Simultaneous Equations | 联立方程

📚 Simultaneous Equations | 联立方程

Simultaneous equations are a fundamental concept in IGCSE Mathematics, testing your ability to solve problems involving two or more unknown variables. This revision guide will walk you through the key methods — substitution, elimination, and graphical approaches — along with common pitfalls and practice strategies to ensure you achieve top marks.

联立方程是 IGCSE 数学中的基础概念，考验你求解两个或更多未知变量问题的能力。本复习指南将带你掌握关键方法——代入法、消元法和图像法——同时涵盖常见陷阱与练习策略，助你冲击高分。

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

Simultaneous equations are a set of equations containing two or more unknown quantities. The solution is the set of values that satisfy all equations at the same time. For example, if we have two equations in x and y, we need to find values of x and y that make both equations true simultaneously.

联立方程是一组包含两个或更多未知量的方程。解是同时满足所有方程的未知量值。例如，如果有两个关于 x 和 y 的方程，我们需要找到同时使两个方程成立的 x 和 y 的值。

In IGCSE, you will encounter up to two equations with two unknowns, usually linear (x and y terms only) or a combination of one linear and one quadratic equation. Understanding how to solve these is essential for many exam questions.

在 IGCSE 中，你会遇到最多含两个未知量的两个方程，通常是线性方程（只含 x 和 y 项），或一个线性一个二次方程的组合。掌握其解法对于回答许多考试题目至关重要。

2. The Substitution Method | 代入法

The substitution method involves rearranging one equation to make one variable the subject, then substituting this expression into the other equation. This reduces the system to a single equation in one unknown, which can be solved directly.

代入法通过重新排列一个方程，使一个变量成为主项，然后将该表达式代入另一个方程。这会将方程组简化为只含一个未知量的单一方程，从而直接求解。

Let’s illustrate with an example:

让我们通过示例说明：

y = 2x + 1 (1)
x + y = 7 (2)

Since equation (1) already gives y in terms of x, we substitute it into equation (2):

由于方程 (1) 已经给出了 y 关于 x 的表达式，我们将其代入方程 (2)：

x + (2x + 1) = 7
3x + 1 = 7
3x = 6
x = 2

Then we substitute x = 2 back into equation (1) to find y:

然后我们将 x = 2 代回方程 (1) 以求出 y：

y = 2(2) + 1 = 5

So the solution is x = 2, y = 5.

因此解为 x = 2，y = 5。

This method is particularly useful when one equation is easy to rearrange, such as when one variable has a coefficient of 1.

当某个方程容易重新排列时，例如其中一个变量的系数为 1，此方法尤为实用。

3. The Elimination Method | 消元法

The elimination method involves adding or subtracting the equations to eliminate one variable. First, you may need to multiply one or both equations by constants so that the coefficients of one variable are equal in magnitude (but opposite in sign for subtraction).

消元法通过将方程相加或相减来消除一个变量。首先，你可能需要将一个或两个方程乘以常数，使某个变量的系数大小相等（若做减法则需要符号相反）。

Let’s use a different example:

我们使用另一个示例：

2x – 3y = 4 (1)
4x + 3y = 8 (2)

Here, the y terms have coefficients -3 and +3, which are opposite. Adding the two equations eliminates y:

这里，y 项的系数分别为 -3 和 +3，是相反数。将两个方程相加可以消去 y：

(2x + 4x) + (-3y + 3y) = 4 + 8
6x = 12
x = 2

Now substitute x = 2 into equation (1):

现在将 x = 2 代入方程 (1)：

2(2) – 3y = 4
4 – 3y = 4
-3y = 0
y = 0

Thus, the solution is x = 2, y = 0.

因此，解为 x = 2，y = 0。

If the coefficients are not aligned, for instance, if we had 2x + 3y = 5 and 3x + 2y = 6, we would multiply the equations appropriately to make the coefficients of x or y the same.

如果系数并不对齐，例如 2x + 3y = 5 和 3x + 2y = 6，我们需要适当乘以方程，使 x 或 y 的系数相同。

4. The Graphical Method | 图像法

Graphical methods involve plotting both equations on the same set of axes. The point where the two lines intersect represents the solution to the simultaneous equations.

图像法是在同一坐标系中绘制两个方程的图像。两条直线相交的点即为联立方程的解。

For example, to solve y = x + 1 and y = -x + 3, we plot both lines. The intersection point is (1, 2), meaning x = 1, y = 2.

例如，求解 y = x + 1 和 y = -x + 3，我们绘制两条直线。交点坐标为 (1, 2)，即 x = 1，y = 2。

Graphical methods are useful for estimating solutions, but for exact answers you should use algebraic methods. However, the graph can confirm your solution or reveal special cases:

图像法有助于估算解，但若需精确答案则应当使用代数方法。但是，图像可以验证你的解，并发掘特殊情况：

If the lines are parallel, there is no solution (inconsistent equations).

如果两条直线平行，则无解（方程组矛盾）。
If the lines coincide (are exactly the same), there are infinitely many solutions.

如果两条直线重合（完全相同），则有无数个解。

Make sure to draw your axes carefully and label each line.

绘制坐标轴时务必仔细，并标出每条直线的名称。

5. Solving Systems with a Quadratic Equation | 含二次方程的方程组的求解

When one equation is linear and the other is quadratic (e.g., y = x² + 2x + 1 and y = x + 3), you cannot solve by simple elimination of x or y. Instead, use substitution: replace the y in the quadratic equation with the linear expression.

当一个方程为一次方程，另一个为二次方程（例如 y = x² + 2x + 1 和 y = x + 3）时，不能通过简单消元来求解。这时应使用代入法：将线性方程中的 y 表达式代入二次方程。

Consider:

考虑以下方程组：

y = x + 3 (1)
y = x² + 2x + 1 (2)

Substitute y = x + 3 into equation (2):

将 y = x + 3 代入方程 (2)：

x + 3 = x² + 2x + 1
0 = x² + x – 2
0 = (x + 2)(x – 1)

So x = -2 or x = 1. Then find y for each x using the linear equation:

因此 x = -2 或 x = 1。然后使用线性方程求出对应的 y：

If x = -2, then y = -2 + 3 = 1.

若 x = -2，则 y = -2 + 3 = 1。
If x = 1, then y = 1 + 3 = 4.

若 x = 1，则 y = 1 + 3 = 4。

There are two solution pairs: (-2, 1) and (1, 4).

有两组解：(-2, 1) 和 (1, 4)。

Note that a linear-quadratic system can have 0, 1, or 2 solutions. A graph can show whether the line intersects the curve, touches it (one solution), or never touches it (no solution).

注意，线性-二次方程组可能有 0 个、1 个或 2 个解。图像可以显示直线与曲线相交、相切（1 个解）或永不相交（无解）。

6. Setting Up Equations from Word Problems | 从应用题中建立方程

Many exam questions present real-world scenarios. The key skill is translating words into algebraic equations. Look for relationships between variables.

许多考试题目给出实际情境。关键技能是将文字转换为代数方程。寻找变量之间的关系。

Example: The sum of two numbers is 12. The difference between twice the first number and the second number is 3. Find the numbers.

示例：两个数之和为 12。第一个数的两倍与第二个数的差为 3。求这两个数。

Let the numbers be a and b. We have:

设这两个数为 a 和 b，则有：

a + b = 12 (1)
2a – b = 3 (2)

Add the two equations to eliminate b:

将两个方程相加以消去 b：

(a + 2a) + (b – b) = 12 + 3
3a = 15
a = 5

Now substitute a = 5 into equation (1):

将 a = 5 代入方程 (1)：

5 + b = 12
b = 7

The numbers are 5 and 7.

这两个数分别是 5 和 7。

Always define your variables clearly at the start, and check your final answers against the original problem.

始终在开始时清楚地定义未知量，并检查最终答案是否符合原题条件。

7. Special Cases: No Solution or Infinite Solutions | 特殊情况：无解或无穷多解

Not all simultaneous equations have a unique solution. Two special cases arise when one equation is a multiple of the other.

并非所有联立方程都有唯一解。当一个方程是另一个方程的倍数时，会出现两种特殊情况。

Consider the system:

考虑以下方程组：

x + y = 5 (1)
2x + 2y = 7 (2)

If we multiply equation (1) by 2, we get 2x + 2y = 10. This contradicts equation (2), which says 2x + 2y = 7. Therefore, there is no solution — the lines are parallel.

若将方程 (1) 乘以 2，得到 2x + 2y = 10。这与方程 (2) 的 2x + 2y = 7 矛盾。因此无解——两直线平行。

Now consider:

再考虑：

x + y = 5 (1)
2x + 2y = 10 (2)

Here, equation (2) is exactly twice equation (1). Both equations represent the same line, so every point on the line satisfies both. There are infinitely many solutions.

这里，方程 (2) 恰好是方程 (1) 的两倍。两个方程表示同一条直线，因此直线上的每个点都同时满足两个方程。有无穷多个解。

In elimination, you can recognise these cases: if the coefficients of x and y are proportional but the constants are not, there is no solution; if everything is proportional, there are infinite solutions.

在消元法中，你可以识别这些情况：若 x 和 y 的系数成比例但常数项不成比例，则无解；若所有项都成比例，则有无穷多解。

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

Students often lose marks due to small errors. Here are the most frequent pitfalls.

学生常因小错误而失分。以下是最常见的陷阱。

Not multiplying every term when applying elimination – always multiply the entire equation.

在消元时未乘以每一项——始终需整个方程乘以常数。
Sign errors when adding or subtracting equations. Write down each step carefully.

相加或相减方程时出现符号错误。仔细写出每一步。
Forgetting to substitute back to find both variables. Always solve for both x and y (or other unknowns).

忘记回代求另一个变量。务必求出所有未知量。
When solving quadratic systems, losing a root by starting with the wrong factorisation. Check your solutions by substituting both pairs into the original equations.

解二次方程组时，因错误因式分解而丢失一个根。通过将两组解代入原方程进行检验。

Top tips for exam day:

考试当日顶尖技巧：

If time permits, always verify your answer by plugging the values into both original equations.

如果时间允许，始终将求得的值代入两个原方程中验证。
Choose the method you are most comfortable with. For linear systems, elimination is generally quicker and more reliable.

选择你最熟悉的方法。对于线性方程组，消元通常更快更可靠。
In graphical questions, use a sharp pencil and a ruler, and plot at least three points for each line.

在图像题中，使用削尖的铅笔和直尺，每条直线至少绘制三个点。

9. Practice Questions | 练习题

Try these questions, then check your answers at the bottom of the page.

尝试以下练习，然后在位于页面底部的答案中核对。

Solve by substitution: y = 3x – 2, 2x + y = 1.

用代入法求解：y = 3x – 2，2x + y = 1。
Solve by elimination: 3x + 2y = 7, 2x – 2y = 8.

用消元法求解：3x + 2y = 7，2x – 2y = 8。
Find the intersection point of y = x² and y = 2x + 3.

求 y = x² 与 y = 2x + 3 的交点。
Two angles have a sum of 180°. One angle is 20° more than four times the other. Set up equations and solve for the angles.

两个角的和为 180°。一个角比另一个角的四倍多 20°。列出方程组并求解。

Answers:

答案：

Substitute y = 3x – 2 into 2x + y = 1 gives 2x + 3x – 2 = 1, so 5x = 3, x = 0.6, y = 3(0.6) – 2 = –0.2.

将 y = 3x – 2 代入 2x + y = 1 得 2x + 3x – 2 = 1，故 5x = 3，x = 0.6，y = 3(0.6) – 2 = –0.2。
Add the equations: (3x + 2y) + (2x – 2y) = 7 + 8, so 5x = 15, x = 3. Then 3(3) + 2y = 7, so 2y = –2, y = –1.

将方程相加：(3x + 2y) + (2x – 2y) = 7 + 8，故 5x = 15，x = 3。那么 3(3) + 2y = 7，所以 2y = –2，y = –1。
Set x² = 2x + 3, so x² – 2x – 3 = 0, (x – 3)(x + 1) = 0. Thus x = 3 or –1. When x = 3, y = 9; when x = –1, y = 1. Intersection points: (3, 9) and (–1, 1).

令 x² = 2x + 3，则 x² – 2x – 3 = 0，(x – 3)(x + 1) = 0。因此 x = 3 或 –1。当 x = 3 时 y = 9；当 x = –1 时 y = 1。交点：(3, 9) 和 (–1, 1)。
Let the angles be a and b. a + b = 180, a = 4b + 20. Substitute a into the first equation: (4b + 20) + b = 180, so 5b = 160, b = 32°, a = 4(32) + 20 = 148°. So the angles are 32° and 148°.

设两个角为 a 和 b。a + b = 180，a = 4b + 20。将 a 代入第一个方程：(4b + 20) + b = 180，所以 5b = 160，b = 32°，a = 4(32) + 20 = 148°。因此两个角分别为 32° 和 148°。

With consistent practice, simultaneous equations will become second nature. Remember to review your work thoroughly and understand the reasoning behind each method.

通过持续练习，联立方程将变得如鱼得水。记得仔细检查你的解题过程，并理解每种方法背后的逻辑。

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Simultaneous Equations | 联立方程