Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the augmented matrix corresponding to each system of equations. \(\begin{aligned}-y+2 z &=6 \\ 2 x+2 y-8 z &=7 \\ 3 y+4 z &=0 \end{aligned}\)

Step-by-Step Solution

Answer

The augmented matrix corresponding to the given system of equations is \(\left[\begin{array}{ccc|c}0 & -1 & 2 & 6 \\ 2 & 2 & -8 & 7 \\ 0 & 3 & 4 & 0\end{array}\right]\).

Step 1: Identify the given equations

We are already given three linear equations in the problem. They are: 1. \(-y + 2z = 6\) 2. \(2x + 2y - 8z = 7\) 3. \(3y +4z = 0\)

Step 2: Rewrite the equations with all the variables

In order to create the augmented matrix, we need to have all the variables present in every equation. If a variable is missing, we simply rewrite the equation with that variable with a coefficient of 0. The given equations with all variables are: 1. \(0x - y + 2z = 6\) 2. \(2x + 2y - 8z = 7\) 3. \(0x + 3y + 4z = 0\)

Step 3: Write the augmented matrix

Now we can create the augmented matrix by writing the coefficients of the variables and the constants in a matrix. The first column will have the coefficients of x, the second column will have the coefficients of y, the third column will have the coefficients of z, and the fourth column will have the constants. The augmented matrix is: \[ \left[ \begin{array}{ccc|c} 0 & -1 & 2 & 6 \\ 2 & 2 & -8 & 7 \\ 0 & 3 & 4 & 0 \end{array} \right] \]

Key Concepts

System of Linear Equations

A system of linear equations comprises multiple linear equations with multiple variables. In our example, the system contains three equations with three variables: \(x\), \(y\), and \(z\). These equations can represent various situations, such as intersecting lines or planes in a multi-dimensional space.
To solve a system of linear equations means to find values for the variables that satisfy all equations simultaneously. In many cases, systems of equations can be solved using substitution, elimination, or graphically.

• Substitution involves solving one equation for one variable and substituting this expression into the other equations.
• Elimination involves adding or subtracting equations to eliminate a variable.
• Graphically, the solution can often be seen as the intersection of lines or planes represented by the equations.

When written out, each equation can be understood as a condition or restriction placed on the possible solutions for the variables. Consider the constraints imposed as each equation set conditions for a solution.

Matrix Representation

Matrix representation is a powerful tool for visualizing and solving systems of linear equations. It involves organizing the coefficients of the variables and constant terms into a matrix. Each row represents an equation, and each column represents the coefficients of a particular variable. This standard structure simplifies complex calculations in linear algebra.
In our scenario, an augmented matrix was used. This matrix setup will include both coefficients and constants, facilitating an easier row operation process later. Notice that:

• The first column (0, 2, 0) corresponds to the \(x\)-coefficients.
• The second column (-1, 2, 3) corresponds to the \(y\)-coefficients.
• The third column (2, -8, 4) captures the \(z\)-coefficients.
• The fourth column (6, 7, 0) contains the constants from the right-hand side of the equations.

The use of a matrix helps with visual simplification and manipulations, such as using Gaussian elimination to solve for variable values efficiently.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and solving systems of linear equations. It extends beyond just finding solutions, providing insights into the properties of the equations and transformations.
One key aspect of linear algebra is the use of vectors and matrices to model and solve problems. With matrices, operations such as matrix multiplication and determinants become essential tools for manipulating and understanding data structures.

• Vectors and matrices are convenient for modeling linear transformations, where a change of basis can transform one set of coordinates into another.
• Concepts like linear independence, rank, and eigenvalues are deeply rooted in linear algebra, providing a framework for understanding the dimensions and structures of matrix spaces.

Through these concepts, linear algebra allows us to approach not just simple solutions but delve deeper into the nature and implications of various systems. Its applications range from simple arithmetic to complex real-world problems in engineering, physics, and computer science.