Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+y+z &=3 \\ 3 x-4 y+2 z &=-7 \\ x+y+z &=2 \end{aligned}$$

Step-by-Step Solution

Answer

The augmented matrix is \(\begin{bmatrix} 2 & 1 & 1 & | & 3 \\ 3 & -4 & 2 & | & -7 \\ 1 & 1 & 1 & | & 2 \end{bmatrix}\).

Step 1: Understand the System of Equations

First, examine the given system of equations: \[\begin{aligned} 2x + y + z &= 3 \ 3x - 4y + 2z &= -7 \ x + y + z &= 2 \end{aligned}\]We are going to construct an augmented matrix that represents this system.

Step 2: Set Up the Coefficients Matrix

List the coefficients of the variables \(x\), \(y\), and \(z\) from each equation as rows in a matrix. The coefficient matrix becomes:\[\begin{bmatrix}2 & 1 & 1 \3 & -4 & 2 \1 & 1 & 1\end{bmatrix}\]

Step 3: Add the Constants to Form the Augmented Matrix

Add the constants from the right-hand side of each equation as an additional column to the coefficients matrix. This final form is the augmented matrix:\[\begin{bmatrix}2 & 1 & 1 & | & 3 \3 & -4 & 2 & | & -7 \1 & 1 & 1 & | & 2\end{bmatrix}\]The vertical line (|) separates the coefficients from the constants in the augmented matrix.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. The goal when working with these systems is often to find the values of the unknowns that satisfy all the equations simultaneously.
Here's how to think about them:

• Variables (unknowns) are often represented as letters like \(x\), \(y\), and \(z\).
• Each equation in the system represents a specific condition that the variables must satisfy.

In our exercise, we have a system of three equations:

• \(2x + y + z = 3\)
• \(3x - 4y + 2z = -7\)
• \(x + y + z = 2\)

These equations are combined to form a system and can be analyzed to make the process of solving them more structured. It is important to interpret the meaning behind each equation and identify the relationships between the unknowns through these conditions.

Matrix Representation

In mathematics, the matrix representation is a preferred method for working with systems of equations. It simplifies the manipulation and solving of the equations. Here’s how a matrix helps in this context:

• Matrices are rectangular arrays of numbers that represent the coefficients from the system of equations.
• The augmented matrix includes an additional column for the constants from the equations.

The given system of equations can be expressed as an augmented matrix:\[\begin{bmatrix} 2 & 1 & 1 & | & 3 \3 & -4 & 2 & | & -7 \1 & 1 & 1 & | & 2 \end{bmatrix}\]Each row of this matrix corresponds to one of the equations, and each column corresponds to the coefficients of one variable. The vertical line is simply a visual separator to indicate the boundary between the coefficients and the constant terms of the equations. This matrix allows us to apply various matrix operations to find the solution for the system more efficiently.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations among other things. It provides the foundational tools needed to work with systems of equations systematically and efficiently. Here’s how it connects to our problem:

• Linear algebra uses concepts like matrices, determinants, and vector spaces to help understand and solve systems of linear equations.
• One of the main uses of linear algebra is to convert systems of equations into matrix form, a process which often leads to easier-to-find solutions.

In the context of solving systems of equations, linear algebra enables the conversion of complex systems into a structured form, such as our augmented matrix. This structured approach allows for the implementation of various solution techniques, such as Gaussian elimination, to efficiently solve the equations. Understanding linear algebra is crucial for grasping these methods and appreciating the power of matrix representation.