Problem 30 - Precalculus Mathematics for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{r} x \quad+3 z=3 \\ 2 x+y-2 z=5 \\ -y+8 z=8 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent.

Step 1: Write the system in matrix form

The given system of equations can be expressed in matrix form as: \[\begin{bmatrix}1 & 0 & 3 & | & 3 \2 & 1 & -2 & | & 5 \0 & -1 & 8 & | & 8\end{bmatrix}\]

Step 2: Use Gaussian elimination

Perform row operations to simplify the matrix. First, eliminate the entry below the first pivot (2 from the second row) by replacing Row 2 with Row 2 minus 2 times Row 1:\[\begin{bmatrix}1 & 0 & 3 & | & 3 \0 & 1 & -8 & | & -1 \0 & -1 & 8 & | & 8\end{bmatrix}\]

Step 3: Simplify second and third rows

To eliminate the second entry in the third row, add Row 2 to Row 3:\[\begin{bmatrix}1 & 0 & 3 & | & 3 \0 & 1 & -8 & | & -1 \0 & 0 & 0 & | & 7\end{bmatrix}\]

Step 4: Identify the system type

Observe the last row: \(0x + 0y + 0z = 7\), which is a contradiction (0 ≠ 7). This means the system of equations is inconsistent.

Key Concepts

Inconsistent System

An inconsistent system of equations is a collection of linear equations that do not have any solutions. This occurs when the equations represent parallel lines, which never intersect. Therefore, it is impossible to find a common solution that satisfies all equations simultaneously. In the context of the exercise, the system is confirmed to be inconsistent because the final matrix row translates to the statement \(0 = 7\). Since this statement is always false, no set of values for \(x\), \(y\), and \(z\) would make the equation true.
When working with systems of equations, an inconsistent system is easily recognized by such contradictions arising during simplifying processes like Gaussian elimination. These contradictions indicate when equations essentially "fight" each other, representing parallel but non-intersecting planes or lines in geometrical contexts.

Dependent System

A dependent system is the opposite of an inconsistent system. Here, the system has an infinite number of solutions and is characterized by equations that are scalar multiples of each other, or where one equation can be derived from another. In essence, each equation in a dependent system does not contribute additional information, as any solution for one equation will automatically satisfy the other equations.
Mathematically, this often results in a row of zeros (0 = 0) within the augmented matrix, showing no new restrictions are imposed by that row. If, in our exercise, we found a row reading \(0x + 0y + 0z = 0\), it would indicate dependency. We would then parameterize the solutions to express them in terms of free variables, giving us a family of solutions.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It transforms a set of equations into a simpler form to easily find solutions or determine if solutions exist. The process involves performing row operations to achieve a row-echelon form or even a reduced row-echelon form. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another.
In the exercise, Gaussian elimination was used to determine the system's inconsistency. By simplifying the augmented matrix, we ended up with a row that dictated a contradiction. This is a classic use of Gaussian elimination, which shows its power not only in finding solutions but also in determining the nature of the system—whether it is inconsistent, dependent, or consistent with a unique solution. Overall, mastering Gaussian elimination equips one with a flexible tool for tackling varied linear systems efficiently.