Problem 56 - College Algebra with Modeling and Visualization

Question

(Refer to Example \(6 .\) ) The augmented matrix is in reduced row-echelon form and represents a system of linear equations. If possible, solve the system. $$ \left[\begin{array}{lll|r} 1 & 0 & 1 & -2 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution is \(x = -2-z\), \(y = 5-3z\), \(z = z\) for any real number \(z\).

Step 1: Identify the Variables

Start by associating the columns of the matrix (before the vertical line) with variables. We will assign: Column 1 to variable \(x\), Column 2 to variable \(y\), and Column 3 to variable \(z\).

Step 2: Interpret the Augmented Matrix

The augmented matrix represents the system of linear equations where:Equation 1: \(x + z = -2\)Equation 2: \(y + 3z = 5\).

Step 3: Solve for One Variable from Each Equation

From Equation 1: \(x + z = -2\). Express \(x\) in terms of \(z\), so \(x = -2 - z\).From Equation 2: \(y + 3z = 5\). Express \(y\) in terms of \(z\), so \(y = 5 - 3z\).

Step 4: Express the General Solution

Since there are three variables and only two equations, \(z\) is a free variable. Express the solution in terms of \(z\) as follows: \[ x = -2 - z \]\[ y = 5 - 3z \]\[ z = z \]This indicates that \((x, y, z) = (-2 - z, 5 - 3z, z)\), where \(z\) can be any real number.

Key Concepts

Reduced Row-Echelon Form

Reduced Row-Echelon Form (RREF) is an important concept in solving systems of linear equations using matrices. When a matrix is in RREF, each leading entry is the only non-zero number in its row and is equal to 1, with all the columns containing a leading entry consisting entirely of zeros, except for that leading entry itself.

To reach RREF, systematic row operations are applied to simplify the matrix. The operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another.

• Each leading 1 in RREF appears to the right of the leading 1 in the row above.
• All entries in a column containing a leading 1 are zero except for the leading 1 itself.
• Any rows of zeros are at the bottom of the matrix.

Interpreting RREF makes it convenient to solve linear equations, as it reveals insights into solutions readily, allowing us to easily determine whether solutions are unique, dependent on free variables, or nonexistent.

Augmented Matrix

An Augmented Matrix is a crucial tool in representing systems of linear equations in a compact form. It consists of the coefficient matrix prepared from the variables' coefficients and an additional column representing the constants from the equations separated by a vertical line.

This visual layout simplifies the manipulation process, allowing for efficient use of elementary row operations. When an augmented matrix is brought to row-echelon or reduced row-echelon form, it greatly simplifies back-substitution and finding solutions.

• The left side (before the vertical line) includes the variable coefficients.
• The right-most column includes the constants from the right-hand side of the equations.

For example in our exercise, it allows transitioning directly from the matrix form to a set of simplified equations. Rows represent individual equations, while the columns (before the line) correspond to variables in the system.

Free Variable

A Free Variable arises in systems of equations when there are more variables than the number of equations, indicating that some variables cannot be solved uniquely. Instead, these variables can assume any value, providing infinite solutions, often expressed in terms of the free variable(s).

In the context of row-echelon or reduced row-echelon form, if a column associated with a variable does not have a leading 1 (pivot), then that variable is free.

• Free variables give flexibility in the solution, resulting in parametric representation.
• They appear when systems are underdetermined, leading to infinitely many solutions.

In the given solution, the variable \( z \) is identified as a free variable. It remains unsolved for a specific value and can be any real number, letting us express other variables \( x \) and \( y \) in terms of \( z \), thus representing a family of solutions rather than a single point.
Understanding and identifying free variables enables us to express the system's solutions comprehensively.