Problem 13 - A Graphical Approach to College Algebra

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 \(\begin{bmatrix} 1 & 1 & 0 & \vert & 2 \\ 0 & 2 & 1 & \vert & -4 \\ 0 & 0 & 1 & \vert & 2 \end{bmatrix}\).

Step 1: Identify Coefficients

Examine each equation to identify the coefficients of the variables \(x\), \(y\), and \(z\). For any variable not present in the equation, its coefficient is 0. The equations are: \(x + y = 2\), \(2y + z = -4\), and \(z = 2\).

Step 2: Write the Coefficient Matrix

Construct a matrix using the coefficients found in Step 1. Each row of the matrix corresponds to an equation, and each column corresponds to the coefficients of \(x\), \(y\), and \(z\) in that order. The matrix is: \[\begin{bmatrix}1 & 1 & 0 \0 & 2 & 1 \0 & 0 & 1\end{bmatrix}\].

Step 3: Identify Constant Terms

Note the constant term (the right-hand side of each equation): 2 for the first equation, -4 for the second, and 2 for the third.

Step 4: Write the Augmented Matrix

The augmented matrix is formed by adding an extra column to the coefficient matrix containing the constant terms. The augmented matrix looks like this: \[\begin{bmatrix}1 & 1 & 0 & \vert & 2 \0 & 2 & 1 & \vert & -4 \0 & 0 & 1 & \vert & 2\end{bmatrix}\].

Key Concepts

Matrix Algebra

Matrix algebra is a powerful mathematical tool used in various areas, like solving systems of linear equations. It involves performing operations on matrices, which are rectangular arrays of numbers. In our exercise, we extract coefficients from linear equations to form a matrix. This initial step simplifies handling complex equations as it allows the use of matrix-specific techniques.

The augmented matrix combines both coefficients and constants of the equations into one single matrix. This provides a visual and organized method to approach systems of equations:

• The rows represent each equation.
• The columns correspond to each variable, followed by the constants on the right side.

It's a structured and efficient approach, making further operations like matrix row reduction applicable. Understanding these foundations makes it easier to address more complex algebraic problems.

Coefficients

Coefficients are the numerical factors of variables in an equation. Identifying coefficients is a fundamental step when converting equations into their matrix forms. In matrix algebra, each coefficient is critical as it forms the elements of the matrix used in calculations.

For the given system:

• The equation \(x + y = 2\) has coefficients 1 for \(x\), 1 for \(y\), and 0 for \(z\) since \(z\) is absent.
• For \(2y + z = -4\), the coefficients are 0 for \(x\), 2 for \(y\), and 1 for \(z\).
• Finally, \(z = 2\) has coefficients 0 for \(x\), 0 for \(y\), and 1 for \(z\).

This highlights the importance of understanding coefficients, as they directly impact the construction and manipulation of matrices. Accurate extraction of coefficients ensures the solution process is correct from the start.

Linear Equations

Linear equations form the backbone of many algebra problems. They consist of terms that are either constants or products of constants and a single variable. In our example, the equations are simple expressions of \(x\), \(y\), and \(z\).

The goal is often to find values of these variables that satisfy all equations simultaneously. This is where converting them into an augmented matrix becomes advantageous. A set of linear equations is systematically organized into a matrix form that makes analytical or computational solutions more efficient.

A good understanding of linear equations helps in recognizing patterns and simplifying complex mathematical problems. When each equation is correctly interpreted and translated into matrix form, solving the system becomes a matter of applying the right matrix method, like Gaussian elimination or using matrix inversions.