Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Write the augmented matrix for the system of linear equations. Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} 9 w-3 x+20 y+z= & 13 \\ 12 w-8 y= & 5 \\ w+2 x+3 y-4 z= & -2 \\ -w-x+y+z= & 1 \end{array}\right. $$

Step-by-Step Solution

Answer

The augmented matrix for the given system of equations is: \[ \left[ \begin{array}{cccc|c} 9 & -3 & 20 & 1 & 13 \ 12 & 0 & -8 & 0 & 5 \ 1 & 2 & 3 & -4 & -2 \ -1 & -1 & 1 & 1 & 1 \end{array} \right] \]

Step 1: Setting up the augmented matrix

Start by identifying the coefficients of the variables in each equation. Line them up vertically with the same variables. The last column will be the constant on the other side of the equals sign. (If a variable is not in an equation, it's as if it multiplies 0 since it has no effect on that equation. Hence, 0 will be filled in its position.)

Step 2: Write down the coefficients of the equations

Following the order, the augmented matrix will be written as follows: \[ \left[ \begin{array}{cccc|c} 9 & -3 & 20 & 1 & 13 \ 12 & 0 & -8 & 0 & 5 \ 1 & 2 & 3 & -4 & -2 \ -1 & -1 & 1 & 1 & 1 \end{array} \right] \]

Key Concepts

Understanding System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables that need to be solved together. In this case, we have equations involving four variables: \( w, x, y, \) and \( z \). Each equation provides information about how these variables relate to each other. The goal is to find the values of these variables that satisfy all the equations simultaneously.

To solve such systems, we can use several methods, such as substitution, elimination, or matrix techniques. By aligning the equations in a specific order, we can simplify our work, making it easier to solve using these methods. Each equation is like a piece of a puzzle that must fit perfectly with others to resolve the entire system.

Basics of Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors, vector spaces, and linear equations. It provides powerful tools for solving large systems of equations, like those in the example provided. Linear algebra relies on concepts such as vectors and matrices, as well as operations like addition and scalar multiplication.

The greatest strength of linear algebra is its ability to simplify complex systems into a form that is more manageable. This is achieved through and other tools like determinants and eigenvalues. Whether dealing with simple systems or more complex ones involving multiple variables, linear algebra helps break down the equations into simpler terms.

• Vectors are lists of numbers representing a point in space.
• Matrices are rectangular arrays of numbers that represent systems of linear equations.

These concepts allow for systematic approaches to solving equations, proving to be vital in engineering, physics, computer science, and economics.

Using Matrix Representation

Matrix representation transforms a system of linear equations into a straightforward array of numbers called a matrix. With each row representing an equation and each column corresponding to a variable, the matrix method organizes the information into a compact form.

An augmented matrix includes both the coefficients of the variables and the constant terms from the system of equations. It looks like this: \[\left[ \begin{array}{cccc|c} 9 & -3 & 20 & 1 & 13 \12 & 0 & -8 & 0 & 5 \1 & 2 & 3 & -4 & -2 \-1 & -1 & 1 & 1 & 1 \end{array} \right]\]
This matrix represents the given system of equations, clearly showing the relationships without rewriting the entire system. Using matrices, we can apply methods like row reduction to find solutions efficiently. The augmented matrix approach is a cornerstone of linear algebra, linking algebraic concepts with computational methods.