Problem 12 - Intermediate Algebra

Question

Fill in the blanks. Fill in the blank. If the denominator determinant \(D\) for a system of equations is \(0,\) the equations of the system are dependent or the system is ___.

Step-by-Step Solution

Answer

inconsistent

Step 1: Understanding the System of Equations

A system of equations with a denominator determinant \( D = 0 \) means the system lacks an invertible matrix form. This indicates that the equations are not independent.

Step 2: Defining Dependent and Inconsistent Systems

A system with determinant \( D = 0 \) can either have equations that are multiples of one another (dependent) or they could never intersect at a common point (inconsistent).

Step 3: Finalizing the Blank

Considering the nature of a zero determinant, if the equations are not independent, the system will either be dependent or inconsistent. Thus, the blank should be filled with 'inconsistent.'

Key Concepts

Determinant

In the realm of systems of equations, the determinant plays a pivotal role. A determinant is a special number that can be calculated from a square matrix. It provides valuable information about the matrix itself. In the context of solving systems of equations, particularly linear systems, the determinant helps us understand whether the system has unique solutions or not. When the determinant (denoted as \( D \)) is nonzero, the system usually has a unique solution, as this means the matrix is invertible.
However, if \( D = 0 \), it signifies a crucial change. A zero determinant reveals that the matrix cannot be inverted, thereby indicating a lack of unique solutions.

• This could mean the equations are dependent, as they are just different forms of the same equation.
• It might also mean they do not intersect, making the system inconsistent.

Understanding the determinant helps us foresee these possibilities and choose the right methods to analyze the system further.

Dependent System

A dependent system of equations arises when the equations in the system describe the same line or plane. This means the equations are basically overlapping or coinciding. In mathematical terms, their graphs lie one over the other, showing infinite solutions.
A typical feature of dependent systems is that they can be derived from one another by multiplying by a constant. When dealing with matrices, this situation corresponds to a determinant of zero, indicating that the rows (or equations) are linearly dependent.

• For example, if you have two equations: \( 2x + 3y = 6 \) and \( 4x + 6y = 12 \), the second equation is just a multiple of the first.
• This results in the system having infinite solutions because every solution to one equation is also a solution to the other.

Recognizing dependent systems is crucial in simplifying and solving complex systems efficiently.

Inconsistent System

When grappling with systems of equations, an inconsistent system is one that does not have any solutions. This occurs when the equations in the system represent parallel lines or planes. These do not intersect at any point, reflecting the absence of a common solution.
From a linear algebra perspective, an inconsistent system is indicated by a zero determinant and a conflict of values or constants that the equations should equal. In simpler terms, the system tries to satisfy equations that are unsatisfiable together.

• Consider the equations \( x + y = 2 \) and \( x + y = 5 \). These equations do not have any shared solutions because they basically propose different intercepts under the same slope.
• This disparity highlights why the equations are inconsistent, resulting in no points of intersection.

Identifying and understanding inconsistent systems allows one to avoid futile attempts at finding non-existent solutions and focus on potential areas for adjustments or modifications.