Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are one of the most important topics in IGCSE Mathematics. They appear in Paper 2 and Paper 4, often as a standalone question or as part of a problem-solving context. Mastering this topic not only earns direct marks but also builds the algebraic foundation needed for coordinate geometry, calculus, and even physics.

联立方程是 IGCSE 数学中最重要的话题之一。它出现在 Paper 2 和 Paper 4 中，通常作为独立题目出现，或作为解题应用的一部分。掌握这一主题不仅能直接得分，还能为坐标几何、微积分甚至物理打下扎实的代数基础。

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

A single equation with two unknowns, such as x + y = 5, has infinitely many solutions. To find a unique solution, we need at least two independent equations that are true at the same time. These are called simultaneous equations.

一个含有两个未知数的方程，例如 x + y = 5，有无数多组解。要找到唯一解，我们至少需要两个同时成立且彼此独立的方程。这样的方程组称为联立方程组。

In IGCSE, you will meet two main types:

在 IGCSE 中，你会遇到两种主要类型：

Linear simultaneous equations – both equations are straight lines, e.g. 2x + y = 7 and x − y = 2.
Linear and quadratic systems – one line and one curve, e.g. y = 3x + 1 and y = x² − 2x + 4.

线性联立方程 – 两个方程均为直线，例如 2x + y = 7 和 x − y = 2。
线性与二次方程组 – 一条直线与一条曲线，例如 y = 3x + 1 和 y = x² − 2x + 4。

The solution of a system is the point (x, y) that satisfies every equation simultaneously.

方程组的解是同时满足每个方程的坐标点 (x, y)。

2. The Elimination Method | 消元法

Elimination means adding or subtracting the equations to cancel one variable. This is often the fastest method for linear systems, especially when coefficients are already aligned.

消元法是指通过将两个方程相加或相减来消去一个未知数。对于线性方程组，尤其是系数已经对齐时，这通常是最快的方法。

Consider this example:

请看以下例子：

2x + 3y = 12 (1)
x − 3y = −3 (2)

Notice that the y-coefficients are +3 and −3. Adding the equations eliminates y:

请注意 y 的系数分别为 +3 和 −3。将两式相加即可消去 y：

(2x + 3y) + (x − 3y) = 12 + (−3)
3x = 9
x = 3

Substitute x = 3 into equation (1):

把 x = 3 代入方程 (1)：

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

So the solution is (3, 2). Always check in the other equation: 3 − 3(2) = 3 − 6 = −3 ✔

因此解为 (3, 2)。务必代入另一个方程检验：3 − 3(2) = 3 − 6 = −3 ✔

If coefficients are not the same, multiply the equations first. For example, to eliminate x, multiply equation (1) by one number and equation (2) by another so that the x-coefficients match.

如果系数不相同，先对方程进行倍乘。例如，要消去 x，就把方程 (1) 和 (2) 分别乘以适当的数，使 x 的系数相等。

3. The Substitution Method | 代入法

Substitution is essential when one variable is already the subject, or when the system contains a quadratic. The idea is to replace one variable using an expression from the other equation.

当一个未知数已经是公式的对象，或方程组中含有二次方程时，代入法是必不可少的。其思想是用一个方程中的表达式去替换另一个方程中的未知数。

Example: solve

例子：解方程组

y = 2x − 1 (1)
3x + 2y = 12 (2)

Since y is already the subject in (1), substitute 2x − 1 for y in (2):

由于 (1) 中已经写出 y 的表达式，将 2x − 1 代入 (2) 中的 y：

3x + 2(2x − 1) = 12
3x + 4x − 2 = 12
7x = 14
x = 2

Now substitute x = 2 back into (1): y = 2(2) − 1 = 3. The solution is (2, 3).

再把 x = 2 代回 (1)：y = 2(2) − 1 = 3。解为 (2, 3)。

Key tip: after finding one variable, always substitute into the simpler equation to avoid unnecessary arithmetic.

关键提示：求出其中一个变量后，应代入较简单的方程中，以避免不必要的运算。

4. The Graphical Method | 图象法

Graphically, each equation represents a line (or curve). The solution is the point where they intersect. This method is slower and less precise, so it is usually reserved for checking answers or for questions that explicitly ask you to draw graphs.

在图象上，每个方程代表一条直线（或曲线）。方程组的解就是它们的交点。这种方法较慢且精度较低，因此通常只用于检验答案，或题目明确要求作图的情形。

To use this method:

使用该方法的步骤如下：

Rearrange each equation into the form y = mx + c.
Plot both lines on the same axes using a ruler, or use a graphic calculator if permitted.
Read the coordinates of the intersection point.

把每个方程改写成 y = mx + c 的形式。
用直尺在同一坐标系中画出两条直线（若允许，也可使用图形计算器）。
读出交点的坐标。

For the system y = x + 1 and y = −2x + 4, the graph shows intersection at (1, 2).

对于方程组 y = x + 1 和 y = −2x + 4，图象显示交点为 (1, 2)。

Check: 2 = 1 + 1 ✔ 2 = −2(1) + 4 ✔

检验：2 = 1 + 1 ✔ 2 = −2(1) + 4 ✔

Remember: a graph can only give approximate answers. Use algebra when exact values are required.

请记住：图象只能给出近似答案。当题目要求精确值时，应使用代数方法。

5. Linear + Quadratic Systems | 线性与二次方程组

IGCSE frequently tests systems where one equation is linear and the other is quadratic. The method is always substitution: replace y in the quadratic using the linear equation, then solve the resulting quadratic.

IGCSE 经常考查一个线性方程与一个二次方程组成的方程组。解题方法始终是代入法：用线性方程表示 y，代入二次方程，再求解得到的二次方程。

Example: solve

例子：解方程组

y = x + 1 (1)
y = x² − 2x + 3 (2)

Substitute (1) into (2):

将 (1) 代入 (2)：

x + 1 = x² − 2x + 3
0 = x² − 3x + 2
(x − 1)(x − 2) = 0

So x = 1 or x = 2. Find the matching y-values from the linear equation:

因此 x = 1 或 x = 2。再从线性方程中求出相应的 y 值：

When x = 1, y = 1 + 1 = 2 → solution (1, 2)
When x = 2, y = 2 + 1 = 3 → solution (2, 3)

当 x = 1 时，y = 1 + 1 = 2 → 解为 (1, 2)
当 x = 2 时，y = 2 + 1 = 3 → 解为 (2, 3)

A linear-quadratic system can give 0, 1, or 2 solutions. The discriminant Δ = b² − 4ac tells you: if Δ > 0 there are two solutions, if Δ = 0 there is one (the line is tangent), and if Δ < 0 there are none.

线性与二次方程组可能有 0 个、1 个或 2 个解。判别式 Δ = b² − 4ac 可以判断：若 Δ > 0 有两个解，若 Δ = 0 有一个解（直线与曲线相切），若 Δ < 0 则无解。

6. Setting Up Equations from Word Problems | 从应用题中列出方程

Exam questions often hide simultaneous equations inside a story. The skill is to translate words into algebra using two clear variables.

考试题常常把联立方程隐藏在文字情境中。关键技能是用两个明确的未知数将文字转化为代数式。

Example: At a cinema, 3 adult tickets and 2 child tickets cost £44. 1 adult ticket and 3 child tickets cost £31. Find the price of each ticket.

例子：在电影院，3 张成人票和 2 张儿童票共 44 英镑；1 张成人票和 3 张儿童票共 31 英镑。求每种票的价格。

Let a = price of an adult ticket and c = price of a child ticket. Then:

设 a 为成人票价，c 为儿童票价，则：

3a + 2c = 44 (1)
a + 3c = 31 (2)

Use elimination. Multiply (2) by 3:

使用消元法。将 (2) 乘以 3：

3a + 9c = 93 (3)

Subtract (1) from (3): 7c = 49, so c = 7. Substitute into (2): a + 21 = 31, so a = 10.

(3) − (1)：7c = 49，故 c = 7。代入 (2)：a + 21 = 31，得 a = 10。

So adult tickets cost £10 and child tickets cost £7. Always write a concluding sentence in your answer.

因此成人票 10 英镑，儿童票 7 英镑。作答时一定要写一句结论性陈述。

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

Not every system has a unique solution. Recognising these cases saves time and prevents wrong answers.

并非每个方程组都有唯一解。识别这些特殊情况可以节省时间，避免答错。

If the lines are parallel, they never meet, so there is no solution. Algebraically, you will get a contradiction such as 0 = 5.

如果两条直线平行，它们永不相交，因此无解。在代数上，你会得到类似 0 = 5 的矛盾式。

Example: x + y = 4 and 2x + 2y = 12. The second equation simplifies to x + y = 6, which contradicts x + y = 4. No solution.

例子：x + y = 4 与 2x + 2y = 12。第二个方程化简为 x + y = 6，与 x + y = 4 矛盾，因此无解。

If the two equations represent the same line, there are infinitely many solutions. Algebraically, you will get an identity like 0 = 0.

如果两个方程表示同一条直线，则有无穷多解。在代数上，你会得到 0 = 0 这样恒成立的式子。

Example: x + y = 4 and 2x + 2y = 8. The second is just double the first, so every point on the line is a solution.

例子：x + y = 4 与 2x + 2y = 8。第二个方程只是第一个的两倍，因此直线上的每个点都是解。

8. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

Many students lose marks on simultaneous equations due to small errors. Here are the most common pitfalls.

许多学生在联立方程上失分，都是因为一些小错误。以下是最常见的陷阱。

Sign errors in elimination: when subtracting equations, subtract every term. Write brackets: (2x + 3y) − (2x − y) = 12 − 5.
Forgetting to multiply every term: if you multiply an equation by 3, multiply both sides and every term.
Not checking the answer: always substitute into both original equations.
Writing coordinates in the wrong order: the solution is (x, y), not (y, x).

消元中的符号错误：做减法时，要减去每一项。写成括号形式：(2x + 3y) − (2x − y) = 12 − 5。
忘记每一项都乘：如果方程要乘以 3，左右两边及每一项都要乘。
没有检验答案：一定要代入两个原方程中进行验证。
坐标顺序写反：解应写成 (x, y)，而不是 (y, x)。

Also, read the question: if it asks for x and y, give both values; if it asks for x² + y², compute that expression after solving.

另外，务必审题：如果题目要求 x 和 y，就给出两个值；如果要求 x² + y²，则在解出后再计算该表达式的值。

9. Exam Tips and Markscheme Strategies | 考试技巧与评分标准策略

In the IGCSE exam, method marks are as important as final answers. Show every step clearly, because even a wrong final answer can earn many marks if the working is correct.

在 IGCSE 考试中，方法分与最终答案同样重要。清晰地写出每一步，因为即使最终答案有误，只要计算过程正确，仍可获得不少分数。

Label your equations (1) and (2). This makes your working easy to follow.
State which method you are using: “Elimination” or “Substitution”.
When solving quadratics, show factorisation or use the quadratic formula explicitly.
Give answers as exact fractions or in the form required (e.g. to 3 significant figures if the question says so).
Use a graphical check only as rough verification, never as a replacement for algebra in non-calculator papers.

给方程标号 (1)、(2)，使解题过程更清晰。
写清所用方法：”消元法”或”代入法”。
解二次方程时，写出因式分解过程，或明确使用二次公式。
答案要写成精确分数，或按题目要求的形式（例如要求保留 3 位有效数字时照做）。
图象法只作为粗略检验，在非计算器试卷上绝不能替代代数方法。

Finally, practise with past papers. Cambridge IGCSE questions often combine simultaneous equations with inequalities, geometry, or proportion. The more contexts you see, the more confident you will be on exam day.

最后，多做真题。剑桥 IGCSE 题目经常把联立方程与不等式、几何或比例结合起来。你见过的情境越多，考试时就会越自信。

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Solving Simultaneous Equations | 解联立方程组