Problem 9 - Contemporary Precalculus

Question

In Exercises \(9-12\), the reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of the system. $$\left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 3 / 2 \\ 0 & 1 & 0 & 0 & 5 \\ 0 & 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right)$$

Step-by-Step Solution

Answer

Answer: The values of the variables are: x1 = 3/2, x2 = 5, x3 = -2, and x4 = 0.

Step 1: Formulate Equations

First, let's write down the system of linear equations from the given matrix: $$ \begin{cases} x_1 + 0x_2 + 0x_3 + 0x_4 = \frac{3}{2}\\ 0x_1 + x_2 + 0x_3 + 0x_4 = 5\\ 0x_1 + 0x_2 + x_3 + 0x_4 = -2\\ 0x_1 + 0x_2 + 0x_3 + x_4 = 0 \end{cases} $$ Which simplifies to the following system of equations: $$ \begin{cases} x_1 = \frac{3}{2}\\ x_2 = 5\\ x_3 = -2\\ x_4 = 0 \end{cases} $$

Step 2: Read the Solution

The system of linear equations is already solved and the variables have their corresponding solutions. The solution of the system is given by the values of the variables: $$ x_1 = \frac{3}{2},\ x_2 = 5,\ x_3 = -2,\ x_4 = 0 $$

Key Concepts

Augmented Matrix

An augmented matrix is a useful tool that represents a system of linear equations in a condensed form. It combines the coefficients of the variables and the constants from each equation of the system into a single matrix. This matrix is often used to simplify solving the system of equations by applying row operations.
Imagine a set of equations, each with variables like \(x_1, x_2, x_3,\) and so on, with constants on the other side of the equality. In matrix form, the numbers in front of these variables are the entries of the matrix, while the constants are placed in the final column of the augmented matrix.

• The first part of the augmented matrix includes all the coefficients of the variables.
• The last column contains the constants.

With the example from the exercise, consider how much easier it is to handle four equations at once in matrix form instead of writing out each equation separately. This is especially beneficial for larger systems, as the matrix form allows for straightforward manipulation and analysis using linear algebra techniques.

System of Linear Equations

A system of linear equations is a collection of equations that you deal with simultaneously. Each equation shares common variables, which must satisfy all equations in the system. Solving the system means finding values for these variables that make all the equations true at the same time.
Let's break down the importance and characteristics of a system of linear equations:

• The system can have various outcomes such as a single solution, infinitely many solutions, or no solution at all.
• Each equation is linear, meaning the highest power of the variable is one, allowing it to be graphically represented as a straight line.#
• These equations often appear in real-world problems, such as in economics, engineering, and science, where multiple conditions must be met simultaneously.

Consider the set of equations extracted from the reduced echelon form of the matrix in the original problem. Each equation isolates one variable, showing perfect independence and making it easy to solve. This clarity is a direct result of converting the system into its reduced row echelon form.

Solution of a System

Finding the solution of a system of linear equations means identifying the common values for all variables that satisfy each individual equation. This process often involves simplifying the system until each equation can be easily solved for one variable.
The solution could manifest in multiple forms:

• A single unique solution, meaning one specific set of values that works for all the equations.
• Infinitely many solutions often occur when the equations describe the same line or plane.
• No solution arises when equations represent parallel lines or planes that never intersect.

In the original problem, the system's solution, \(x_1 = \frac{3}{2}, x_2 = 5, x_3 = -2, x_4 = 0\), is straightforward due to the reduced row echelon form. Each variable corresponds neatly with a single equation, allowing a clear and direct interpretation of the outcomes. This clear separation of variables is one of the key benefits of using reduced row echelon form, as witnessed in the solved exercise.