Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a core topic in IGCSE Mathematics and are tested in both Paper 2 and Paper 4. They appear in many forms: linear–linear systems, linear–quadratic systems, word problems, and coordinate geometry questions. A solid grasp of the solving methods is essential for reaching the highest grades.

联立方程是 IGCSE 数学的核心主题，在 Paper 2 和 Paper 4 中都会考到。它们以多种形式出现：线性–线性方程组、线性–二次方程组、应用题以及坐标几何问题。扎实掌握求解方法对于冲击高分至关重要。

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

A simultaneous equation system consists of two or more equations that share the same variables. The solution to the system is the set of values that satisfies every equation at the same time. For example, if two variables x and y appear in two equations, we need to find values of x and y that make both equations true simultaneously.

联立方程组由两个或两个以上含有相同变量的方程组成。该方程组的解是同时满足每一个方程的一组数值。例如，如果一个变量 x 和一个变量 y 出现在两个方程中，我们需要找到使两个方程同时成立的 x 和 y 的值。

Consider this pair:

考虑下面这一对：

x + y = 10
x − y = 2

Here, x = 6 and y = 4 is the only pair of values that satisfies both equations. In IGCSE examinations, you will need to solve such systems using algebraic or graphical methods.

在这里，x = 6 和 y = 4 是唯一同时满足这两个方程的一组值。在 IGCSE 考试中，你需要用代数方法或图象法来解此类方程组。

2. The Elimination Method | 消元法

Elimination is the most powerful algebraic method for solving simultaneous linear equations. The idea is to add or subtract the equations so that one variable disappears, leaving a single equation in the other variable.

消元法是解联立线性方程最强有力的代数方法。其思路是将两个方程相加或相减，使其中一个变量消失，从而得到只含另一个变量的一元方程。

Steps:

步骤：

Arrange both equations with like terms in the same order.
Make the coefficients of one variable the same (multiply if necessary).
Add or subtract the equations to eliminate that variable.
Solve the resulting one-variable equation.
Substitute back into one original equation to find the other variable.

将两个方程按同类项对齐排列。
将某一个变量的系数化为相同（必要时乘一个倍数）。
将两方程相加或相减，消去该变量。
解得到的一元方程。
代回原方程之一，求出另一个变量。

Example: Solve 3x + 2y = 12 and 5x − 2y = 4.

例：解 3x + 2y = 12 与 5x − 2y = 4。

3x + 2y = 12
5x − 2y = 4

Since the coefficients of y are +2 and −2, adding the equations eliminates y:

由于 y 的系数分别为 +2 和 −2，将两个方程相加即可消去 y：

8x = 16 → x = 2

Substitute x = 2 into 3x + 2y = 12:

将 x = 2 代入 3x + 2y = 12：

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

So the solution is x = 2, y = 3. Always check both equations: 5(2) − 2(3) = 10 − 6 = 4. ✓

所以解为 x = 2，y = 3。务必代入两个方程检验：5(2) − 2(3) = 10 − 6 = 4。✓

3. The Substitution Method | 代入法

Substitution is especially useful when one equation is already written in the form y = … or x = …, or when one variable has coefficient 1. We replace that variable in the other equation with its equivalent expression.

代入法特别适用于某个方程已经写成 y = … 或 x = … 的形式，或者某个变量的系数为 1 的情形。我们把该变量在另一个方程中替换成与之等价的表达式。

Steps:

步骤：

Make one variable the subject of one equation.
Substitute this expression into the other equation.
Solve the resulting equation in one variable.
Substitute back to find the second variable.

把一个方程中的一个变量用另一个变量表示。
把这个表达式代入另一个方程。
解所得的一元方程。
代回求出第二个变量。

Example: Solve y = 2x + 1 and 3x + y = 11.

例：解 y = 2x + 1 与 3x + y = 11。

Substitute y = 2x + 1 into 3x + y = 11:

将 y = 2x + 1 代入 3x + y = 11：

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

Then y = 2(2) + 1 = 5. The solution is x = 2, y = 5. This pair also satisfies the second equation: 3(2) + 5 = 11. ✓

于是 y = 2(2) + 1 = 5，解为 x = 2，y = 5。这组值同样满足第二个方程：3(2) + 5 = 11。✓

4. Graphical Method | 图象法

The graphical method involves plotting both equations on the same axes. Each linear equation produces a straight line. The point where the two lines intersect gives the solution, because it lies on both lines and therefore satisfies both equations.

图象法是指在同一个坐标系中画出两个方程的图象。每个线性方程对应一条直线。两条直线的交点就是方程组的解，因为该点同时在两条直线上，因此同时满足两个方程。

For the system y = 2x + 1 and y = −x + 7, the lines cross at (2, 5). You can verify this: 2(2) + 1 = 5 and −2 + 7 = 5.

对于方程组 y = 2x + 1 与 y = −x + 7，两条直线相交于 (2, 5)。你可以验证：2(2) + 1 = 5，且 −2 + 7 = 5。

The graphical method is useful for estimating solutions or checking your answers, but in an exam it is less accurate than algebra. If the question asks for exact values, always use elimination or substitution.

图象法适合估计解或检验答案，但在考试中精度不如代数方法。如果题目要求精确值，应使用消元法或代入法。

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

Not every pair of linear equations has exactly one solution. There are two special cases you must recognise.

并非所有线性方程组都恰有一个解。有两种特殊情况你必须能够识别。

If the two lines are parallel, they never intersect, so there is no solution. If the two lines are the same line, they intersect at every point, so there are infinitely many solutions.

如果两条直线平行，它们永不相交，因此无解。如果两条直线重合，它们在每一点都相交，因此有无穷多解。

Case 情形	Lines 直线	Number of Solutions 解的数量
Different gradients 斜率不同	Intersecting 相交	Exactly one 恰有一个
Same gradient, different intercept 斜率相同、截距不同	Parallel 平行	None 无解
Same gradient and intercept 斜率与截距都相同	Coincident 重合	Infinitely many 无穷多

For example, 2x + 3y = 6 and 4x + 6y = 12 are the same line, because the second equation is exactly twice the first. In contrast, 2x + 3y = 6 and 4x + 6y = 14 have no solution, since the left-hand sides are proportional but the constants are not.

例如，2x + 3y = 6 与 4x + 6y = 12 是同一条直线，因为第二个方程正好是第一个的两倍。相反，2x + 3y = 6 与 4x + 6y = 14 无解，因为左边成比例而常数项不成比例。

6. Solving a Linear and a Quadratic Equation Together | 解线性–二次方程组

At Extended level, you may be asked to solve one linear equation together with one quadratic equation. The method is always substitution: use the linear equation to express one variable, then substitute into the quadratic equation.

在 Extended 层面，你可能需要联立一个线性方程和一个二次方程求解。方法始终是代入法：先用线性方程表示一个变量，再代入二次方程。

Example: Solve x + y = 7 and x² + y² = 25.

例：解 x + y = 7 与 x² + y² = 25。

From the linear equation, y = 7 − x. Substitute into x² + y² = 25:

由线性方程得 y = 7 − x。代入 x² + y² = 25：

x² + (7 − x)² = 25

x² + 49 − 14x + x² = 25

2x² − 14x + 24 = 0

Divide both sides by 2 and factorise:

两边同时除以 2 并因式分解：

x² − 7x + 12 = 0 → (x − 3)(x − 4) = 0

So x = 3 or x = 4. Substitute back to find y: when x = 3, y = 4; when x = 4, y = 3. The solutions are (3, 4) and (4, 3). Notice that a linear–quadratic system can give two solutions.

所以 x = 3 或 x = 4。代回求 y：当 x = 3 时 y = 4；当 x = 4 时 y = 3。解为 (3, 4) 和 (4, 3)。注意线性–二次方程组可能给出两个解。

7. Forming Equations from Word Problems | 从应用题列方程

Many IGCSE questions present a real-world context and ask you to form and solve the equations yourself. The key skills are choosing sensible variables and translating sentences into mathematical statements.

许多 IGCSE 题目会给出实际情境，要求你自己列方程并求解。关键技能是选择合适的变量，并把文字描述转化为数学表达式。

Example: The sum of two numbers is 10 and their difference is 2. Find the numbers.

例：两个数的和为 10，差为 2。求这两个数。

Let the two numbers be x and y. Then:

设这两个数为 x 和 y，则：

x + y = 10
x − y = 2

Add the equations: 2x = 12 → x = 6. Substitute back: 6 + y = 10 → y = 4. The numbers are 6 and 4.

两式相加：2x = 12 → x = 6。代回：6 + y = 10 → y = 4。这两个数是 6 和 4。

When forming equations, define your variables clearly at the start, and always check your final answers against the wording of the question.

列方程时，一开始就要清楚定义变量，并且始终把最终答案放回题干中检验。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

Here are the mistakes that most frequently cost students marks in IGCSE simultaneous equation questions.

下面列出在 IGCSE 联立方程题中最常让学生失分的错误。

Forgetting to multiply every term when scaling an equation. If you multiply one side by 3, you must multiply every term on that side by 3.
Making subtraction errors when eliminating a negative term. Always write down the new equation after adding or subtracting.
Substituting the found value into the wrong equation or the wrong variable.
Not checking the answer in both original equations.
When solving a quadratic by substitution, forgetting to find both x-values and both y-values.

对方程进行放大时，忘记将每一项都乘以倍数。如果一边乘以 3，那么该边每一项都必须乘以 3。
消去负项时出现减法错误。加减之后一定要把新方程写出来。
把求出的值代入错误的方程或错误的变量。
没有把答案放回两个原方程中检验。
用代入法求解二次方程时，忘记求两个 x 值和对应的两个 y 值。

Always write “let x = …” and state your final answer as a clear coordinate pair or sentence. Examiners reward working marks, so do not skip steps.

务必写明“设 x = …”，并把最终答案用清晰的坐标对或完整句子写出。阅卷官按步骤给分，因此不要跳步。

9. Practice Questions | 练习

Try these examples on your own, then check your answers.

请先独立完成下列题目，再核对答案。

Q1. Solve 4x + y = 9 and 2x + 3y = 7.

Q1. 解 4x + y = 9 与 2x + 3y = 7。

Q2. Solve 2x − 3y = 5 and x + 2y = 8.

Q2. 解 2x − 3y = 5 与 x + 2y = 8。

Q3. Solve y = x² and y = 2x + 3.

Q3. 解 y = x² 与 y = 2x + 3。

Answers: Q1: x = 2, y = 1. Q2: x = 4, y = 1. Q3: x = −1, y = 1 and x = 3, y = 9.

答案：Q1：x = 2，y = 1。Q2：x = 4，y = 1。Q3：x = −1，y = 1 与 x = 3，y = 9。

10. Summary | 总结

To solve simultaneous equations successfully, choose the right method: elimination works well when coefficients are already equal or opposites; substitution works well when one equation is already solved for a variable or when the system contains one quadratic equation. The graphical method helps you visualise the solution but is rarely exact. Always check your answers, and remember the special cases of parallel and coincident lines.

要成功解出联立方程，必须选择正确的方法：当系数已经相同或互为相反数时，消元法很有效；当某个方程已经化成某个变量的表达式，或方程组中含有一个二次方程时，代入法很合适；图象法帮助你直观理解解，但很少能得到精确值。始终检验答案，并牢记平行线与重合线这两种特殊情况。

Mastering this topic will give you a strong advantage across the algebra, geometry and problem-solving sections of the IGCSE Mathematics papers.

熟练掌握这一主题，将使你在 IGCSE 数学试卷的代数、几何和问题解决部分占据明显优势。

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Solving Simultaneous Equations | 解联立方程组