Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.2 x+0.1 y-0.3 z=0.2 \\ 0.8 x+0.4 y-1.2 z=0.1 \\ 1.6 x+0.8 y-2.4 z=0.2 \end{array} $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Write the Augmented Matrix

First, construct the augmented matrix from the given system of equations:\[\begin{bmatrix}0.2 & 0.1 & -0.3 & | & 0.2 \0.8 & 0.4 & -1.2 & | & 0.1 \1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]

Step 2: Make the First Pivot 1

To simplify the first row, we want to make the first pivot (0.2) into 1. We do this by dividing the entire first row by 0.2:\[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1 \0.8 & 0.4 & -1.2 & | & 0.1 \1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]

Step 3: Eliminate First Column Below Pivot

To zero out the elements below the first pivot, perform row operations. For \( R_2 \), use: \( R_2 - 0.8R_1 \). For \( R_3 \), use: \( R_3 - 1.6R_1 \):\[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1 \0 & 0 & 0 & | & -0.7 \0 & 0 & 0 & | & -1.4 \\end{bmatrix}\]

Step 4: Analyze the Reduced Rows

Notice the second and third rows became zero in the coefficients part but differ in the augmented component. This indicates no solutions exist as the system is inconsistent.

Key Concepts

Augmented Matrix

An augmented matrix is a very useful tool when solving systems of equations, particularly using methods like Gaussian elimination.
To form an augmented matrix, you take the coefficients of each variable from all the equations and write them in matrix form. Then, you differentiate the augmented part with a vertical line and add the constants on the right side of this line.
For the given exercise, the augmented matrix looked like this: \[\begin{bmatrix}0.2 & 0.1 & -0.3 & | & 0.2 \0.8 & 0.4 & -1.2 & | & 0.1 \1.6 & 0.8 & -2.4 & | & 0.2\end{bmatrix}\]
This matrix format allows us to apply operations more systematically and track changes needed for Gaussian elimination methods.

System of Equations

A system of equations is essentially a set of two or more equations with the same variables. When we talk about solving systems, we aim to find values for these variables that satisfy all given equations simultaneously.
The given problem involves a system with three equations and three unknowns: \(x\), \(y\), and \(z\). By using Gaussian elimination, we apply row operations to simplify this system until it's easiest to solve or identify its nature.

• The equations interact to describe potential lines or planes in geometric space.
• Our goal is either a unique solution, infinitely many solutions, or to determine no solution exists.

Understanding these dynamics is crucial for choosing appropriate solution methodologies like Gaussian elimination.

Inconsistent System

An inconsistent system, such as the one encountered in this exercise, refers to a set of equations with no possible set of solutions. This occurs when equations represent parallel lines or planes that never intersect.
During Gaussian elimination, inconsistency becomes apparent when rows in the matrix become contradictory, like in our case: \[\begin{bmatrix}0 & 0 & 0 & | & -0.7 \0 & 0 & 0 & | & -1.4\end{bmatrix}\]
This reduced row form shows zero coefficients with non-zero constants, signaling a logical inconsistency — implying no intersection exists amongst the defined mathematical surfaces described by the equations.
Therefore, no solutions satisfy this system, highlighting its inconsistency.

Pivot Element

A pivot element is a non-zero element in a matrix that is used to clear out other elements in the column, essentially making solving easier.
In Gaussian elimination, you transform selected elements into a row-echelon form, establishing pivot elements in each row sequentially. For example, in the exercise, the first pivot was initially 0.2, which was transformed into 1 by dividing the entire row by this number: \[\begin{bmatrix}1 & 0.5 & -1.5 & | & 1 \end{bmatrix}\]
As a rule of thumb, make the pivot into 1, and eliminate other elements in its column to 0, simplifying further computations in the elimination process.

• Different rows have their distinct pivot elements.
• Maintaining them helps progress towards reducing the matrix.

It’s through these organized steps that each variable can eventually be systematically solved or determined unsolvable.