Solving Simultaneous Equations | 解联立方程

📚 Solving Simultaneous Equations | 解联立方程

Simultaneous equations are a fundamental topic in IGCSE Mathematics. They involve finding the values of two or more unknown variables that satisfy multiple equations at the same time. Mastering this topic is essential for success in both Paper 2 and Paper 4 of the IGCSE exam.

联立方程是IGCSE数学中的一个基础主题。它涉及找到同时满足多个方程的两个或更多未知变量的值。掌握这个主题对于在IGCSE考试的Paper 2和Paper 4中取得好成绩至关重要。

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

Simultaneous equations are a set of equations that share the same variables. For example, the system 2x + y = 7 and x – y = 2 is a pair of simultaneous equations. The solution is the pair of x and y values that makes both equations true at the same time.

联立方程是一组共享相同变量的方程。例如，方程组2x + y = 7和x – y = 2是一对联立方程。解就是使两个方程同时成立的x和y值对。

In IGCSE Mathematics, you will encounter two main types: linear simultaneous equations (where both equations represent straight lines) and non-linear systems (where one equation is quadratic). The number of equations must equal the number of unknown variables for a unique solution to exist. If there are fewer equations than unknowns, the system is underdetermined and may have infinitely many solutions.

在IGCSE数学中，你会遇到两种主要类型：线性联立方程（两个方程都表示直线）和非线性方程组（其中一个方程是二次方程）。方程的数量必须等于未知变量的数量，才能存在唯一解。如果方程少于未知数，方程组是欠定的，可能有无穷多组解。

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2. The Graphical Method | 图像法

The graphical method involves drawing the graphs of both equations on the same set of axes. The point where the two lines intersect represents the solution to the simultaneous equations.

图像法涉及在同一坐标系中绘制两个方程的图像。两条直线相交的点就代表联立方程的解。

For the equations y = 2x + 1 and y = -x + 4, plotting both lines shows they intersect at the point (1, 3). Therefore, the solution is x = 1 and y = 3. You can verify this by substituting both values into each equation.

对于方程y = 2x + 1和y = -x + 4，绘制两条直线显示它们在点(1, 3)处相交。因此，解是x = 1和y = 3。你可以将这两个值代入每个方程进行验证。

Steps for the graphical method:

图像法的步骤：

• Rearrange each equation into the form y = mx + c.

  将每个方程重新整理为y = mx + c的形式。

• Plot at least three points for each line to ensure accuracy.

  每条线至少绘制三个点以确保准确性。

• Identify the intersection point and read off the x and y coordinates.

  确定交点并读出x和y坐标。

• Check the solution by substituting into both original equations.

  通过代入两个原始方程来检查解。

The graphical method is useful for estimation and visual understanding, but it is often less accurate than algebraic methods, especially when the solution involves fractions or decimals. In IGCSE exams, you may be asked to use this method to estimate solutions, so a sharp pencil and a ruler are essential.

图像法对估算和直观理解很有用，但通常不如代数方法准确，特别是当解涉及分数或小数时。在IGCSE考试中，你可能会被要求使用这种方法估算解，因此一支削尖的铅笔和一把直尺是必不可少的。

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3. 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 method works for both linear and non-linear systems.

代入法涉及重新整理一个方程，使一个变量成为主题，然后将这个表达式代入另一个方程。这种方法对线性和非线性方程组都适用。

Consider the system: x + y = 10 and y = 2x + 1.

考虑方程组：x + y = 10和y = 2x + 1。

Step 1: Substitute y = 2x + 1 into the first equation. Since the second equation already gives y in terms of x, no rearrangement is needed.

步骤1：将y = 2x + 1代入第一个方程。由于第二个方程已经用x表示y，无需重新整理。

x + (2x + 1) = 10

Step 2: Simplify and solve for x:

步骤2：化简并求解x：

3x + 1 = 10 → 3x = 9 → x = 3

Step 3: Substitute x = 3 back into y = 2x + 1:

步骤3：将x = 3代回y = 2x + 1：

y = 2(3) + 1 = 7

Therefore, the solution is x = 3, y = 7. Check: 3 + 7 = 10 ✓, and 7 = 2(3) + 1 ✓.

因此，解为x = 3，y = 7。验证：3 + 7 = 10 ✓，7 = 2(3) + 1 ✓。

The substitution method is particularly useful when one equation already has a variable as the subject, or when dealing with a linear equation combined with a quadratic equation, as you will see in Section 7.

代入法特别适用于一个方程已经有一个变量作为主题的情况，或者在处理线性方程与二次方程组合时，你将在第7节中看到。

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4. The Elimination Method | 消元法

The elimination method requires adding or subtracting the equations to eliminate one of the variables. This is often the fastest method for linear systems because it avoids fractions until the final step.

消元法需要通过加减方程来消去其中一个变量。对于线性方程组，这通常是最快的方法，因为它直到最后一步才涉及分数。

Consider the system:

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