Problem 34 - 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. $$ \left\\{\begin{array}{l} -6 x+12 y=10 \\ 2 x-4 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions.

Step 1: Formulate the System as a Matrix Equation

First, represent the given system of equations in the form of a matrix equation. The system is: \[ \begin{align*} -6x + 12y &= 10, \2x - 4y &= 8. \end{align*} \] This can be written in matrix form \( A\mathbf{x} = \mathbf{b} \), where: \[ A = \begin{bmatrix} -6 & 12 \ 2 & -4 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 10 \ 8 \end{bmatrix}. \]

Step 2: Find the Determinant of the Coefficient Matrix

Calculate the determinant of the matrix \( A \) to determine if it is invertible. The determinant is calculated as follows: \[ \det(A) = (-6)(-4) - (12)(2) = 24 - 24 = 0. \] Since the determinant is zero, the matrix \( A \) is not invertible.

Step 3: Interpret the Determinant Result

Since the determinant of \( A \) is zero, it means the matrix is singular, and thus, the system of equations does not have a unique solution. This implies that the system could either be dependent or inconsistent.

Step 4: Analyze Consistency by Comparing Rows

To determine if the system is dependent or inconsistent, observe that the second equation is a scalar multiple of the first: \[ 2(2x - 4y) = 8x - 16y = 0, \] and \[ (-6)(2x - 4y) = -12x + 24y = 0. \] These equivalent forms suggest that the equations in the system are dependent rather than inconsistent.

Step 5: Conclude the Nature of the System

Since both equations are scalar multiples, they represent the same line. Therefore, the system is dependent, indicating there are infinitely many solutions.

Key Concepts

Dependent System of Equations

In the context of systems of equations, a dependent system is one where all equations represent the same line or plane. This means that rather than intersecting at a unique point or having no intersection at all, the graphs of the equations overlap completely. This is notable because each equation in the dependent system is essentially a scaled version of the others.
In the exercise given, the equations are:

• \(-6x + 12y = 10\)
• \(2x - 4y = 8\)

By restructuring or scaling these equations, you can see they represent the same line. Therefore, instead of intersecting at one unique point, they lie on the same line with infinitely many solutions that satisfy both equations simultaneously. This characteristic is central to understanding dependent systems.

Determinant of a Matrix

The determinant is a special number calculated from a square matrix. It provides critical information about the matrix, such as whether it is invertible or singular. Specifically, for a 2x2 matrix \(A\) of the form \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is computed as \( ad - bc \).
The determinant tells us about the linear dependence of the rows or columns:

• A non-zero determinant means the matrix is invertible, indicating a unique solution to the system of equations.
• A zero determinant indicates the matrix is singular, meaning it may have either no solution or infinitely many, as it lacks "full rank."

In the given exercise, the determinant was found to be zero, signaling that the system cannot have a unique solution because the matrix is not invertible.

Matrix Representation of Systems

To solve a system of equations using matrices, those equations must be expressed in matrix form. This involves organizing the coefficients into a matrix, and the variables and constants into vectors. The general representation is:
\(A\mathbf{x} = \mathbf{b}\), where:

• \(A\) is the coefficient matrix, containing the coefficients of the variables.
• \(\mathbf{x}\) is a column vector representing the variables.
• \(\mathbf{b}\) is a column vector of the constants from the right-hand side of the equations.

In the case of the exercise above, matrix \(A\) is \(\begin{bmatrix} -6 & 12 \ 2 & -4 \end{bmatrix}\), \(\mathbf{x}\) is \(\begin{bmatrix} x \ y \end{bmatrix}\), and \(\mathbf{b}\) is \(\begin{bmatrix} 10 \ 8 \end{bmatrix}\).
This clear organization aids in analyzing and solving the system using matrix operations.

Inconsistent Systems of Equations

An inconsistent system of equations is one that has no solution. This typically occurs when the equations represent parallel lines or planes that never meet – in other words, they have no points in common.
A key tool in identifying inconsistent systems is analyzing the matrix determinant. For instance:

• If the determinant of a coefficient matrix is non-zero, the system has a unique solution, so it's consistent.
• A zero determinant can imply a lack of a unique solution and requires further inspection to determine if the system is dependent or inconsistent.

In the given exercise, despite the zero determinant suggesting the possibility of an inconsistent system, further analysis revealed that the equations were dependent, representing the same line, thus pointing to infinite solutions rather than no solutions.