Problem 13 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The augmented matrix is \( \begin{bmatrix} 1 & 1 & 0 & | & 2 \\ 0 & 2 & 1 & | & -4 \\ 0 & 0 & 1 & | & 2 \end{bmatrix} \).

Step 1: Identify the System of Equations

First, we identify the given system of equations. The system is: \( x + y = 2 \), \( 2y + z = -4 \), and \( z = 2 \).

Step 2: Recognize the Variables and Constants

Next, recognize the variables \(x\), \(y\), and \(z\), and the constants are \(2\), \(-4\), and \(2\). We will use these to form the matrix.

Step 3: Write the Coefficients of the Equations

For each equation, extract the coefficients of the variables: \(x + y = 2\) has coefficients \([1, 1, 0]\), \(2y + z = -4\) has coefficients \([0, 2, 1]\), and \(z = 2\) has coefficients \([0, 0, 1]\).

Step 4: Form the Augmented Matrix

Using the coefficients identified in Step 3, write the augmented matrix where each row represents an equation and the last column represents the constant terms. This gives us the matrix: \[ \begin{bmatrix} 1 & 1 & 0 & | & 2 \ 0 & 2 & 1 & | & -4 \ 0 & 0 & 1 & | & 2 \end{bmatrix} \]

Key Concepts

System of Equations

A system of equations is a group of two or more equations that contain the same set of variables. These systems can be solved to find the values of these variables that satisfy all equations simultaneously. In the given original exercise, the system of equations consists of three equations:

• Equation 1: \(x + y = 2\)
• Equation 2: \(2y + z = -4\)
• Equation 3: \(z = 2\)

Each equation represents a linear relationship between the variables \(x\), \(y\), and \(z\). While this example does not solve the system, understanding how to write it in a useful form like an augmented matrix is an important step.
These systems can be represented graphically, but more complex systems benefit from algebraic methods, especially when using matrices.

Coefficients

Coefficients are the numerical values that multiply the variables in an equation. They tell us how much of each variable is present in an equation. In our system of equations, each equation has its own set of coefficients:

• The first equation, \(x + y = 2\), has coefficients \([1, 1, 0]\) for \(x\), \(y\), and \(z\), respectively.
• The second equation, \(2y + z = -4\), has coefficients \([0, 2, 1]\).
• The third equation, \(z = 2\), has coefficients \([0, 0, 1]\).

It is important to note that even variables absent from an equation still have an implied coefficient of zero. Understanding coefficients is crucial because they allow the construction of the augmented matrix by filling rows with these values. This simplifies the handling of linear systems, especially in computational applications.

Variables in Matrices

When dealing with matrices, we often talk about variables in terms of their placement relative to other variables and constants. In an augmented matrix, variables are translated into a structured format that aligns neatly with their coefficients:

• The variables become part of a matrix where each row corresponds to one equation.
• Columns represent each variable \(x\), \(y\), and \(z\), with the final column standing for the constants on the right-hand side of each equation.

To illustrate, the matrix forms as follows:\[ \begin{bmatrix} 1 & 1 & 0 & | & 2 \ 0 & 2 & 1 & | & -4 \ 0 & 0 & 1 & | & 2 \end{bmatrix} \]This matrix setup captures all the information needed to eventually solve the system using methods such as Gaussian elimination or matrix row operations. By augmenting the matrix with the constants, you've got a compact, efficient way to work through and solve complex systems of equations.