Problem 10 - College Algebra with Modeling and Visualization

Question

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{rrr|r} 1 & -1 & 3 & 2 \\ -2 & 1 & 1 & -2 \\ -1 & 0 & -2 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

\(x - y + 3z = 2\), \(-2x + y + z = -2\), \(-x - 2z = 1\)

Step 1: Understand the structure of the augmented matrix

The augmented matrix represents a system of linear equations. Each row of the matrix corresponds to a linear equation, with the columns representing the coefficients of the variables and the constant on the right side of the equation.

Step 2: Identify the coefficients of the variables

In the given matrix, the first column represents the coefficients for the variable \(x\), the second column for \(y\), the third column for \(z\), and the numbers on the right side of the vertical bar represent the constants of each equation.

Step 3: Write the first equation from the first row

The first row of the matrix \([1, -1, 3|2]\) corresponds to the equation: \(1x - 1y + 3z = 2\). Simplifying, we can write it as: \(x - y + 3z = 2\).

Step 4: Write the second equation from the second row

The second row of the matrix \([-2, 1, 1|-2]\) corresponds to the equation: \(-2x + 1y + 1z = -2\). Simplifying, we can write it as: \(-2x + y + z = -2\).

Step 5: Write the third equation from the third row

The third row of the matrix \([-1, 0, -2|1]\) corresponds to the equation: \(-1x + 0y - 2z = 1\). Simplifying, we can write it as: \(-x - 2z = 1\).

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra to deal with systems of linear equations. It combines the coefficients of variables in equations and their respective constants into a single, organized table. This method allows us to streamline the process of solving the system. In an augmented matrix, each equation in the system is represented as a row. The columns correspond to the coefficients of each variable.

For instance, consider the matrix given in the original problem:

• The first column contains the coefficients for the variable \(x\).
• The second column represents the coefficients for the variable \(y\).
• The third column shows the coefficients for the variable \(z\).

The vertical bar in an augmented matrix separates these coefficients from the constants of each equation. This organization makes it easier to visualize how altering one element affects the entire system. Using an augmented matrix helps in efficient problem-solving through operations like row reduction, which can simplify, and eventually solve systems of equations.

Systems of Equations

A system of equations is a collection of two or more equations with multiple variables that you want to solve simultaneously. These systems can arise in various practical scenarios, such as determining the intersection point of lines or optimizing conditions in operations.

To work with systems of equations, you generally aim to find values for each variable that satisfy all equations at the same time. There are several methods to solve these, but augmented matrices are particularly beneficial when dealing with multiple variables and equations.

• Each equation in a system can be written in standard form with variables aligned vertically across these equations.
• This structured representation makes it easy to convert the system into an augmented matrix format.

Solving systems of equations requires strategies such as substitution, elimination, or using matrix operations to manipulate the equations. The solution set can be unique, infinite, or might not exist, depending on the specific relations between the equations.

Variables in Equations

In mathematics, especially when dealing with linear equations, variables are symbols involved in operations and calculations. They act as placeholders for numbers and are essential for formulating equations that describe real-world phenomena.

Variables allow us to set rules and relationships between different quantities. In the case of linear equations, they are typically represented by letters like \(x\), \(y\), and \(z\). Each variable may have one or multiple coefficients associated with it, influencing its overall effect within an equation.

• The coefficients multiply the variables and serve as indicators of the variables’ influence on the respective equations.
• In systems of equations, we solve for these variables to find their values that make each equation true simultaneously.

Understanding the role of variables within equations is key, as it lays the groundwork for solving complex mathematical problems. By effectively manipulating variables and their coefficients, one can transform intricate systems into solvable ones.