Problem 51 - College Algebra and Calculus: An Applied Approach

Question

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & -8 \\ 0 & 0 & 1 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

Therefore, the solution to the system of equations represented by the augmented matrix is \(x = -4\), \(y = -8\) and \(z = 2\)

Step 1: Understanding the matrix format

The augmented matrix represents a system of equations. Each row of the matrix corresponds to a linear equation, and the numbers in the row are the coefficients of the variables in the equation. The number in the last column corresponds to the right side of the equation.

Step 2: Translating the matrix to a system of equations

From the matrix, we can write each row as a linear equation: From Row 1: \(1*x + 0*y + 0*z = -4\),\nFrom Row 2: \(0*x + 1*y + 0*z = -8\),\nFrom Row 3: \(0*x + 0*y + 1*z = 2\).

Step 3: Simplifying the equations

Simplify each equation by removing the zero terms: From Row 1: \(x = -4\),From Row 2: \(y = -8\),From Row 3: \(z = 2\)

Key Concepts

Augmented Matrix

In linear algebra, an augmented matrix is a powerful tool that helps in solving systems of linear equations. It is essentially a combination of the coefficient matrix and the constants from the equations' right sides. The augmented matrix represents the entire system of equations in a compact form that simplifies calculations.
For a system of linear equations, each row in the augmented matrix corresponds to an equation. The elements on the left side of the vertical line represent the coefficients of the variables, while the elements on the right side are the constants from each equation.
Using an augmented matrix alongside methods like Gauss-Jordan elimination allows for systematic operations that can simplify and solve complex systems of equations. It provides an organized way to perform row operations, ultimately leading to a matrix where the solution can be easily identified.

System of Linear Equations

Understanding a system of linear equations involves recognizing a collection of equations with multiple variables. Each equation in the system represents a straight line in the coordinate space, and the solution to the system is the point where all these lines intersect.
A typical system involving three variables, such as \(x\), \(y\), and \(z\), can be written as:

• Equation 1: \(a_1x + b_1y + c_1z = d_1\)
• Equation 2: \(a_2x + b_2y + c_2z = d_2\)
• Equation 3: \(a_3x + b_3y + c_3z = d_3\)

The coefficients (a, b, c) and constants (d) define how the equations relate to each other. These systems can be solved by expressing them in matrix form and applying systematic methods like Gauss-Jordan elimination to reach a consistent and manageable solution format.

Matrix Representation

Matrix representation translates a system of equations into a visual structure consisting of numbers arranged in rows and columns. This method simplifies complex calculations by organizing all information into a matrix format. For our system involving three variables \(x, y, z\), the matrix representation of the system may look like this:

• Row 1: Coefficients of \(x\), \(y\), \(z\) and constant
• Row 2: Coefficients of \(x\), \(y\), \(z\) and constant
• Row 3: Coefficients of \(x\), \(y\), \(z\) and constant

The matrix for the system turns into the following format:\[\begin{bmatrix}a_{11} & a_{12} & a_{13} & | & b_1 a_{21} & a_{22} & a_{23} & | & b_2 a_{31} & a_{32} & a_{33} & | & b_3\end{bmatrix}\]The vertical line in the matrix distinguishes the coefficients from the constants. Matrix representation is a foundational concept in linear algebra used for analysis, solutions, and transformations of linear systems efficiently.