Problem 47 - Precalculus Mathematics for Calculus

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 $$ \left\\{\begin{array}{l} a_{1} x+b_{1} y+c_{1} z=d_{1} \\ a_{2} x+b_{2} y+c_{2} z=d_{2} \\ a_{3} x+b y+c_{3} z=d_{3} \end{array}\right. $$ 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

No, a linear system cannot have exactly two solutions. If it has two, it has infinitely many.

Step 1: Understanding the Problem

We need to analyze whether a given system of linear equations can have exactly two solutions. To explore this, we will start by showing that the arithmetic mean of two solutions is also a solution.

Step 1: Show the Mean of Solutions is a Solution

We have two solutions to the given system: \[ \begin{align*} a_1 x_0 + b_1 y_0 + c_1 z_0 &= d_1, \ a_2 x_0 + b_2 y_0 + c_2 z_0 &= d_2, \ a_3 x_0 + b_3 y_0 + c_3 z_0 &= d_3, \end{align*} \]\[ \begin{align*} a_1 x_1 + b_1 y_1 + c_1 z_1 &= d_1, \ a_2 x_1 + b_2 y_1 + c_2 z_1 &= d_2, \ a_3 x_1 + b_3 y_1 + c_3 z_1 &= d_3. \end{align*} \]Now we compute the arithmetic mean of the solutions:\[ \left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2}\right). \]Plugging these into the equations of the system, verify each equation.

Step 2: Verify Each Equation for the Mean

Substitute \(\left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2}\right)\) into the first equation:\[ a_1 \left(\frac{x_0 + x_1}{2}\right) + b_1 \left(\frac{y_0 + y_1}{2}\right) + c_1 \left(\frac{z_0 + z_1}{2}\right) = \frac{d_1 + d_1}{2} = d_1. \]Do the same for the second and third equations:\[ \begin{align*} a_2 \left(\frac{x_0 + x_1}{2}\right) + b_2 \left(\frac{y_0 + y_1}{2}\right) + c_2 \left(\frac{z_0 + z_1}{2}\right) &= d_2, \ a_3 \left(\frac{x_0 + x_1}{2}\right) + b_3 \left(\frac{y_0 + y_1}{2}\right) + c_3 \left(\frac{z_0 + z_1}{2}\right) &= d_3. \end{align*} \]This confirms the arithmetic mean is indeed a solution.

Step 3: Infinitely Many Solutions Conclusion

Having shown that the mean of two solutions is also a solution, it follows that any linear combination of the two solutions is a solution. Since you can choose arbitrary coefficients for such combinations, we have infinitely many solutions whenever there exist two distinct solutions.

Key Concepts

Infinite Solutions

In the context of linear systems, infinite solutions mean that there are countless solutions that can satisfy all equations in the system simultaneously. This typically occurs when the equations are not all independent and form a geometric plane that overlaps with others in multiple ways. In simpler terms, each equation represents a plane in three-dimensional space. When two or more planes coincide neatly on top of one another, we find an infinite number of intersections or solutions.
For instance, imagine two infinite sheets of paper lying perfectly atop one another; any point along their overlapping area represents a solution. Infinite solutions often emerge when the equations describe the same line or plane in space, or when the equations are linearly dependent, meaning one equation is a multiple of the others.

• The combination of two distinct solutions leads to a spectrum of possible solutions.
• If a system can have more than one solution, often, it can stretch to infinity.

Linear Equations

Linear equations are fundamental components in algebra, representing relationships where terms are linearly related. These include equations like 3x + 2y = 5, where x and y are variables. Each linear equation represents a line when plotted in a two-dimensional space, or a plane in three-dimensional space. Key characteristics of linear equations include:

• Variables are typically first degree, meaning they have no exponents or powers beyond one.
• They form straight lines or flat planes, as opposed to curves or more complex shapes.

Understanding linear equations is crucial for analyzing linear systems. They are the building blocks of more complex structures in algebra. By solving these equations simultaneously, we determine points where these lines or planes intersect.
This intersection often reveals crucial insights into the relationships between different variables in a given problem. They are called 'linear' because of the constancy in their rate of change, allowing predictability and simple calculations.

Algebraic Systems

An algebraic system is a set of equations that can be solved together, showing how variables interconnect. These systems often arise from practical problems, where several conditions need to be satisfied simultaneously.
For example, an algebraic system may model the relationship between time, speed, and distance in physics. Solving algebraic systems involves finding all possible values of the variables that satisfy every equation in the system.

• Systems can be solved using various methods like substitution, elimination, and matrix operations.
• Understanding the structure of the equations is essential for effective solution strategies.

In more complex systems, especially those involving multiple variables or non-linear equations, breaking down the components can guide the solving process. Recognizing patterns, such as dependence or inconsistency among equations, is key. Sometimes, a system will have no solution, one unique solution, or infinitely many solutions.
Each outcome provides different insights: no solution implies a conflict among conditions, while infinite solutions point to underlying redundancies within the system.