Solving Simultaneous Equations | 解联立方程组

📚 Solving Simultaneous Equations | 解联立方程组

Simultaneous equations are a pair (or more) of equations that share the same unknown variables. Solving them means finding a set of values for the unknowns that satisfy every equation at the same time. This topic appears in almost every IGCSE Mathematics paper, and it is also a powerful tool for solving real-life problems involving two unknowns.

联立方程组是指一组（两个或更多）含有相同未知数的方程。解联立方程组，就是找到一组同时满足所有方程的未知数取值。这一内容几乎出现在每份 IGCSE 数学试卷中，也是解决生活中涉及两个未知量问题的有力工具。

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

A linear simultaneous equation system contains two linear equations with two unknown variables, usually called x and y. The solution is an ordered pair (x, y) that satisfies both equations at the same time. Graphically, each equation represents a straight line, and the solution is the intersection point of the two lines.

线性联立方程组通常指两个含有两个未知数（通常为 x 和 y）的线性方程。解是一个有序数对 (x, y)，它必须同时满足两个方程。在图像上，每个方程代表一条直线，而方程组的解就是两条直线的交点。

For example | 例如：

x + y = 10
x − y = 4

The solution is x = 7 and y = 3, because 7 + 3 = 10 and 7 − 3 = 4. When you check your answer, always substitute into both original equations.

这个方程组的解是 x = 7，y = 3，因为 7 + 3 = 10 且 7 − 3 = 4。验算时，务必把答案同时代入两个原方程进行检验。

There are three main methods for solving simultaneous equations: substitution, elimination, and graphical. You should be comfortable with all three, as different situations favour different methods.

解联立方程组主要有三种方法：代入法、消元法和图像法。你需要熟练掌握这三种方法，因为不同情境下各有优势。

2. The Substitution Method | 代入法

The substitution method works best when one equation already has a variable alone on one side, or when it is very easy to rearrange it that way. You replace one unknown in the other equation with its expression from the first equation, which reduces the system to a single equation in one unknown.

代入法最适合其中一个方程中某个未知数已经单独在一边的情况，或者很容易通过变形做到这一点的情况。把第一个方程中该未知数的表达式代入另一个方程，从而把方程组转化为一个只含一个未知数的方程。

Steps | 步骤如下：

Rearrange one equation to make either x or y the subject.

先将其中一个方程变形，把 x 或 y 单独放在等号一边。
Substitute this expression into the other equation.

将这个表达式代入另一个方程。
Solve the resulting equation in one unknown.

解这个只含一个未知数的新方程。
Substitute the value back into any original equation to find the other unknown.

把求出的值代回任一个原方程，求另一个未知数。

Worked example | 典例：Solve | 解：

y = 2x + 1
3x + y = 11

Substitute y = 2x + 1 into the second equation:

将 y = 2x + 1 代入第二个方程：

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

Then y = 2(2) + 1 = 5. The solution is x = 2, y = 5.

因此 y = 2(2) + 1 = 5。解为 x = 2，y = 5。

You can verify: 3(2) + 5 = 6 + 5 = 11, which matches the second equation.

可以验算：3(2) + 5 = 6 + 5 = 11，与第二个方程一致。

3. The Elimination Method | 消元法

The elimination method is often the fastest way to solve two linear equations. You align the equations and then add or subtract them so that one of the variables disappears, leaving a single equation in the other variable.

消元法通常是解两个线性方程最快的办法。把两个方程按相同变量对齐，然后通过相加或相减使其中一个变量消失，剩下一个只含另一个变量的方程。

Steps | 步骤如下：

Multiply one or both equations so that the coefficients of one variable become equal.

把一个或两个方程乘以适当的数，使其中一个变量的系数相同。
If the signs of those equal coefficients are the same, subtract the equations; if they are opposite, add them.

如果相同系数的符号相同，就用减法；如果符号相反，就用加法。
Solve for the remaining variable, then substitute back to find the other one.

解出剩下的变量，再代回求另一个变量。

Worked example | 典例：Solve | 解：

2x + 3y = 12
2x − y = 4

Both equations have the term 2x, so subtract the second equation from the first:

两个方程都含有 2x 项，因此用第一个方程减去第二个方程：

(2x + 3y) − (2x − y) = 12 − 4
4y = 8
y = 2

Now substitute y = 2 into 2x − y = 4:

再把 y = 2 代入 2x − y = 4：

2x − 2 = 4
2x = 6
x = 3

Check: 2(3) + 3(2) = 6 + 6 = 12, correct.

验算：2(3) + 3(2) = 6 + 6 = 12，结果正确。

If neither variable has the same coefficient, multiply one or both equations first. For example, to eliminate y from 3x + 2y = 13 and 2x − y = 4, double the second equation to get 4x − 2y = 8, then add the two equations together.

如果两个方程中没有相同系数的变量，就先对其中一个或两个方程作倍乘。例如，要从 3x + 2y = 13 和 2x − y = 4 中消去 y，可把第二个方程乘以 2，得到 4x − 2y = 8，然后将两个方程相加。

4. The Graphical Method | 图像法

Each linear equation can be drawn as a straight line on the same coordinate plane. The coordinates of the point where the two lines cross give the solution to the simultaneous equations.

每个线性方程都可以在同一坐标平面上画成一条直线。两条直线交点的坐标就是联立方程组的解。

To draw a line accurately, rearrange the equation into the form y = mx + c, then plot the y-intercept (0, c) and use the gradient m to find another point. Alternatively, make a table with two or three x-values.

要准确画直线，可以先把方程化成 y = mx + c 的形式，标出 y 轴截距 (0, c)，再根据斜率 m 找出另一个点；也可以列一个取两三组 x 值的表格。

Worked example | 典例：Solve graphically | 用图像法解：

y = x + 1
y = −2x + 4

Line 1 has gradient 1 and intercept 1. Line 2 has gradient −2 and intercept 4. Plot both lines on the same axes. They intersect at the point (1, 2), so the solution is x = 1, y = 2.

第一条直线斜率为 1、截距为 1；第二条直线斜率为 −2、截距为 4。在同一坐标轴上画出两条直线，它们的交点是 (1, 2)，所以解为 x = 1，y = 2。

The graphical method is good for estimating solutions and for checking answers, but if a question requires exact values, use an algebraic method instead.

图像法适合估算解，也适合检验答案；但如果题目要求精确值，则应改用代数方法。

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

In IGCSE extended mathematics, you may meet a system containing one linear equation and one quadratic equation. These systems usually have two pairs of solutions, because a straight line can intersect a curve at two points.

在 IGCSE 扩展数学中，你可能会遇到一个线性方程与一个二次方程组成的方程组。这类方程组通常有两组解，因为一条直线可能与一条曲线相交两次。

The method is always substitution: rearrange the linear equation to make x or y the subject, then substitute into the quadratic equation. This produces a quadratic equation that you can solve by factorisation or the quadratic formula.

解题方法始终是代入：先将线性方程变形，把 x 或 y 单独列出，再代入二次方程。这样会得到一个二次方程，可用因式分解或求根公式求解。

Worked example | 典例：Solve | 解：

y = 2x + 1
y = x² + x + 1

Since both expressions equal y, set them equal to each other:

因为两个等号右边都等于 y，令它们相等：

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

So x = 0 or x = 1. Substitute each value into y = 2x + 1:

所以 x = 0 或 x = 1。分别代回 y = 2x + 1：

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Solving Simultaneous Equations | 解联立方程组