Problem 39 - Algebra and Trigonometry

Question

Can a Linear System Have Exactly Two Solutions? (a) Suppose that \(\left(x_{0}, y_{0}, z_{0}\right)\) and \(\left(x_{1}, y_{1}, z_{1}\right)\) are solutions of the system $$\left\\{\begin{array}{l}{a_{1} x+b_{1} y+c_{1} z=d_{1}} \\ {a_{2} x+b_{2} y+c_{2} z=d_{2}} \\ {a_{3} x+b_{3} y+c_{3} z=d_{3}}\end{array}\right.$$ Show that \(\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right)\) is also a solution. b) Use the result of part (a) to prove that if the system has two different solutions, then it has infinitely many solutions.

Step-by-Step Solution

Answer

A linear system cannot have exactly two solutions; it has none, one, or infinitely many.

Step 1: Understanding System of Equations

To determine if a linear system can have exactly two solutions, we must understand that a system of linear equations can have either zero, one, or infinitely many solutions under normal circumstances.

Step 2: Checking Given Solutions

Given two solutions, \(x_0, y_0, z_0\) and \(x_1, y_1, z_1\), we are tasked to show that their midpoint is also a solution to the system. This will demonstrate that a linear system can derive additional solutions through those it already possesses.

Step 3: Midpoint Calculation

The midpoint of the solutions \(x_0, y_0, z_0\) and \(x_1, y_1, z_1\) is given by the point \(\left(\frac{x_0+x_1}{2}, \frac{y_0+y_1}{2}, \frac{z_0+z_1}{2}\right)\). We must check if this point satisfies all equations in the system.

Step 4: Verification Process

Substitute the midpoint \(\left(\frac{x_0+x_1}{2}, \frac{y_0+y_1}{2}, \frac{z_0+z_1}{2}\right)\) into each equation, ensuring each equation is satisfied using the linearity principle: - For equation 1: \ a_1 \left(\frac{x_0+x_1}{2}\right) + b_1 \left(\frac{y_0+y_1}{2}\right) + c_1 \left(\frac{z_0+z_1}{2}\right) = \frac{d_1+d_1}{2} = d_1 \- Similarly verify for equations 2 and 3.

Step 5: Conclusion from Part (a)

The midpoint is also a solution, establishing that if two points are solutions, their connecting line is filled with solutions. Hence, it suggests an infinite number of solutions lying on this line.

Step 6: Implication for Infinitely Many Solutions

Since connecting any two distinct solutions on a line generates a continuous stretch of solutions (a line), the system has infinitely many solutions. A system can't just stop at two solutions without violating linearity of space.

Key Concepts

Understanding a System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. Such systems are often written in the form: \[\begin{array}{cccccc} a_1 x &+& b_1 y &+& c_1 z &= d_1 \ a_2 x &+& b_2 y &+& c_2 z &= d_2 \ a_3 x &+& b_3 y &+& c_3 z &= d_3 \end{array}\] Here, each equation represents a plane in three-dimensional space. Linear equations can intersect or overlap, defining various solution possibilities. The main scenarios we encounter are:

• Zero solutions: The planes do not intersect at a common point.
• One unique solution: All planes intersect at a single point.
• Infinitely many solutions: The planes overlap along a line or coincide entirely.

Understanding these possibilities helps us explore how solutions relate in systems of linear equations, especially when considering more than one solution.

Exploring Infinitely Many Solutions

A fascinating aspect of linear systems is that having two distinct solutions can lead to having an infinite number. This occurs because linear equations are based on lines or planes, which can contain numerous points.
When two solutions exist, we can find a whole line of solutions between them. To understand this, let's consider the logic behind it. If \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) are solutions of the same system, then not only are these particular points valid solutions, but every point along the line connecting them will be as well.
This is because linear equations maintain a consistent direction and spread through their space, without introducing breaks or interruptions of "stopping" after two solutions. So, if a system has two solutions, it's indicative of the potential for this continuous line of solutions, indicating infinitely many solutions.

Finding the Midpoint of Solutions

The midpoint of two points in space represents the average of their coordinates. This concept is essential in explaining why a linear system cannot have exactly two solutions. If you have two solutions, \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\), you can calculate the midpoint with:\[\left(\frac{x_0+x_1}{2}, \frac{y_0+y_1}{2}, \frac{z_0+z_1}{2}\right)\]By substituting this midpoint into the linear equations of the system, we validate that it, too, satisfies each equation. It is because the principles of linearity ensure that the average remains on the line defined by the two original points.
This exercise reinforces the idea that if a system has two distinct solutions, it not only supports many solutions but signifies an entire line or plane of them, leading us to understand why a system with precisely two solutions is not possible.