Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are a set of equations with multiple unknown variables, and solving them means finding values that satisfy all equations at the same time. In IGCSE Edexcel Mathematics, you will encounter both linear and non‑linear cases, along with word problems that require setting up your own equations.

联立方程是一组包含多个未知数的方程，解联立方程就是找到同时满足所有方程的值。在 Edexcel IGCSE 数学中，你会遇到线性和非线性的情况，以及需要自己建立方程的应用题。

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

When you have two or more equations that share the same variables, they form a system. For IGCSE, the most common system is two equations with two unknowns, usually \(x\) and \(y\). The solution is an ordered pair \((x, y)\) that makes both equations true.

当你拥有两个或更多含有相同变量的方程时，它们就形成了一个方程组。在 IGCSE 中，最常见的是两个方程、两个未知数，通常用 \(x\) 和 \(y\) 表示。解是一组有序数对 \((x, y)\)，使两个方程同时成立。

2. Graphical Interpretation | 图形解释

Each linear equation represents a straight line on a Cartesian plane. The point where the two lines intersect is the solution to the system. If the lines are parallel, there is no solution; if they coincide, there are infinitely many solutions.

每个线性方程在笛卡尔平面上代表一条直线。两条直线交点的坐标就是方程组的解。如果两条线平行，则无解；如果两条线重合，则有无数个解。

y = 2x + 1 and y = −x + 4 → intersection (1, 3)

In the example above, substituting \(x=1\) into either equation gives \(y=3\), so the solution is \((1,3)\).

在上面的例子中，将 \(x=1\) 代入任一方程可得 \(y=3\)，所以解是 \((1,3)\)。

3. The Substitution Method | 代入法

Substitution works by rearranging one equation to isolate a variable, then substituting that expression into the other equation. This method is especially useful when one coefficient is 1 or −1.

代入法的步骤是：先从一个方程中解出一个变量（用另一个变量表示），然后将该表达式代入另一个方程。当一个变量的系数为 1 或 −1 时特别方便。

Example: Solve \(x + y = 7\) and \(2x − y = 5\).

例：解方程组 \(x + y = 7\) 和 \(2x − y = 5\)。

From the first equation, \(y = 7 − x\). Substitute into the second: \(2x − (7 − x) = 5\) → \(3x − 7 = 5\) → \(3x = 12\) → \(x = 4\). Then \(y = 3\).

由第一个方程得 \(y = 7 − x\)。代入第二个方程：\(2x − (7 − x) = 5\)，即 \(3x − 7 = 5\)，所以 \(3x = 12\)，\(x = 4\)。于是 \(y = 3\)。

4. The Elimination Method | 消元法

Elimination involves adding or subtracting the equations to remove one variable. You may first need to multiply one or both equations by constants so that the coefficients of a variable are opposites or equal.

消元法通过相加或相减两个方程来消去一个变量。你可能需要先将一个或两个方程乘以适当的常数，使某个变量的系数相反或相等。

Example: Solve \(3x + 2y = 11\) and \(2x + 3y = 9\).

例：解 \(3x + 2y = 11\) 和 \(2x + 3y = 9\)。

Multiply the first equation by 3 and the second by 2: \(9x + 6y = 33\) and \(4x + 6y = 18\). Subtract: \(5x = 15\) → \(x = 3\). Then substitute to find \(y = 1\).

将第一个方程乘以 3，第二个方程乘以 2：\(9x + 6y = 33\)，\(4x + 6y = 18\)。两式相减得 \(5x = 15\)，所以 \(x = 3\)。再代入求得 \(y = 1\)。

5. Choosing the Right Method | 选择合适的方法

Both methods work for all linear systems, but some systems are easier with one method. If a variable already has coefficient 1, substitution is often simpler. If coefficients are small and can be aligned easily, elimination is quick.

两种方法对所有线性方程组都适用，但有些题目用其中一种更简单。如果某个变量系数已经是 1，代入法通常更直接；如果系数较小且容易对齐，消元法更快捷。

Situation	Recommended Method
A variable has coefficient 1	Substitution
Coefficients are multiples of each other	Elimination
Both equations are in the form \(y = mx + c\)	Equate or substitute

Always check your answer by substituting both values into the original equations.

务必将求得的数值代回原方程进行检验。

6. Solving Non‑Linear Systems | 解非线性方程组

In Edexcel IGCSE, you may be asked to solve a system where one equation is linear and the other is quadratic, e.g. \(y = x^2 + 2x − 1\) and \(y = x + 5\). The substitution method is the standard approach: replace \(y\) in the quadratic with the linear expression.

在 Edexcel IGCSE 中，你可能会遇到一个线性方程和一个二次方程组成的方程组，例如 \(y = x^2 + 2x − 1\) 和 \(y = x + 5\)。标准做法是代入法：用线性表达式代替二次方程中的 \(y\)。

x² + 2x − 1 = x + 5 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0

This gives \(x = −3\) or \(x = 2\). Substituting back into the linear equation gives two solutions: \((−3, 2)\) and \((2, 7)\).

由此得到 \(x = −3\) 或 \(x = 2\)。将它们代回线性方程，得到两组解：\((−3, 2)\) 和 \((2, 7)\)。

7. Word Problems: Setting Up Equations | 应用题：建立方程

Read the problem carefully, define variables for the unknown quantities, then translate the given conditions into two equations. Pay attention to units and relationships such as “sum”, “difference”, “product” or “ratio”.

仔细阅读题目，为未知量定义变量，然后将题目中的条件转化为两个方程。注意单位以及“和”、“差”、“积”、“比”等关系词。

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

例：两个数的和是 15，差是 3，求这两个数。

Let the numbers be \(a\) and \(b\). Then \(a + b = 15\) and \(a − b = 3\). Adding the equations: \(2a = 18\) → \(a = 9\), so \(b = 6\).

设两数为 \(a\) 和 \(b\)，则 \(a + b = 15\)，\(a − b = 3\)。两式相加得 \(2a = 18\)，所以 \(a = 9\)，进而 \(b = 6\)。

8. Problems Involving Money and Mixtures | 涉及金钱与混合物的题目

Many real‑world problems involve total cost or quantity. Define variables for each item, then form equations for the total amount and for the total value (or weight).

许多实际问题涉及总价或总量。为每种物品设变量，然后分别建立关于总数和总价值（或总重量）的方程。

Example: 50 tickets were sold, some at £2 and some at £5, raising £160. How many of each were sold?

例：共售出 50 张票，一部分每张 2 英镑，一部分每张 5 英镑，共收入 160 英镑。两种票各售出多少张？

Let \(x\) be the number of £2 tickets and \(y\) the number of £5 tickets. Then \(x + y = 50\) and \(2x + 5y = 160\). Solving gives \(x = 30\), \(y = 20\).

设 2 英镑票为 \(x\) 张，5 英镑票为 \(y\) 张。则 \(x + y = 50\)，\(2x + 5y = 160\)。解得 \(x = 30\)，\(y = 20\)。

9. Checking Solutions | 检验解

Always substitute your values back into both original equations. For linear systems, a quick mental check is often enough. For non‑linear systems, ensure that each pair satisfies both equations, because extraneous solutions can appear during substitution.

始终将求出的数值代回原方程组进行检验。对于线性方程组，通常心算即可；对于非线性方程组，必须确保每一组解都同时满足两个方程，因为在代入过程中可能出现增根。

If (p, q) is a solution, then both LHS₁ = RHS₁ and LHS₂ = RHS₂ must hold.

如果 \((p, q)\) 是解，那么两个方程的左、右两边都必须分别相等。

10. Common Mistakes to Avoid | 常见错误避坑

Forgetting to multiply every term when scaling an equation.
当对方程整体乘以一个数时，忘记每一项都要乘。
Mixing up signs when subtracting equations.
相减两个方程时搞错符号。
After substitution, solving only for one variable and not finding the other.
代入后只求出一个变量而忘了求另一个变量。
Not checking the solution in both equations.
没有把解代回两个方程中去检验。
For non‑linear systems, losing one of the two quadratic solutions.
解非线性方程组时，丢失二次方程的两个解中的一个。

11. Practice Questions | 练习题目

Try these typical exam‑style questions to build confidence.

尝试以下典型的考试风格题目，以增强信心。

Solve \(4x + 3y = 10\) and \(2x − y = 0\).
解方程组 \(4x + 3y = 10\) 和 \(2x − y = 0\)。
Solve \(y = x^2 − 4\) and \(y = 2x − 1\).
解方程组 \(y = x^2 − 4\) 和 \(y = 2x − 1\)。
The perimeter of a rectangle is 30 cm. The length is 3 cm more than twice the width. Find the dimensions.
一个长方形的周长是 30 cm。长比宽的 2 倍多 3 cm。求长和宽。

Answers: 1. \(x = 1\), \(y = 2\) 2. \((3, 5)\) and \((−1, −3)\) 3. width = 4 cm, length = 11 cm.

答案：1. \(x = 1\), \(y = 2\) 2. \((3, 5)\) 和 \((−1, −3)\) 3. 宽 4 cm，长 11 cm。

12. Summary | 总结

Simultaneous equations are a core skill in IGCSE Mathematics. Master the substitution and elimination methods for linear systems, learn to handle one linear + one quadratic system, and practise translating word problems into equations. Always show clear steps and check your answers.

联立方程是 IGCSE 数学的核心技能。掌握线性方程组的代入法与消元法，学会处理“一个线性 + 一个二次”的方程组，并练习将文字题转化为方程。解题时写出清晰的步骤，并主动检验答案。

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Solving Simultaneous Equations | 解联立方程