Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are a cornerstone of IGCSE Mathematics. They appear in algebra, geometry, and even real-world problem solving. Mastering these methods not only secures exam marks but builds a strong foundation for advanced studies.

联立方程是 IGCSE 数学的核心内容。它们出现在代数、几何，甚至实际生活问题的求解中。掌握这些方法不仅能在考试中稳稳得分，更会为后续高阶学习打下坚实基础。

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

Simultaneous equations are a set of equations with multiple variables that share the same values. In IGCSE, you will most often solve two linear equations with two unknowns, $x$ and $y$, but you may also see non-linear systems.

联立方程是一组含有多个变量、且变量取值相同的方程。在 IGCSE 中，最常见的是含有两个未知数 $x$ 和 $y$ 的两个一次方程，但也可能遇到非线性方程组。

For example:

例如：

2x + y = 7
x − y = 2

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

解为 $x = 3$、$y = 1$，因为将这两组值代入后，两个方程都成立。

2. The Elimination Method | 消元法

Elimination is usually the fastest method when the coefficients of one variable are opposites or easy to match. You add or subtract the equations to remove one variable, then solve for the other.

当某个变量的系数互为相反数或很容易配成相同系数时，消元法通常最快。通过相加或相减两个方程，消去一个变量，再求另一个变量。

Consider the system:

思考以下方程组：

3x + 2y = 12
x − 2y = −4

Add the two equations to eliminate $y$:

将两个方程相加以消去 $y$：

4x = 8 → x = 2

Substitute $x = 2$ into the first equation:

将 $x = 2$ 代入第一个方程：

3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3

Thus the solution is $x = 2$, $y = 3$. Always check by substituting into the second equation as well.

因此解为 $x = 2$，$y = 3$。务必同时代入第二个方程进行检验。

3. The Substitution Method | 代入法

When one equation has a variable with coefficient 1, substitution is often cleaner. Solve for that variable and substitute into the other equation.

当某个变量的系数为 1 时，代入法往往更简洁。先解出该变量，再代入另一个方程。

Example:

示例：

y = 2x + 1
3x + y = 11

Substitute $y = 2x + 1$ into the second equation:

将 $y = 2x + 1$ 代入第二个方程：

3x + (2x + 1) = 11 → 5x + 1 = 11 → 5x = 10 → x = 2

Then $y = 2(2) + 1 = 5$. The solution is $x = 2$, $y = 5$.

于是 $y = 2(2) + 1 = 5$。解为 $x = 2$，$y = 5$。

4. Choosing the Right Method | 如何选择合适的方法

Both methods work for any linear system, but choosing wisely saves time. Use elimination when coefficients are already aligned or easy to format. Use substitution when a variable is already isolated or has coefficient 1.

两种方法对任何线性方程组都适用，但合理选择可以节省时间。当系数已对齐或容易整理时，用消元法；当变量已被单独表示或系数为 1 时，用代入法。

You can also decide based on avoiding large fractions. Smaller numbers mean fewer mistakes.

你也可以根据是否会出现大分数来进行选择。数字越小，出错越少。

5. Solving Non-Linear Systems | 解非线性方程组

Edexcel IGCSE Higher Tier includes systems where one equation is linear and the other is quadratic. The strategy is substitution, leading to a quadratic in one variable.

Edexcel IGCSE 高阶考卷包含一个线性方程和一个二次方程组成的方程组。解题策略是代入法，最终得到一个关于某个变量的二次方程。

Example:

示例：

y = x + 2
x² + y² = 10

Substitute $y = x + 2$ into the circle equation:

将 $y = x + 2$ 代入圆的方程：

x² + (x + 2)² = 10 → x² + x² + 4x + 4 = 10 → 2x² + 4x − 6 = 0

Divide by 2 and factor:

两边除以 2 并因式分解：

x² + 2x − 3 = 0 → (x + 3)(x − 1) = 0

So $x = −3$ or $x = 1$. Corresponding $y$ values are $y = −1$ or $y = 3$. The solutions are (−3, −1) and (1, 3).

所以 $x = −3$ 或 $x = 1$。对应的 $y$ 值分别为 $y = −1$ 或 $y = 3$。解为 (−3, −1) 和 (1, 3)。

6. Word Problems | 应用题

Real-life problems often translate into simultaneous equations. Set variables for unknown quantities, form two equations from the given conditions, then solve.

实际问题通常可以转化为联立方程。设变量表示未知量，根据已知条件列出两个方程，再求解。

Example: Two pens and three pencils cost $2.40. One pen and two pencils cost $1.40. Find the price of each.

示例：两支笔和三支铅笔共需 2.40 美元；一支笔和两支铅笔共需 1.40 美元。求每支笔和每支铅笔的价格。

Let $p$ be pen price and $c$ be pencil price. Then:

设 $p$ 为笔的价格，$c$ 为铅笔的价格，则：

2p + 3c = 2.40
p + 2c = 1.40

Multiply the second equation by 2 and subtract:

第二个方程乘以 2 后相减：

(2p + 3c) − (2p + 4c) = 2.40 − 2.80 → −c = −0.40 → c = 0.40

Then $p + 2(0.40) = 1.40 → p = 0.60$. So a pen costs $0.60 and a pencil $0.40.

于是 $p + 2(0.40) = 1.40 → p = 0.60$。所以笔每支 0.60 美元，铅笔每支 0.40 美元。

7. Graphical Interpretation | 图形含义

Each linear equation represents a straight line on the coordinate plane. The point where the two lines intersect is the simultaneous solution. Parallel lines have no solution, and coincident lines have infinitely many solutions.

每个线性方程在坐标平面上代表一条直线。两条直线的交点就是联立方程的解。平行线没有解，重合线有无穷多解。

For example, the system:

例如，方程组：

y = x + 1
y = x − 3

has no solution because both lines have slope 1 and different intercepts. They never meet.

无解，因为两条直线斜率都是 1 且截距不同，永远不会相交。

For the system $y = 2x$ and $y = 2x$, every point is common. Hence infinitely many solutions.

对于方程组 $y = 2x$ 和 $y = 2x$，所有点都是公共的，因此有无穷多解。

8. Common Mistakes | 常见错误

Many errors come from sign mistakes, arithmetic slips, or forgetting to check the solution.

很多错误来自符号问题、算术失误，或者忘记检验解。

When subtracting equations, distribute the minus sign to every term.
减去一个方程时，要把负号分配到每一项。
Check your solution using both original equations.
用两个原方程分别检验你的解。
In substitution, remember to place brackets around the substituted expression.
在代入时，记得给被代入的表达式加上括号。
Write out each step clearly; a neat layout prevents careless mistakes.
每一步都要书写清楚；整齐的格式能避免粗心错误。

9. Exam Tips & Practice | 考试技巧与练习

In the Edexcel IGCSE exam, simultaneous equation questions are worth around 3–5 marks. Show full working to earn method marks even if your final answer is wrong.

在 Edexcel IGCSE 考试中，联立方程题通常占 3–5 分。即使最终答案错误，也要写出完整过程以获得方法分。

Here is a quick practice problem:

这里有一个快速练习题：

5x − 2y = 8
3x + y = 7

Try solving it before reading on. Use substitution: $y = 7 − 3x$, then $5x − 2(7 − 3x) = 8 → 5x − 14 + 6x = 8 → 11x = 22 → x = 2$. Then $y = 1$.

先尝试自行求解再往下读。用代入法：$y = 7 − 3x$，然后 $5x − 2(7 − 3x) = 8 → 5x − 14 + 6x = 8 → 11x = 22 → x = 2$，所以 $y = 1$。

Practice daily with different coefficients to build speed and confidence.

每天用不同系数进行练习，能提升速度与信心。

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Solving Simultaneous Equations | 解联立方程