Solving Simultaneous Equations | 联立方程的解法

📚 Solving Simultaneous Equations | 联立方程的解法

Simultaneous equations are a cornerstone of IGCSE Mathematics. They appear in almost every paper, often as a straightforward algebraic question, but also hidden inside problem-solving and graph interpretation tasks. Mastering them not only boosts your exam score but also builds the algebraic confidence needed for A-level Mathematics.

联立方程是 IGCSE 数学的核心考点之一，几乎每年考试都会出现。它有时以纯代数题直接考查，有时则隐藏在实际应用题或图像题中。掌握联立方程不仅能帮助你在考试中拿分，更能为将来 A-level 数学打下坚实的代数基础。

---

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

Simultaneous equations are a set of two or more equations that share the same unknown variables. To solve them, we must find values of the variables that satisfy all equations at the same time.

联立方程是指包含两个或两个以上相同未知数的方程组。解联立方程，就是要找到一组变量值，使方程组中每一个方程都同时成立。

For example:

3x + y = 14
2x − y = 1

Here, both equations contain the same x and y, so we need one pair (x, y) that works in both.

例如：

3x + y = 14
2x − y = 1

这里两个方程含有相同的 x 和 y，因此我们需要找到一组 (x, y)，同时能满足两个方程。

---

2. The Elimination Method | 消元法

The elimination method works by adding or subtracting the two equations to cancel out one variable. This leaves a single linear equation in one unknown, which we can solve directly.

消元法的核心思想是通过两个方程相加或相减，消去其中一个未知数，从而得到一个只含一个未知数的一元一次方程，然后直接求解。

Using the example above, add the two equations:

(3x + y) + (2x − y) = 14 + 1
5x = 15
x = 3

Now substitute x = 3 into the first equation:

3(3) + y = 14
9 + y = 14
y = 5

Thus the solution is x = 3, y = 5.

仍然用上面的例子，将两个方程相加：

(3x + y) + (2x − y) = 14 + 1
5x = 15
x = 3

再把 x = 3 代入第一个方程：

3(3) + y = 14
9 + y = 14
y = 5

所以解为 x = 3，y = 5。

Sometimes you need to multiply one or both equations by a constant before adding or subtracting, so that the coefficients of one variable become equal in magnitude.

有时需要先将一个或两个方程乘以常数，使某个未知数的系数绝对值相等，再进行加减消元。

Example: Solve 2x + 3y = 8 and 3x − 2y = −1.

Multiply the first equation by 2 and the second by 3 to match the y coefficients (6):

4x + 6y = 16
9x − 6y = −3

Adding gives 13x = 13, so x = 1. Substituting back, 2(1) + 3y = 8 → 3y = 6 → y = 2.

例如：解 2x + 3y = 8 和 3x − 2y = −1。

将第一个方程乘以 2，第二个方程乘以 3，使 y 的系数绝对值相同（均为 6）：

4x + 6y = 16
9x − 6y = −3

相加得 13x = 13，因此 x = 1。代回原方程：2(1) + 3y = 8 → 3y = 6 → y = 2。

---

3. The Substitution Method | 代入法

The substitution method is especially useful when one equation has a variable with coefficient 1, or when you are dealing with a linear equation combined with a quadratic equation.

代入法特别适合某个方程中某个未知数系数为 1 的情况，也常用于一个线性方程与一个二次方程联立的情形。

Step 1: Rearrange one equation to make one variable the subject.

Step 2: Substitute this expression into the other equation.

Step 3: Solve the resulting single-variable equation.

Step 4: Substitute back to find the other variable.

步骤一：将其中一个方程变形，用另一个变量表示该未知数。

步骤二：将此表达式代入另一个方程。

步骤三：解出这个一元方程。

步骤四：代回求出另一个未知数。

Example: Solve y = 2x + 1 and x + y = 10.

Since y is already the subject, substitute into the second equation:

x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3

Then y = 2(3) + 1 = 7. Solution: x = 3, y = 7.

例如：解 y = 2x + 1 和 x + y = 10。

因为 y 已经被表示为以 x 为主题的表达式，直接代入第二个方程：

x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3

于是 y = 2(3) + 1 = 7。解为 x = 3，y = 7。

---

4. Choosing Between Elimination and Substitution | 如何选择消元法或代入法

Both methods always work for linear systems. However, choosing the right one can save you time and reduce errors.

这两种方法对于线性方程组都适用。但选择合适的方法可以节省时间，减少错误。

• Use elimination when both equations are in the form ax + by = c and the coefficients line up neatly.
• Use substitution when one variable is already isolated or has a coefficient of 1.
• For mixed linear-quadratic systems, substitution is usually the only algebraic option.

• 当两个方程均为 ax + by = c 的标准形式，且系数对齐较好时，优先用消元法。
• 当某个未知数已经被单独表示，或系数为 1 时，用代入法更直接。
• 对于线性方程与二次方程混合的方程组，代入法通常是唯一的代数解法。

---

5. Solving Linear and Quadratic Systems | 线性与二次联立方程组

At IGCSE level, you must also solve a system where one equation is linear and the other is quadratic. For example:

y = x² − 3x + 2
y = 2x − 1

Since both expressions equal y, set them equal to each other:

x² − 3x + 2 = 2x − 1
x² − 5x + 3 = 0

At IGCSE level, you might use the quadratic formula:

x = (−b ± √(b² − 4ac)) / 2a

Substituting a = 1, b = −5, c = 3 gives:

x = (5 ± √13) / 2

Then substitute each x value back into the linear equation to find the corresponding y values.

在 IGCSE 阶段，你还需要掌握一个线性方程与一个二次方程联立的解法。例如：

y = x² − 3x + 2
y = 2x − 1

由于两个式子都等于 y，将它们相等：

x² − 3x + 2 = 2x − 1
x² − 5x + 3 = 0

在 IGCSE 阶段，可使用求根公式：

x = (−b ± √(b² − 4ac)) / 2a

代入 a = 1，b = −5，c = 3：

x = (5 ± √13) / 2

然后将每个 x 值代回线性方程，求出对应的 y 值。

---

6. The Graphical Method | 图像法

You should also understand how to solve simultaneous equations graphically. Plot both lines (or curves) on the same axes; the solution is the point of intersection.

你还需要理解如何通过图像法解联立方程。在同一坐标系中画出两条线（或曲线），它们的交点即为方程组的解。

For two straight lines, there are three possibilities:

• One unique solution: the lines intersect at exactly one point.
• No solution: the lines are parallel with different intercepts.
• Infinitely many solutions: the lines coincide.

对于两条直线，有三种可能：

• 唯一解：两条线相交于一个点。
• 无解：两条线平行且截距不同。
• 无穷多解：两条线完全重合。

In the exam, graphical methods are most often tested by giving the graph and asking you to read off the intersection point. Always check whether your reading is consistent with an algebraic result.

在考试中，图像法通常以“给出图像、读出交点坐标”的形式出现。务必核对读出的坐标是否与代数运算结果一致。

---

7. Forming Equations from Word Problems | 从实际问题建立方程组

Word problems are a classic exam style. The key is to define your variables clearly and translate the information into two equations.

应用题是经典考题类型。关键是明确设未知数，并把题目信息翻译成两个方程。

Example: Two pens and three pencils cost $7. Four pens and one pencil cost $9. Find the cost of one pen and one pencil.

Let p = cost of a pen, c = cost of a pencil.

2p + 3c = 7
4p + c = 9

Using elimination, multiply the first equation by 2:

4p + 6c = 14
4p + c = 9

Now subtract: 5c = 5, so c = 1. Then 4p + 1 = 9 → p = 2.

So a pen costs $2 and a pencil costs $1.

例如：两支笔和三个笔记本共 7 元。四支笔和一个笔记本共 9 元。求一支笔和一个笔记本各多少钱。

设 p = 笔的单价，c = 笔记本的单价。

2p + 3c = 7
4p + c = 9

用消元法，将第一个方程乘以 2：

4p + 6c = 14
4p + c = 9

两式相减：5c = 5，于是 c = 1。再代入得 4p + 1 = 9 → p = 2。

因此一支笔 2 元，一个笔记本 1 元。

When solving word problems, always state what your variables represent and include units in your final answer. Check that your solution makes sense in the context of the question.

解应用题时，务必说明未知数的含义，并在最终答案中写明单位。同时检查答案是否符合题意。

---

8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Below are the most frequent errors students make when solving simultaneous equations.

以下是学生在解联立方程时最常犯的错误。

Mistake | 错误	Why It Happens | 出错原因	Fix | 解决办法
Sign errors when subtracting	Forgetting to subtract every term	Write out the full brackets and remove them carefully
Substituting back into the wrong equation	Not checking both equations	Always substitute into the simpler equation and verify in the other
Forgetting to multiply all terms when scaling	Rushing through multiplication	Tick each term as you multiply
Losing solutions in quadratic cases	Only recognising one value of x	Remember a quadratic has up to two roots; find both

---

9. Exam Tips and Final Strategy | 考试技巧与最终策略

Finally, here are some practical tips to help you perform well on simultaneous equations questions in the IGCSE exam.

最后，这里有一些实用技巧，帮助你在 IGCSE 考试中做好联立方程题目。

• Always show your working: the examiner awards method marks even if your final answer is wrong.
• Check your answer by substituting both values into both original equations.
• If the coefficients are large, simplify by dividing any common factor first.
• In word problems, define variables clearly before writing equations.
• When an answer looks odd (like a fraction that does not fit the context), go back and re-check.

• 一定要写出解题过程：即使最终答案错误，考官也会给方法分。
• 把两个解代回两个原方程中验证。
• 如果系数较大，先约去公因数再计算。
• 在应用题中，先明确设未知数，再列方程。
• 若答案看起来奇怪（如与实际不符的分数），请回头检查。

Simultaneous equations reward systematic thinking. With the right method, careful arithmetic, and a quick verification step, you can secure full marks on this topic every time.

联立方程考查的是系统性的思维。只要方法得当、计算仔细、并养成验证的习惯，你完全可以在每次考试中稳定拿到全分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）