Problem 45 - Intermediate Algebra

Question

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. $$ \left\\{\begin{array}{l} 8 x-2 y=4 \\ 4 x-y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Write the System in Matrix Form

Convert the given system of equations into the matrix equation form \( A \mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constants matrix. For the given system, this will be:\[ A = \begin{bmatrix} 8 & -2 \ 4 & -1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 4 \ 2 \end{bmatrix} \]

Step 2: Calculate the Determinant of Coefficient Matrix

Determine the determinant of the coefficient matrix \( A \). This will help us analyze the system for dependency or inconsistency. Calculate it as follows:\[ \text{det}(A) = \begin{vmatrix} 8 & -2 \ 4 & -1 \end{vmatrix} = (8)(-1) - (-2)(4) = -8 + 8 = 0 \]

Step 3: Analyze the Determinant

The determinant of the matrix \( A \) is zero. This means that the coefficient matrix does not have an inverse, indicating that the system of equations may be either dependent or inconsistent.

Step 4: Verify Dependency or Inconsistency

Since the determinant is zero, we further analyze the system by checking if the equations are scalar multiples of each other. Divide the first equation by 2 to get: \[ 4x - y = 2 \]This is exactly the second equation, which means the system is dependent and has infinitely many solutions.

Key Concepts

Dependent System

A dependent system of equations is a set of equations where all the equations relate to each other or are essentially the same equation written in various forms. In simpler terms, each equation does not add new information to the system but rather restates the previous ones in different ways.
Here's how you can identify a dependent system:

• If you can write one equation as a scalar multiple of another, then the system is dependent.
• A dependent system often results from parallel lines if graphed, indicating they are the same line and thus have infinitely many solutions.

In the exercise provided, after rewriting the first equation, we found it was the same as the second equation, proving the system is dependent with infinitely many solutions. This happens when the equations are not independent, leading to overlap in the information they provide.

Inconsistent System

An inconsistent system of equations has no solution at all. This occurs when the equations represent parallel lines that never intersect.
The main features of an inconsistent system are:

• Equations in an inconsistent system will not share a common solution.
• Graphically, the lines represented by these equations are parallel and do not meet.

In detecting an inconsistent system, examine the coefficients of the variables. If the lines are parallel (the ratios of the coefficients of x and y are the same, but the constants differ), the system is inconsistent. However, in our original problem set, we observed a dependent system rather than an inconsistent one.

Determinant of a Matrix

The determinant of a matrix provides a scalar value that offers insights into the matrix's properties. For a 2x2 matrix:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The determinant is given by the formula: \[ ext{det}(A) = ad - bc\]The determinant is crucial when working with systems of equations:

• A non-zero determinant implies the matrix has an inverse and the system is independent with a unique solution.
• A zero determinant indicates the matrix does not have an inverse, and the system may be either dependent or inconsistent.

In our exercise, we computed the determinant and found it to be zero, signaling the system might be dependent or inconsistent. Further analysis showed the equations were multiples of one another, confirming dependency.