Problem 53 - 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. $$ \left[\begin{array}{rrrrr} 1 & 0 & 2 & \vdots & -4 \\ 0 & 1 & 1 & \vdots & 6 \\ 0 & 0 & 0 & \vdots & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution to the system of equations represented by the augmented matrix is \(x = -4 - 2z\), \(y = 6 - z\), and \(z\) can be any real number.

Step 1: Interpret the first row

The first row of the matrix corresponds to the equation \(1x + 0y + 2z = -4\), which simplifies to \(x + 2z = -4\). This equation tells us that the value of \(x\) is equal to the result of \(-4 - 2z\).

Step 2: Interpret the second row

The second row of the matrix corresponds to the equation \(0x + 1y + 1z = 6\), which simplifies to \(y + z = 6\). This equation tells us that the value of \(y\) is equal to the result of \(6 - z\).

Step 3: Interpret the third row

The third row of the matrix corresponds to the equation \(0x + 0y + 0z = 0\), which simplifies to \(0 = 0\). This equation tells us that there are no restrictions on the value of \(z\), meaning \(z\) can be any real number.

Key Concepts

Augmented Matrix

In solving a system of linear equations using matrix methods, the concept of an augmented matrix is central. An augmented matrix allows for the efficient representation of a system by combining the coefficients of the variables and constants into a single matrix. This is particularly useful in methods such as Gauss-Jordan elimination.

• The left side of the vertical divider (\( \vdots \)) in an augmented matrix contains the coefficients of the variables ( e.g., \(x, y, z\)).
• The right side contains the constants, which the equations equal.
• By transforming the augmented matrix into reduced row echelon form (RREF), we can easily find the solutions for the variables.

In our example, the augmented matrix represents the equations and helps identify relationships between the variables. This representation is crucial because it simplifies the manipulation required to find solutions.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. Systems can be consistent (with at least one solution) or inconsistent (no solutions). The Gauss-Jordan elimination process is one way to solve such systems by simplifying the relations between variables via row operations.

• Each equation in the system corresponds to a row in the matrix. It codifies how the variables relate to one another.
• In the example, each row represents an equation derived from the system we want to solve.
• The relations deduced from the matrix after elimination let us express certain variables in terms of others, often simplifying calculations.

Understanding these relationships as equations rather than rows of numbers is key to translating the matrix back into useful solutions.

Parametric Solution

An elegant aspect of solving systems of linear equations using matrices is the discovery of parametric solutions. When a variable is free, meaning it can take any real value, its behavior can be described using parameters.

• A free variable often arises in matrices with rows like \(0 = 0\), indicating a lack of constraints on that variable. In our example, \(z\) is such a variable.
• With a parametric solution, we express dependent variables in terms of the free variables. Here, \(x = -4 - 2z\) and \(y = 6 - z\), with \(z\) acting as the parameter.
• This form of solution is powerful as it encompasses infinitely many solutions all expressed compactly through the parameter(s).

Parametric solutions are not just about finding numbers; they provide a generalized form to predict how solutions change with differing parameters.