Problem 19 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{aligned} &4 a-b+2 c=0\\\ &\begin{array}{l} 2 a+b-c=-11 \\ 2 a-2 b+c=3 \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Write Augmented Matrix

First, express the system of equations as an augmented matrix. The system \(\begin{aligned} 4a-b+2c&=0\ 2a+b-c&=-11\ 2a-2b+c&=3 \end{aligned}\) can be represented by the augmented matrix:\[ \begin{bmatrix} 4 & -1 & 2 & | & 0 \ 2 & 1 & -1 & | & -11 \ 2 & -2 & 1 & | & 3 \end{bmatrix} \].

Step 2: Row Reduce to Row Echelon Form

Perform row operations to reduce the matrix to row echelon form: First, swap Row 1 with Row 2 and adjust Row 2 and Row 3 to eliminate 2 under the leading 4 in Row 1. The operations yield:\[ \begin{bmatrix} 2 & 1 & -1 & | & -11 \ 0 & -3 & 2 & | & 11 \ 0 & -3 & 2 & | & 3 \end{bmatrix} \].

Step 3: Simplify Further for Upper Triangular Form

Continue row operations to obtain an upper triangular form. Subtract Row 3 from Row 2, yielding:\[ \begin{bmatrix} 2 & 1 & -1 & | & -11 \ 0 & 0 & 0 & | & 8 \ 0 & -3 & 2 & | & 3 \end{bmatrix} \]. Since the second row simplifies to all zeros except for 8, this indicates an inconsistency.

Step 4: Analyze the System for Consistency

The row \([ 0 \, 0 \, 0 \, | \, 8 ]\) indicates that the system is inconsistent. This means there is no solution as it leads to a contradiction (0 cannot equal 8).

Key Concepts

Augmented Matrix

An augmented matrix is a useful tool for solving systems of linear equations. When setting up an augmented matrix, we compile the coefficients of variables and constants into a matrix form. This allows us to use matrix operations to find solutions efficiently.

For example, consider the system of equations:

• \(4a - b + 2c = 0\)
• \(2a + b - c = -11\)
• \(2a - 2b + c = 3\)

It can be represented as an augmented matrix:\[\begin{bmatrix} 4 & -1 & 2 & | & 0 \ 2 & 1 & -1 & | & -11 \ 2 & -2 & 1 & | & 3 \end{bmatrix}\]The vertical bar separates the coefficients of the variables on the left from the constants on the right.

Augmented matrices simplify systems of equations, making row operations and other modifications straightforward.

Row Echelon Form

Transforming a matrix into row echelon form makes it easier to solve the corresponding system of equations. In this form, each row starts with one or more zeros followed by a 'leading' entry, which is non-zero. The leading entry in each row is to the right of the leading entry in the row above it.

For our example matrix:\[\begin{bmatrix} 2 & 1 & -1 & | & -11 \ 0 & -3 & 2 & | & 11 \ 0 & -3 & 2 & | & 3 \end{bmatrix}\]To reach this form, we used row operations such as swapping rows and eliminating entries below pivot positions, focusing on creating zeros below the leading entries.

This step is crucial because reaching a row echelon form helps identify whether a solution exists or if the system is inconsistent, as further simplification can pinpoint contradictions in the equations.

Row Operations

Row operations are a series of manipulations performed on matrices that help to solve systems of linear equations. They maintain matrix equivalence while aiming to achieve a more 'solved' form. The fundamental row operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting multiples of one row to another row

In our exercise, these operations were applied to achieve row echelon form. For instance, swapping rows helped us position a simpler pivot at the top.

Once transformed, such operations can reveal critical properties of the job they perform, such as inconsistencies: in this case, simplifying a row to all zeros with a non-zero constant indicates there's no valid solution to the system. This is essential knowledge, as it saves time in problem solving by clearly showing when pursuing a solution is futile due to inherent contradictions.