Mastering Simultaneous Equations | 掌握联立方程组

📚 Mastering Simultaneous Equations | 掌握联立方程组

Simultaneous equations lie at the heart of algebra. Whenever a real-life problem involves two unknown quantities connected by two separate conditions, you need a pair of equations — and a strategy to solve them together. This topic is examined in both Foundation and Higher tiers of the Edexcel IGCSE Mathematics specification, so mastering it is essential for securing full marks in the algebra sections of your paper.

联立方程组是代数的核心内容。每当一个实际问题涉及两个未知量，并且由两个独立条件相联系时，你就需要一对方程，以及一种把它们放在一起求解的策略。该知识点在 Edexcel IGCSE 数学大纲的基础级和高级试卷中都会考查，掌握它对在代数部分拿到满分至关重要。

1. Understanding Simultaneous Equations | 理解联立方程组

A linear equation in two unknowns, such as 2x + y = 7, has infinitely many pairs of solutions. For example, (0, 7), (1, 5), (3, 1) and (5, −3) all satisfy this equation. However, when a second equation is added, the combined constraint usually narrows the possibilities down to exactly one ordered pair (x, y) that satisfies both equations simultaneously.

一个含两个未知数的线性方程，例如 2x + y = 7，有无数对解。例如，(0, 7)、(1, 5)、(3, 1) 和 (5, −3) 都满足该方程。然而，当加入第二个方程时，组合起来的约束通常会把可能性缩小到恰好一个有序对 (x, y)，使其同时满足两个方程。

Consider the pair:

2x + y = 7
x − y = 2

Testing the candidate (3, 1): in the first equation, 2(3) + 1 = 7 ✓; in the second, 3 − 1 = 2 ✓. Therefore the solution is x = 3, y = 1. This is the only pair that works for both equations, which is why these are called simultaneous equations — you solve them at the same time.

检验候选解 (3, 1)：在第一个方程中，2(3) + 1 = 7 ✓；在第二个方程中，3 − 1 = 2 ✓。因此解为 x = 3, y = 1。这是唯一一对同时满足两个方程的解，这也是为什么它们被称为”联立”方程组——你需要同时求解。

2. The Elimination Method | 消元法

Elimination is the most direct algebraic technique. The goal is to add or subtract the two equations so that one variable cancels out, leaving a single equation in the other variable. This method is particularly efficient when the coefficients of one variable are equal in magnitude.

消元法是最直接的代数技巧。目标是通过相加或相减两个方程，使其中一个变量相互抵消，剩下一个只含另一个变量的一元方程。当一个变量的系数大小相等时，这种方法尤其高效。

Worked example 1: Solve 2x + y = 7 and x − y = 2.

例 1：解 2x + y = 7 和 x − y = 2。

Step 1: Observe that the y terms are +y and −y. Adding the equations cancels y.

第一步：观察 y 项分别为 +y 和 −y。将两个方程相加即可消去 y。
Step 2: Add the left sides and the right sides separately: (2x + x) + (y − y) = 7 + 2, giving 3x = 9.

第二步：左边相加、右边相加：(2x + x) + (y − y) = 7 + 2，得到 3x = 9。
Step 3: Divide by 3 to get x = 3.

第三步：两边除以 3，得到 x = 3。
Step 4: Substitute x = 3 into either original equation. Using x − y = 2 gives 3 − y = 2, so y = 1.

第四步：将 x = 3 代入任意一个原方程。代入 x − y = 2 得 3 − y = 2，所以 y = 1。
Step 5: Check: 2(3) + 1 = 7 ✓ and 3 − 1 = 2 ✓. Solution: (3, 1).

第五步：检验：2(3) + 1 = 7 ✓，且 3 − 1 = 2 ✓。解为 (3, 1)。

Worked example 2: Solve 3x + 2y = 13 and x + 2y = 7.

例 2：解 3x + 2y = 13 和 x + 2y = 7。

Here the y coefficients are identical (+2y), so subtract the second equation from the first: (3x − x) + (2y − 2y) = 13 − 7, which gives 2x = 6, hence x = 3. Substituting x = 3 into x + 2y = 7 gives 3 + 2y = 7, so 2y = 4 and y = 2. The solution is (3, 2). Check: 3(3) + 2(2) = 13 ✓.

这里 y 的系数相同（都是 +2y），所以用第一个方程减去第二个方程：(3x − x) + (2y − 2y) = 13 − 7，得到 2x = 6，故 x = 3。将 x = 3 代入 x + 2y = 7，得 3 + 2y = 7，于是 2y = 4，y = 2。解为 (3, 2)。检验：3(3) + 2(2) = 13 ✓。

3. The Substitution Method | 代入法

Substitution is the method of choice when one variable is already isolated or has a coefficient of 1. You replace that variable with its algebraic expression in the other equation, turning the pair into a single linear equation.

当一个变量已被单独表示出来，或它的系数为 1 时，代入法是最佳选择。你把该变量替换成它在另一个方程中的代数表达式，从而把方程组转化为一个一元线性方程。

Worked example: Solve y = 2x − 1 and x + y = 8.

例：解 y = 2x − 1 和 x + y = 8。

Step 1: Label the equations: y = 2x − 1 (1), x + y = 8 (2).

第一步：给方程编号：y = 2x − 1 (1)，x + y = 8 (2)。
Step 2: Substitute (1) into (2): replace y with (2x − 1), giving x + (2x − 1) = 8.

第二步：将 (1) 代入 (2)：用 (2x − 1) 替换 y，得到 x + (2x − 1) = 8。
Step 3: Simplify: 3x − 1 = 8, so 3x = 9 and x = 3.

第三步：化简：3x − 1 = 8，所以 3x = 9，x = 3。
Step 4: Substitute x = 3 back into (1): y = 2(3) − 1 = 5.

第四步：将 x = 3 代回 (1)：y = 2(3) − 1 = 5。
Step 5: Check in (2): 3 + 5 = 8 ✓. Solution: (3, 5).

第五步：代入 (2) 检验：3 + 5 = 8 ✓。解为 (3, 5)。

A practical tip: when substituting, always wrap the expression in brackets first. For example, write x + (2x − 1) = 8, not x + 2x − 1 = 8 — this prevents sign errors when the expression contains negative terms.

实用技巧：代入时，先把表达式用括号括起来。例如，写成 x + (2x − 1) = 8，而不是 x + 2x − 1 = 8——这可以防止当表达式含有负项时出现符号错误。

4. The Graphical Method | 图像法

The graphical method gives a visual interpretation of simultaneous equations. Each equation is drawn as a straight line on the same set of axes, and the solution is the point of intersection of the two lines, where its x and y coordinates satisfy both equations.

图像法为联立方程组提供了直观的几何解释。每个方程在同一个坐标系上被画成一条直线，而解就是两条直线的交点，交点的 x 坐标和 y 坐标同时满足两个方程。

Worked example: Solve y = 2x + 1 and y = −x + 7 graphically.

例：用图像法解 y = 2x + 1 和 y = −x + 7。

First, create a coordinate table for each line by choosing convenient x values:

首先，为每条直线选取方便的 x 值，制作坐标表：

x	0	1	2
y = 2x + 1	1	3	5
y = −x + 7	7	6	5

Plot both sets of points and draw the two straight lines. The lines intersect at the point (2, 5), so the solution is x = 2, y = 5. You can confirm: 2(2) + 1 = 5 and −2 + 7 = 5.

描出两组点并画出两条直线。两条直线相交于点 (2, 5)，因此解为 x = 2，y = 5。你可以验证：2(2) + 1 = 5，且 −2 + 7 = 5。

For IGCSE exams, remember that graphical solutions are only as accurate as your drawing. If the answer is expected to be an integer, the intersection will usually land exactly on grid points; otherwise you must read the coordinates as carefully as possible.

在 IGCSE 考试中，请记住图像解法的精度取决于你的绘图。如果答案预计是整数，交点通常会精确落在网格点上；否则你必须尽可能仔细地读取坐标。

5. Choosing the Best Method | 选择最佳方法

All three methods — elimination, substitution and graphical — will produce the same solution, but some are quicker and safer in specific situations. Experienced students choose the method before they start writing.

三种方法——

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Mastering Simultaneous Equations | 掌握联立方程组