Problem 35 - Algebra for College Students

Question

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{llll:r}1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The solution set is \((x_1, x_2, x_3, x_4) = (-2, 4, -3, 0)\).

Step 1: Understand the Given Matrix

The given matrix is: \[\begin{bmatrix} 1 & 0 & 0 & 0 & -2 \ 0 & 1 & 0 & 0 & 4 \ 0 & 0 & 1 & 0 & -3 \ 0 & 0 & 0 & 1 & 0 \end{bmatrix}\].This matrix is already in reduced row echelon form. The matrix represents the system of equations corresponding to the variables \(x_1, x_2, x_3, \) and \(x_4\).

Step 2: Identify the Solution for Each Variable

Each row in the matrix corresponds to a different equation in the system. The first four columns correspond to the coefficients of the variables, while the fifth column provides the constants.- The first row indicates \(x_1 = -2\).- The second row indicates \(x_2 = 4\).- The third row indicates \(x_3 = -3\).- The fourth row indicates \(x_4 = 0\).

Step 3: Write the Solution Set

The solution to the system is given directly by reading the constants on the right of the vertical line in the matrix. The solution set is:\[ \{ (x_1, x_2, x_3, x_4) \mid x_1 = -2, x_2 = 4, x_3 = -3, x_4 = 0 \} \].

Step 4: Interpret the Solution Set

The solution specifies a single point in four-dimensional space, as all the variables have unique values. This means the system is consistent and has a unique solution.

Key Concepts

Solution Set

When we talk about a solution set, we're referring to all possible combinations of variable values that satisfy a given system of equations. In matrix terms, once we have simplified a matrix to its reduced row echelon form, we can easily read off the values of the variables.

• In this exercise, the reduced row echelon matrix showed us a straightforward solution for each variable.
• This leads to a solution set that includes only one unique set of values for the variables.

The solution set effectively becomes a single point (or vector) in a high-dimensional space. Each specific configuration of variable values corresponds to one point in this multidimensional space, providing clarity and finality to equations.

System of Equations

A system of equations consists of multiple equations involving the same set of variables. In this problem, the system of four equations correlates directly with the reduced row echelon matrix.

• Each row of the matrix corresponds to one equation from the system.
• The goal is to solve the system to find a common solution where all equations are satisfied simultaneously.

Using a matrix simplifies handling multiple equations, allowing us to use specific operations to reach reduced row echelon form, from which solutions are easier to spot.

Unique Solution

A unique solution to a system means there is exactly one set of values for the variables that satisfy all equations. When a matrix is in reduced row echelon form, and each variable has a distinct non-zero coefficient in separate rows, the system typically has a unique solution.

• In our matrix, observing separate leading 1s in each row ensures each variable has a distinct solution.
• This indicates consistency and proves that the matrix and system represent a single solution.

In this exercise, each variable was isolated with clear constants, ensuring consistency and individuality of solutions.

Matrices in Algebra

Matrices are powerful tools in algebra, especially for solving systems of equations like the one in this exercise. They provide a structured approach to manage and solve multiple equations efficiently.

• Matrices help organize coefficients and constants systematically.
• Manipulating matrices through row operations leads us to reduced row echelon form.
• This form makes it easy to identify solutions without messy algebraic manipulations.

By arranging the equations neatly into a matrix, algebraists can apply consistent methodologies to explore solutions, even for complex systems.