Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrr|r}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 1\end{array}\right]\)

Step-by-Step Solution

Answer

The given matrix is in row-reduced form, which indicates a consistent system with a unique solution. Reading directly from the matrix, we have \(x_1 = 3\), \(x_2 = -2\), and \(x_3 = 1\). Therefore, the solution set is \(\{(3, -2, 1)\}\).

Step 1: Identify the type of system

The given matrix is in row-reduced form, which means that the system of linear equations is a consistent system. The matrix is: \(\left[\begin{array}{rrr|r}1 & 0 & 0 & 3 \\\ 0 & 1 & 0 & -2 \\\ 0 & 0 & 1 & 1\end{array}\right]\) The matrix has three non-zero rows and there are no contradictory or dependent equations. This means that the system has a unique solution.

Step 2: Read the solution from the row-reduced matrix

Since the matrix is already in row-reduced form, we can read the solution directly from the matrix. From the matrix: 1. The first row represents the equation \(x_1 = 3\). 2. The second row represents the equation \(x_2 = -2\). 3. The third row represents the equation \(x_3 = 1\). The solution to the system of linear equations is: \(x_1 = 3\), \(x_2 = -2\), and \(x_3 = 1\).

Step 3: Write the solution set

The solution set of the given system of linear equations is: \(\{(3, -2, 1)\}\) So the system has a unique solution, and the solution is: \(x_1 = 3\), \(x_2 = -2\), and \(x_3 = 1\).

Key Concepts

System of Linear Equations

Understanding the basics of a system of linear equations is crucial for anyone diving into algebra. At its heart, a system of linear equations is a collection of two or more linear equations involving the same set of variables. An example would be equations like 2x + 3y = 5 and 4x - y = 11, which can be visualized as straight lines when graphed on a coordinate plane.

The solution to such a system corresponds to the point or points where the lines intersect, representing the values of the variables that satisfy all the equations simultaneously. When dealing with these kinds of systems, we often use methods such as substitution, elimination, and matrix operations to find these intersection points - the solutions.

In matrix terms, we aim to transform the system into a row-reduced form, which displays whether the lines intersect and if so, at what points. This form can also confirm if the lines are parallel, which would mean there's no intersection and thus no solution, or if they coincide, offering an infinite number of solutions.

Consistent System

Moving on to what makes a system 'consistent': a consistent system of linear equations is one that has at least one set of solutions. This contrasts with an inconsistent system, which has no solution.

The row-reduced matrix we use is an excellent tool for determining consistency. It is a streamlined version of the matrix, achieved through a series of operations, that makes identifying solutions easier. If, in this form, there are no rows that lead to an absurd statement (like 0 = 1), the system is consistent.

This matrix form shows us the relationships between variables clearly, and we can often read the solutions right off of it. A consistent and independent system will have a unique solution, as in the case of the given exercise, where the row-reduced matrix shows us direct solutions for each variable, affirming the system's consistency.

Unique Solution

Lastly, let's focus on what we mean by a 'unique solution'. This is the holy grail of a consistent system, indicating that there's one and only one set of values for the variables that satisfies all the equations.

In the context of the row-reduced matrix, a unique solution is easily identifiable. If each variable has a leading one (or pivot) in its own row and column, and there are no free variables, the system's solution is unique.

To put it simply, in our given matrix example, each variable can be solved with no reliance on others, leading to a single solution point (3, -2, 1) - hence, it's unique. Understanding the concept of a unique solution empowers students to quickly assess the type of system they are dealing with and the methods they should use to solve it.