Problem 27 - Intermediate Algebra

Question

Suppose that the augmented matrix of a system of three equations in three variables can be changed to the following matrix. $$ \left[\begin{array}{rrr:r} 1 & 1 & -2 & 4 \\ 0 & -5 & 11 & -13 \\ 0 & 0 & 0 & -9 \end{array}\right] $$ What can be said about the solution set of the system?

Step-by-Step Solution

Answer

The system has no solutions due to the inconsistent matrix row.

Step 1: Identify the Matrix Form

The given matrix is an augmented matrix representing a system of linear equations. Each row in the matrix can be interpreted as an equation of the form ax + by + cz = d.

Step 2: Interpret the Rows

The first row, \( 1x + 1y - 2z = 4 \), represents a valid equation. The second row, \( 0x - 5y + 11z = -13 \), is also a valid equation. The third row, \( 0x + 0y + 0z = -9 \), leads to the equation \( 0 = -9 \), which is a contradiction.

Step 3: Analyze the Contradiction

The equation \( 0 = -9 \) contradicts the nature of the system, as it suggests an inconsistency. Such inconsistency in one of the rows of an augmented matrix implies that the system has no solutions. This is because no combination of values for x, y, and z can satisfy this impossible equation.

Step 4: Conclusion on Solution Set

Since the system includes a row representing a contradiction, it is inconsistent. An inconsistent system means that there is no intersection among the planes represented by the equations, so there are no solutions.

Key Concepts

System of Linear Equations

Understanding a system of linear equations is crucial for solving various mathematical problems. A system of linear equations consists of two or more equations with the same set of variables. These equations can represent many real-world scenarios, such as intersections of lines or planes.
To work with such systems, we use augmented matrices, which are mathematical tools that compactly represent the system's equations. Each row in an augmented matrix corresponds to an equation. For example, in the augmented matrix given in the exercise, each row serves as one of the three linear equations involving variables \(x\), \(y\), and \(z\).

• The first step in analyzing a system is translating the matrix back into its equations, like converting rows into equations (e.g., \(1x + 1y - 2z = 4\)).
• Each equation represents a plane in three-dimensional space. Understanding how these planes interact helps determine the number of solutions the system has.

By examining the matrix structure, you can identify patterns that indicate different types of solutions. The language of matrices provides a streamlined method for this analysis.

Inconsistent System

An inconsistent system refers to a system of equations that can't be solved because the equations contradict each other. In mathematical terms, it means there's no set of values that can simultaneously satisfy all equations in the system.
In the provided exercise, the third row of the augmented matrix is at the heart of the inconsistency. This row corresponds to the equation \(0x + 0y + 0z = -9\). Here, the left side (\(0\)) doesn't equal the right side (\(-9\)), which is a clear contradiction. No values of \(x\), \(y\), or \(z\) can make this equation true.
This type of inconsistency translates visually into non-intersecting planes. Instead of intersecting at a point, line, or plane, the planes represented by the equations in the system do not meet at all.

• Identifying an inconsistent system helps avoid futile attempts at finding solutions.
• Recognizing such rows in an augmented matrix is critical to understanding the system's behavior.

An important skill is to quickly spot these contradictions to conclude that the system has no solutions.

No Solutions

When dealing with systems of linear equations, one possible outcome is that the system has no solutions. This occurs when the equations describe situations where there's no common solution.
In an augmented matrix, no solution arises if you find a contradiction like \(0 = -9\), as shown in the original exercise. Here, mathematical rules tell us that no such condition can exist legitimately. Consequently, the system of equations is considered unsolvable under these terms.
Geometrically, when no solution exists, it means the planes do not intersect in any line, point, or plane, essentially meaning they are completely disjoint in space. It's not just a matter of being far apart but fundamentally unable to meet.

• When analyzing, look for rows that simplify to impossible equations.
• This outcome describes a system so disparate that it confirms the absence of a viable solution set.

Understanding when a system has no solutions is essential for correctly interpreting results and knowing when to stop computations.