Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&3 x+2 y=4\\\&6 x+4 y=8\end{aligned}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions because the equations are identical.

Step 1: Write the System in Matrix Form

The given system of equations is: \(3x + 2y = 4\) and \(6x + 4y = 8\). Write this system in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the variable matrix, and \(\mathbf{b}\) is the constant matrix. We have: \[A = \begin{pmatrix} 3 & 2 \ 6 & 4 \end{pmatrix}, \, \mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix}, \, \mathbf{b} = \begin{pmatrix} 4 \ 8 \end{pmatrix}\]

Step 2: Calculate the Determinant of A

To use Cramer's rule, we first need the determinant of the coefficient matrix \(A\). The determinant \(D\) of \(A\) is given by: \[D = \begin{vmatrix} 3 & 2 \ 6 & 4 \end{vmatrix} = (3)(4) - (2)(6) = 12 - 12 = 0\]

Step 3: Analyze the Determinant

Since the determinant \(D = 0\), Cramer's rule cannot be used to solve this system. A zero determinant indicates that the system may be dependent or inconsistent. We need to use another method to find a solution.

Step 4: Use Another Method - Check for Consistency

Since Cramer's rule doesn't apply, check if the system might have infinitely many solutions or none. Compare the second equation to the first: The second equation \(6x + 4y = 8\) is just twice the first equation \(3x + 2y = 4\). This suggests that the system is dependent, indicating infinitely many solutions. Both equations represent the same line.

Key Concepts

System of Equations

A system of equations consists of multiple equations that are solved together because they share the same variables. In this particular exercise, we are given a system:

• 3x + 2y = 4
• 6x + 4y = 8

Here, both equations have the variables x and y. The goal is to find the values of x and y that satisfy both equations at the same time.
When solving such systems, different methods can be applied including substitution, elimination, and matrix-based methods like Cramer's rule. Cramer's rule specifically uses determinants of matrices to find solutions if the determinant of the coefficient matrix is not zero.

Determinant of a Matrix

The determinant is a special value calculated from a square matrix, often denoted as det(A) or \(|A|\). It provides useful information about the matrix, such as whether the matrix is invertible. To use Cramer's rule in solving a system of equations, one must calculate the determinant of the coefficient matrix.
For the matrix \( A = \begin{pmatrix} 3 & 2 \ 6 & 4 \end{pmatrix} \), the determinant, denoted \( D \), is calculated by using the formula:\[ D = ad - bc \]where \( a = 3 \), \( b = 2 \), \( c = 6 \), and \( d = 4 \). This gives:\[ D = (3)(4) - (2)(6) = 12 - 12 = 0 \]A determinant of zero indicates certain properties about the system that we will discuss next.

Dependent System

A dependent system of equations occurs when the equations in the system are not distinct or independent. For the given system, the second equation is simply a multiple of the first equation:

• The first equation: 3x + 2y = 4
• The second equation: 6x + 4y = 8 (which is 2 times the first equation)

This means that both equations describe the same line. Because of this dependency, there isn't a single unique solution; instead, there is a set of solutions that satisfy both equations. Recognizing dependency is crucial, as it affects the method we use to find solutions.

Infinitely Many Solutions

When a system of equations is dependent, it does not have a unique solution. Instead, it has infinitely many solutions. This happens because both equations describe the same geometric line when plotted on a graph.
In the context of this exercise, since both equations are multiples of each other and represent the same line, any point on that line is a solution. Therefore, all points that lie on the line represented by the equation 3x + 2y = 4 (or 6x + 4y = 8) satisfy the system.
This concept shows us why the determinant being zero highlights a lack of uniqueness in solutions, and it prompts us to seek alternative methods beyond Cramer's rule, such as parameterizing one of the variables to express the infinite set of solutions.