Problem 9 - College Algebra with Modeling and Visualization

Question

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{rrr|r} 3 & 1 & 4 & 0 \\ 0 & 5 & 8 & -1 \\ 0 & 0 & -7 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The system is \(3x + y + 4z = 0\), \(5y + 8z = -1\), \(-7z = 1\).

Step 1: Understand the Augmented Matrix

An augmented matrix represents a system of linear equations, where the columns before the vertical line represent the coefficients of the variables, and the column after the vertical line represents the constants on the right-hand side of each equation.

Step 2: Identify Variables

Assign variables to the columns preceding the vertical line. Commonly, these variables are denoted as \(x\), \(y\), and \(z\) for a 3-column matrix. Thus, the first column corresponds to \(x\), the second to \(y\), and the third to \(z\).

Step 3: Convert Rows to Equations

Convert each row of the matrix into a linear equation. For the given matrix:1. The first row is \(3x + y + 4z = 0\).2. The second row is \(5y + 8z = -1\).3. The third row is \(-7z = 1\).

Step 4: Write the System of Equations

Combine the equations from each row to form the system of linear equations:\[\begin{align*}3x + y + 4z &= 0, \5y + 8z &= -1, \-7z &= 1.\end{align*}\]

Key Concepts

Understanding Augmented Matrices

Augmented matrices are an efficient way to represent a system of linear equations.
They consist of a coefficient matrix merged with a column vector of constants. This is often displayed as a matrix with a vertical line separating the coefficients from the constants.
Here's how it works:

• The columns before the vertical line represent the coefficients for each variable in the system.
• The single column after the vertical line is the constants from the right side of the equations.
• Each row corresponds to a single equation in the system.

For instance, in the matrix example, the first row corresponds to the equation involving the coefficients 3, 1, 4, and the constant 0. This method helps us manipulate systems of equations neatly and solve them efficiently.

What is 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. The goal is to find the values of these variables that satisfy all equations in the system simultaneously.
Let's look at some key aspects:

• Each equation in the system can be plotted as a line (in two dimensions) or a plane (in three dimensions).
• The solution to the system is the point or points where the equations intersect.
• Systems can have zero, one, or infinitely many solutions, depending on whether the lines/planes are parallel, intersect at one point, or coincide completely.

In our specific exercise, the system of linear equations derived from the augmented matrix represents lines in three-dimensional space.

Matrix Representation of Linear Systems

Matrix representation is a compact form of writing down systems of linear equations, making it easier to perform operations and find solutions through various matrix techniques.
Here's how matrix representation works:

• The coefficient matrix holds all the coefficients of the variables of the system in an orderly manner.
• The constant matrix or vector contains the constants from each equation.
• When the coefficient and constant matrices are combined into an augmented matrix, problem-solving becomes more straightforward.

Representing systems with matrices not only simplifies notation but also enables the use of linear algebra techniques, such as Gaussian elimination or matrix inverses, to find solutions effectively.