Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations appear in almost every IGCSE Mathematics paper. Whether you are studying at Core or Extended level, these equations test your ability to work systematically and connect algebra with geometry. In this tutorial, you will learn the three main strategies: graphing, substitution and elimination, and also how to handle quadratic cases and real-world word problems.

联立方程几乎出现在每一份 IGCSE 数学试卷中。无论你学习的是 Core 还是 Extended 级别，这些方程都在考查你系统化处理问题的能力，以及将代数与几何联系起来的能力。在本教程中，你将会学到三种主要策略：图像法、代入法和消元法，并且学会处理二次情形与真实应用题。

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

A simultaneous equation system is a set of equations that share unknown variables. For example, x + y = 10 and x − y = 2 form a system. The solution is the pair (x, y) that satisfies both equations.

联立方程组是一组共享未知数的方程。例如，x + y = 10 和 x − y = 2 构成一个方程组。解就是同时满足两个方程的 (x, y)。

For two linear equations, the solution represents the intersection point of two straight lines on the Cartesian plane.

对于两个线性方程，其解代表笛卡尔平面上两条直线的交点。

Equation 1	Equation 2	Solution
x + y = 10	x − y = 2	x = 6, y = 4

2. The Graphical Method | 图像法

To solve graphically, plot both lines on the same axes. The coordinates of the point where the lines cross give the solution.

用图像法解方程组时，在同一直角坐标系中画出两条直线。两线交点的坐标就是方程组的解。

Reorder each equation into the form y = mx + c.
Draw both lines using a ruler.
Read the intersection coordinates.

将每个方程改写为 y = mx + c 的形式。
用直尺画出两条直线。
读出交点的坐标。

Example: x + y = 6 and y = x + 2. Draw both lines and you will see that they cross at (2, 4).

示例：x + y = 6 和 y = x + 2。画出两条直线后，你会看到它们相交于点 (2, 4)。

x + y = 6, y = x + 2 → (2, 4)

The graphical method is useful for checking answers, but it can be imprecise; always confirm with an algebraic method in the exam.

图像法适合用于检验答案，但可能不精确；考试中请务必用代数方法加以确认。

3. The Substitution Method | 代入法

Substitution works best when one equation has a variable that is already isolated, such as y = 2x + 1, or when rearranging is simple.

当一个方程中的某个变量已经单独放在一边时（例如 y = 2x + 1），或者移项非常简单时，代入法最为有效。

Solve one equation for one variable.
Substitute that expression into the other equation.
Solve the resulting one-variable equation.
Substitute back to find the other variable.

先解出一个未知数。
将该表达式代入另一个方程。
解得到的一元方程。
代回求出另一个未知数。

Example: y = 2x + 1 and x + y = 10. Replace y in the second equation to get x + (2x + 1) = 10.

示例：y = 2x + 1 和 x + y = 10。在第二个方程中替换 y，得到 x + (2x + 1) = 10。

x + (2x + 1) = 10 → 3x + 1 = 10 → 3x = 9 → x = 3, y = 7

Use brackets carefully when substituting an expression that contains more than one term.

当代入的表达式含有多个项时，务必小心使用括号。

4. The Elimination Method | 消元法

Elimination involves aligning variables and adding or subtracting whole equations to cancel one variable.

消元法是将方程对齐，然后通过加减整个方程来消去一个变量。

Align the equations in the form ax + by = c.
Multiply one or both equations to make the coefficients of one variable equal.
Add or subtract the equations to eliminate that variable.
Solve for the remaining variable, then back-substitute.

将方程整理为 ax + by = c 的形式。
将其中一个或两个方程同时乘以适当的数，使某个变量的系数相同。
两式相加或相减，消去这个变量。
解出剩下的变量，再代回求另一个变量。

Example: 2x + 3y = 12 and 4x − 3y = 6. Here the y coefficients are +3 and −3, so adding the equations cancels y.

示例：2x + 3y = 12 和 4x − 3y = 6。这里 y 的系数分别是 +3 和 −3，因此两式相加即可消去 y。

(2x + 3y) + (4x − 3y) = 12 + 6 → 6x = 18 → x = 3, y = 2

When subtracting, remember to change the signs of every term in the expression being subtracted.

作减法时，要记得改变被减表达式中每一项的符号。

5. Fractions and Decimals | 分数与小数处理

When coefficients are fractions, multiply every term by the least common multiple (LCM) of the denominators to obtain integer coefficients.

当系数是分数时，将每一项都乘以分母的最小公倍数（LCM），从而得到整数系数。

Example: x/2 + y/3 = 4 and x/4 − y/2 = 1. Multiply the first equation by 6 and the second by 4.

示例：x/2 + y/3 = 4 和 x/4 − y/2 = 1。将第一个方程乘以 6，第二个方程乘以 4。

×6: 3x + 2y = 24, ×4: x − 2y = 4 → 4x = 28 → x = 7, y = 1.5

Always multiply every single term, including the constant on the right-hand side.

必须乘以每一项，包括等号右边的常数项。

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

Two lines may be parallel, meaning no intersection and therefore no solution.

两条直线可能平行，即没有交点，因此方程组无解。

Example: y = 2x + 3 and y = 2x − 1 have the same slope 2, so they are parallel. No solution.

示例：y = 2x + 3 和 y = 2x − 1 的斜率相同，都是 2，所以它们平行，无解。

If two equations represent the same line, every point on the line is a solution, giving infinitely many solutions.

如果两个方程表示同一条直线，那么直线上的每一个点都是解，因而有无数多个解。

Example: x + y = 5 and 2x + 2y = 10 represent the same line — the second equation is the first doubled.

示例：x + y = 5 和 2x + 2y = 10 表示同一条直线 — 第二个方程只是第一个方程两边乘以 2。

7. Simultaneous Equations with Quadratics | 含二次项的联立方程

Extended-level papers often include one linear and one quadratic equation, such as y = x + 2 and y = x².

Extended 级别的试卷经常包含一个线性方程和一个二次方程，例如 y = x + 2 和 y = x²。

Use substitution to obtain a quadratic equation, then solve by factorising or using the quadratic formula.

使用代入法得到一个二次方程，再通过因式分解或求根公式来求解。

Example: y = x + 2 and y = x². Since both equal y, set x² = x + 2.

示例：y = x + 2 和 y = x²。因为两者都等于 y，所以令 x² = x + 2。

x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = 2, y = 4 or x = −1, y = 1

Remember that the quadratic formula is x = (−b ± √(b² − 4ac)) / (2a).

记住求根公式为 x = (−b ± √(b² − 4ac)) / (2a)。

When arising quadratics do not factorise, always use the quadratic formula rather than guessing.

当出现的二次方程无法因式分解时，应使用求根公式，而不是胡乱猜测。

8. Word Problems | 应用题

Many IGCSE questions present simultaneous equations inside a real-world context. You must define variables, form equations, then solve.

许多 IGCSE 题目会以现实情境来呈现联立方程。你需要先定义变量、建立方程，然后再求解。

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

示例：两个数之和为 15，差为 3，求这两个数。

Let x and y be the two numbers. Then x + y = 15 and x − y = 3. Adding these equations gives 2x = 18.

设两个数为 x 和 y，则 x + y = 15，x − y = 3。两式相加得 2x = 18。

x + y = 15, x − y = 3 → 2x = 18 → x = 9, y = 6

Always write a final sentence stating the answer in the context of the question: ‘The two numbers are 9 and 6.’

最后一定要写出答案句，结合题目情境来表述结果：”这两个数分别是 9 和 6。”

9. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always check your answer by substituting into both original equations.
When multiplying an equation, multiply every term.
Be careful with signs when subtracting equations.
In word problems, clearly state what each variable represents.

始终将答案代回两个原方程进行检验。
方程两边每一项都要乘，不能漏项。
做减法消元时，注意符号变化。
在应用题中，清楚说明每个变量代表什么。

Common Mistake	Correction
Substituting y = 2x + 1 without brackets, writing x + 2x + 1	Write x + (2x + 1) = 10, then simplify to 3x + 1 = 10
Subtracting 4x − 3y from 2x + 3y without changing signs	Switch signs of the bottom equation first: 2x + 3y − 4x + 3y
Forgetting to multiply the constant term by the LCM	Multiply all three terms: ax, by and c

10. Practice Questions | 练习题目

Try these questions on your own, then check your answers below.

请先独立完成以下练习，然后再对照下面的答案。

Solve by elimination: 3x + y = 7 and x − y = 1.
Solve by substitution: y = 3x − 4 and 2x + y = 16.
Solve the quadratic system: y = x² and y = 2x + 3.

用消元法求解：3x + y = 7 和 x − y = 1。
用代入法求解：y = 3x − 4 和 2x + y = 16。
解含二次项的方程组：y = x² 和 y = 2x + 3。

Answers: 1. x = 2, y = 1 | 2. x = 4, y = 8 | 3. x = 3, y = 9 or x = −1, y = 1

For question 3, you should reach x² = 2x + 3, which gives x² − 2x − 3 = 0, then factorise to (x − 3)(x + 1) = 0.

第 3 题中，你应得到 x² = 2x + 3，即 x² − 2x − 3 = 0，然后分解为 (x − 3)(x + 1) = 0。

If you solved all three questions correctly, you are well prepared for simultaneous equations in your IGCSE exam.

如果你三题全部答对，那么你已经为 IGCSE 考试中的联立方程题做好了充分准备。

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Solving Simultaneous Equations | 解联立方程