Problem 22 - College Algebra with Modeling and Visualization

Question

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & 0 & -4 & \frac{3}{4} \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & -3 \end{array}\right] $$

Step-by-Step Solution

Answer

No solution due to the inconsistent equation.

Step 1: Understand the Matrix Form

The given matrix represents a system of linear equations. Since it's in row-echelon form, it corresponds to the equations: \( x - 4z = \frac{3}{4} \), \( y + 2z = 1 \), and \( 0 = -3 \).

Step 2: Analyze the Inconsistency

The third equation, \( 0 = -3 \), is a contradiction and indicates that there is no solution to this system, as a row of zeros equating to a nonzero number is impossible.

Key Concepts

Augmented Matrix

An augmented matrix is a helpful tool when dealing with systems of linear equations. It combines the coefficient matrix with the constants from each equation into one extendable matrix format. This single matrix then helps apply mathematical operations efficiently. By examining the matrix, you also gain insight into the structure of the system.

• The coefficients from each equation fill out the left side of the matrix.
• The constants that the equations equal are placed on the right side, separated by a vertical line.

In our exercise, your matrix:\[\begin{bmatrix}1 & 0 & -4 & | & \frac{3}{4} \0 & 1 & 2 & | & 1 \0 & 0 & 0 & | & -3\end{bmatrix}\]represents a set of linear equations. The numbers to the left of the vertical line are coefficients of variables, and the numbers on the right are the constants.

Row-Echelon Form

The row-echelon form is a specific arrangement of a matrix that simplifies the process of solving linear systems. To be in row-echelon form, a matrix

• has all zero rows, if any, at the bottom,
• has a leading one in each row, meaning the first nonzero entry from the left in a nonzero row is 1,
• ensures the leading ones move further to the right as you progress down the matrix.

This form is crucial because it sets the stage for easy backward substitution. Observing our given matrix, \[\begin{bmatrix}1 & 0 & -4 & \vdots & \frac{3}{4} \0 & 1 & 2 & \vdots & 1 \0 & 0 & 0 & \vdots & -3\end{bmatrix}\] readily identifies the structured system of equations clearly and shows where any inconsistencies may arise.

Backward Substitution

Backward substitution is a method used to solve linear systems once the matrix is in row-echelon form. It offers a step-by-step way to find variable values by starting from the bottom row and moving upwards.

• Begin with the equation containing the fewest variables, usually the last row.
• Use this discovered value to solve for other variables in previous equations.

Unfortunately, in our exercise, the last row presented a row of zeros on the left and a nonzero constant on the right, making substitution unviable. This contradiction leads us to an obvious sign of an issue with solutions.

Inconsistent System

An inconsistent system in linear algebra refers to a system of equations that has no solutions. Such a system arises when the equations contradict each other.

• If you reach a point where an equation reads something impossible like zero equals a nonzero number, the system is inherently flawed and unsolvable.
• In terms of an augmented matrix, this inconsistency appears as a row of zeros on the left equating to a nonzero value on the right side of the augmented part.

In our problem, the row \(0 = -3\) clearly signals this inconsistency, showing us conclusively that there are no solutions for this set of equations.