations by a suitable constant before adding or subtracting.

如果系数不同，先对方程（或两个方程）乘以适当的常数，再进行加减。

4. The Graphical Method | 图像法

You can solve simultaneous equations by drawing both lines on the same axes. The point of intersection gives the solution.

你可以在同一坐标系中画出两条直线，交点对应的坐标就是方程组的解。

Rearrange each equation into the form y = mx + c.
Plot at least two points for each line.
Read the coordinates of the intersection point.

将每个方程改写成 y = mx + c 的形式。
每条直线至少标出两个点。
读出交点的坐标。

For example, y = x + 1 and y = −x + 5 intersect at (2, 3). The graphical method is often less accurate than algebraic methods, so use it as a check.

例如，y = x + 1 与 y = −x + 5 相交于 (2, 3)。图像法通常不如代数方法精确，因此可用作检验。

5. Solving by Substitution with Non-Linear Equations | 非线性方程组的代入法

At IGCSE you may also solve one linear and one quadratic equation simultaneously. The substitution method is the standard approach.

在 IGCSE 中，还可能要求解一个一次方程和一个二次方程组成的方程组。标准方法是代入法。

Example / 示例：

y = x + 2
x² + y² = 10

Substitute y = x + 2 into x² + y² = 10: x² + (x + 2)² = 10 → x² + x² + 4x + 4 = 10 → 2x² + 4x − 6 = 0 → x² + 2x − 3 = 0.

将 y = x + 2 代入 x² + y² = 10：x² + (x + 2)² = 10 → x² + x² + 4x + 4 = 10 → 2x² + 4x − 6 = 0 → x² + 2x − 3 = 0。

Factorise: (x + 3)(x − 1) = 0 → x = −3 or x = 1. Then find y values: when x = −3, y = −1; when x = 1, y = 3. Solutions: (−3, −1) and (1, 3).

因式分解：(x + 3)(x − 1) = 0 → x = −3 或 x = 1。再求 y 值：x = −3 时 y = −1；x = 1 时 y = 3。解为 (−3, −1) 和 (1, 3)。

6. Word Problems Involving Simultaneous Equations | 方程组应用题

Many exam questions describe a real-world situation. You must translate the words into two equations, then solve.

许多考题会描述一个实际情境。你需要把文字翻译成两个方程，然后求解。

Example / 示例：

The sum of two numbers is 15, and their difference is 3. Find the numbers.

两个数之和为 15，之差为 3。求这两个数。

Let the numbers be a and b. Write: a + b = 15 and a − b = 3. Add the equations: 2a = 18 → a = 9. Then b = 15 − 9 = 6. The numbers are 9 and 6.

设两个数为 a 和 b。列出方程：a + b = 15，a − b = 3。两式相加：2a = 18 → a = 9。则 b = 15 − 9 = 6。两个数为 9 和 6。

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

Some systems do not have a single unique solution. If the lines are parallel, there is no solution. If the equations represent the same line, there are infinitely many solutions.

有些方程组没有唯一解。如果两条直线平行，则无解；如果两个方程表示同一条直线，则有无穷多解。

Situation / 情形	Result / 结果
Parallel lines (same gradient, different intercept) / 平行直线（斜率相同，截距不同）	No solution / 无解
Same line (all coefficients proportional) / 同一直线（系数成比例）	Infinite solutions / 无穷多解

For example, y = 2x + 1 and y = 2x − 3 are parallel, so the system has no solution.

例如，y = 2x + 1 与 y = 2x − 3 平行，因此该方程组无解。

8. Practice Questions | 练习题

Try these problems on your own before checking the answers.

先独立尝试以下题目，再核对答案。

Solve by elimination: 4x + y = 10 and 2x − y = 4.
Solve by substitution: y = 3x − 2 and 2x + y = 8.
A shop sells pens and pencils. 3 pens and 5 pencils cost $7, while 2 pens and 3 pencils cost $4.40. Find the price of one pen and one pencil.

用加减消元法解：4x + y = 10 与 2x − y = 4。
用代入消元法解：y = 3x − 2 与 2x + y = 8。
商店出售钢笔和铅笔。3 支钢笔和 5 支铅笔共 7 美元，2 支钢笔和 3 支铅笔共 4.40 美元。求一支钢笔和一支铅笔的价格。

Answers: (1) x = 7/3, y = 2/3; (2) x = 2, y = 4; (3) pen = $1.20, pencil = $0.80.

答案：(1) x = 7/3, y = 2/3；(2) x = 2, y = 4；(3) 钢笔 1.20 美元，铅笔 0.80 美元。

9. Common Mistakes | 常见错误

Avoid these frequent traps in simultaneous equations:

在解方程组时，要避免以下常见陷阱：

Sign errors when subtracting: Always distribute the minus sign correctly, e.g. 3x − (x − 2) = 3x − x + 2.
Not checking your answer: Always substitute both values back into the original equations.
Forgetting to multiply every term: When scaling an equation, multiply every term on both sides.
Mixing up variables: Keep track of which variable you substituted.

相减时符号错误：计算时务必正确分配负号，例如 3x − (x − 2) = 3x − x + 2。
不检验答案：始终将两个值代入原方程检验。
忘记乘以每一项：对方程整体缩放时，两边每一项都要乘。
混淆变量：注意你代入的是哪个变量。

10. Summary | 总结

To solve simultaneous equations, choose the most efficient method: substitution for linear-quadratic systems, elimination for simple linear systems, and graphs as a visual check. Always write your solution as an ordered pair or clearly state the values.

解方程组时，要选择最有效的方法：线性与二次混合系统用代入法，简单线性系统用消元法，图像法用作直观检验。始终把解写成坐标对的形式或清楚地标明各变量的值。

Key skills to practise: rearranging equations, factorising quadratics, and translating word problems into algebra.

需要练习的关键技能：重排方程、二次因式分解，以及把文字题转化为代数式。

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Mastering Simultaneous Equations | 解方程组完全指南