Problem 46 - 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. $$ \left\\{\begin{array}{l} 9 x-3 y=6 \\ 3 x-y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent because the equations represent parallel lines that do not intersect.

Step 1: Write the System in Matrix Form

The given system of equations can be written in matrix form as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the constant matrix. Here, we have \( A = \begin{bmatrix} 9 & -3 \ 3 & -1 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \end{bmatrix} \), and \( B = \begin{bmatrix} 6 \ 8 \end{bmatrix} \). Thus, the matrix equation is \( \begin{bmatrix} 9 & -3 \ 3 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 6 \ 8 \end{bmatrix} \).

Step 2: Check for Matrix Invertibility

To solve for \( X \), the matrix \( A \) must be invertible. Calculate the determinant of \( A \): \( \text{det}(A) = (9)(-1) - (3)(-3) = -9 + 9 = 0 \). A determinant of zero indicates that \( A \) is not invertible, which means the system is either dependent or inconsistent.

Step 3: Analyze System for Dependence or Inconsistency

Given that the matrix is not invertible, check if the equations are multiples of each other. The first equation is \( 9x - 3y = 6 \) and the second equation is \( 3x - y = 8 \). Dividing the first equation by 3, we obtain \( 3x - y = 2 \), which is clearly different from the second equation \( 3x - y = 8 \). Therefore, the system is inconsistent as the two lines are parallel and do not intersect.

Key Concepts

System of Equations

In mathematics, a system of equations is a set of two or more equations with the same variables. The objective is to find the values of these variables that satisfy all the equations simultaneously. For instance, the given system involves two linear equations:

• Equation 1: \(9x - 3y = 6\)
• Equation 2: \(3x - y = 8\)

These types of systems can have several outcomes:

• A single unique solution, where the lines intersect at one point.
• Infinitely many solutions, where the equations describe the same line, meaning all points on the line are solutions.
• No solution, usually because the lines are parallel and never intersect.

Understanding the nature of the solutions helps us decide the next steps in solving the system.

Matrix Form

Transforming a system of equations into a matrix form makes it easier to handle, especially when using computational tools. In our example, the system:

• Equation 1: \(9x - 3y = 6\)
• Equation 2: \(3x - y = 8\)

can be rewritten in matrix form as \(AX = B\), where:

• \(A\) is the coefficient matrix: \(\begin{bmatrix} 9 & -3 \ 3 & -1 \end{bmatrix}\)
• \(X\) is the variable matrix: \(\begin{bmatrix} x \ y \end{bmatrix}\)
• \(B\) is the constant matrix: \(\begin{bmatrix} 6 \ 8 \end{bmatrix}\)

The matrix equation \(AX = B\) concisely summarizes the system, allowing us to use systematic procedures like Gaussian elimination or the inverse matrix method if \(A\) is invertible.

Determinant

The determinant is a special number calculated from a square matrix. It reveals important properties of the matrix and affects the solvability of the associated system. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\). In our case, the matrix is \(\begin{bmatrix} 9 & -3 \ 3 & -1 \end{bmatrix}\). The determinant is calculated as:

• \(9(-1) - (-3)(3) = -9 + 9 = 0\)

A determinant of zero indicates that the matrix is singular, meaning it cannot be inverted. This result leads us to conclude that the system does not have a unique solution, suggesting that the equations are either dependent or inconsistent.

Invertibility

Invertibility is a crucial concept in matrix algebra, particularly in solving systems of equations. A matrix is invertible if there exists another matrix that multiplies with it to yield the identity matrix, much like multiplying a number by its reciprocal equates to one. If a matrix is invertible, the system of equations can be solved exactly. To determine invertibility, calculate the determinant. If the determinant is non-zero, the matrix is invertible. In our problem, the determinant of matrix \(A\) is zero, illustrating that \(A\) is not invertible.
Given that \(A\) is not invertible, the system was examined for dependency. Upon checking, it was found that the equations were parallel and not multiples of one another, confirming the system as inconsistent. This results in no intersection point, and thus, no solution exists for the given system.