Problem 30 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 3 x-4 y=2 \\ 9 x-12 y=6 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has infinitely many solutions since the determinant is zero and equations are dependent.

Step 1: Write the System as a Matrix Equation

The given system of equations can be represented as a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \), where: \[ A = \begin{bmatrix} 3 & -4 \ 9 & -12 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 2 \ 6 \end{bmatrix} \]

Step 2: Calculate the Determinant of Matrix A

To apply Cramer's rule, we first need to find the determinant of matrix \( A \). The determinant of \( A \) is calculated as follows:\[ \det(A) = \begin{vmatrix} 3 & -4 \ 9 & -12 \end{vmatrix} = (3)(-12) - (9)(-4) = -36 + 36 = 0 \].

Step 3: Analyze the Determinant

Since \( \det(A) = 0 \), the matrix \( A \) is singular, and the system of equations does not have a unique solution. This indicates that the system is either dependent (infinitely many solutions) or inconsistent (no solutions).

Step 4: Determine Solution Possibilities

Given that the matrix is singular and determinant is zero, the equations are scalar multiples of one another indicating that the system is dependent. Thus, the system has infinitely many solutions.

Key Concepts

Matrix Equation

A matrix equation is a way to represent a system of linear equations using matrices. It's a compact way to handle and solve equations. You start with the system of equations, for example:

• \(3x - 4y = 2\)
• \(9x - 12y = 6\)

Here, the unknowns are \(x\) and \(y\).
This system can be expressed in the matrix form \( A\mathbf{x} = \mathbf{b} \), where:

• \( A \) is the coefficient matrix: \[A = \begin{bmatrix} 3 & -4 \ 9 & -12 \end{bmatrix}\]
• \( \mathbf{x} \) is the variable matrix: \[\mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}\]
• \( \mathbf{b} \) is the outcome matrix: \[\mathbf{b} = \begin{bmatrix} 2 \ 6 \end{bmatrix}\]

If you multiply \(A\) by \(\mathbf{x}\) in matrix terms, you get \(\mathbf{b}\). This provides a structured way to represent the system.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible. In the case of the matrix \[A = \begin{bmatrix} 3 & -4 \ 9 & -12 \end{bmatrix}\],
the determinant \(\det(A)\) is calculated by a specific formula for 2x2 matrices:

• Multiply the elements on the main diagonal \((3)(-12)\).
• Multiply the elements on the other diagonal \((9)(-4)\).
• Subtract the second product from the first:

This gives:\[\det(A) = (3)(-12) - (9)(-4) = -36 + 36 = 0\]
A determinant of zero implies that the matrix does not have a unique solution when used in a system of equations. This critical step often determines whether we can solve a system using methods like Cramer's rule.

Singular Matrix

A singular matrix is one that does not have an inverse. This generally happens when the determinant of the matrix is zero. In our example:

• \( A = \begin{bmatrix} 3 & -4 \ 9 & -12 \end{bmatrix} \)

We calculated \(\det(A) = 0\), making it a singular matrix.
A singular matrix often indicates dependency in the system it represents:

• Equations might be similar or multiples of each other, suggesting they don't provide unique information.
• With a singular matrix \(A\), calculating the inverse to solve \( A\mathbf{x} = \mathbf{b} \) isn't possible.

Such matrices lead to systems with either no solutions or infinitely many solutions. This is because the equations are insufficiently independent to pinpoint a distinct solution.

System of Equations

A system of equations consists of two or more equations with the same unknowns. Solving such a system means finding values for the variables that satisfy all equations simultaneously. In our given example:

• \(3x - 4y = 2\)
• \(9x - 12y = 6\)

This set can have several different outcomes in terms of solutions:

• Unique Solution: When each equation gives unique information, and their graphs intersect at exactly one point.
• No Solution: When the equations represent parallel lines with no intersection (inconsistent).
• Infinitely Many Solutions: When the equations are the same line, meaning they overlap completely.

In this instance, because the determinant of the coefficient matrix is zero, we find infinitely many solutions. The equations are scalar multiples of one another, leading them to be dependent, which means they represent the same line.