Problem 50 - College Algebra and Calculus: An Applied Approach

Question

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrr} 1 & 0 & 9 \\ 0 & 1 & -3 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution of the system of equations is \(x = -9z\), \(y = 3z\), where \(z\) is a free variable that can have any real number as its value.

Step 1: Analyzing the reduced matrix

Identify the reduced matrix which is presented in a row echelon form. This form of the matrix represents the system of equations as follows:\[\begin{align*}x + 9z &= 0\\y -3z &= 0\end{align*}\]

Step 2: Isolating Variables

Now, isolate the variables \(x\), \(y\), and \(z\) from the two equations, to get their values. From the first equation, isolate \(x\): \(x = -9z\). And from the second equation, isolate \(y\): \(y = 3z\). Note that, since the variable \(z\) isn't represented in the matrix, it's considered free, which means it can have any real value.

Key Concepts

Augmented Matrix

In linear algebra, matrices are powerful tools to simplify systems of equations. An augmented matrix is a type of matrix that combines the coefficients from a system of linear equations with their corresponding constants. This setup creates an easy-to-manage format when performing operations like Gauss-Jordan elimination.

The augmented matrix includes:

• The coefficients of each variable in each equation.
• The constants from each equation, typically placed after a vertical bar or at the end of the row.

For example, a system of equations\[x + 2y = 3\4x - y = 8\]can be represented as an augmented matrix:\[\begin{bmatrix} 1 & 2 & | & 3 \ 4 & -1 & | & 8 \end{bmatrix}\]This format allows for the systematic application of operations to simplify and solve the equations.

Linear Equations

Linear equations are fundamental algebraic expressions that represent straight lines when graphed. They consist of variables raised to the first power and constants. Each equation forms a straight-line relationship between the variables.

Key characteristics of linear equations:

• They do not include variables raised to a power higher than one.
• They can have multiple variables like in the system above: \( x, y, z \).
• Solutions to these equations can be visualized graphically as points or intersections of lines.

The ultimate goal is to find the values of the variables that satisfy all equations in the system. These solutions may appear as single points, infinite solutions, or no solution, depending on the relationship between the lines.

Row Echelon Form

Row Echelon Form (REF) is a matrix state that simplifies solving equations. This form is achieved through operations that create a stair-step pattern of zeros beneath the leading coefficients (the first non-zero number from the left) in each row.

Characteristics of row echelon form:

• All zero rows are at the bottom of the matrix.
• The leading coefficient of a nonzero row is always to the right of the leading coefficient of the row above it.
• Each leading coefficient is 1 (or can be easily changed to 1).

This state significantly facilitates solving a system of equations. By continuing these operations, especially using Gauss-Jordan elimination, one can further reduce the matrix to reduced row echelon form, where variables can directly be isolated, leading to the system solution. This process is evident in the given exercise, where two equations were formed: \(x + 9z = 0\) and \(y - 3z = 0\), allowing us to easily solve for \(x\) and \(y\).