Problem 40 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{c} 3 x-\frac{1}{2} y=6 \\ -2 x+\frac{1}{3} y=-4 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has no unique solution as the determinant is zero.

Step 1: Identify the Coefficient Matrix

First, extract the coefficients from the given system of equations to form the coefficient matrix \( A \):\[ A = \begin{pmatrix} 3 & -\frac{1}{2} \ -2 & \frac{1}{3} \end{pmatrix} \]

Step 2: Find the Determinant of the Coefficient Matrix

Calculate the determinant of matrix \( A \). The determinant \( \det(A) \) is given by:\[ \det(A) = (3)(\frac{1}{3}) - (-2)(-\frac{1}{2}) \]Simplifying, we get:\[ \det(A) = 1 - 1 = 0 \]

Step 3: Check Determinant for Solution Existence

Since the determinant \( \det(A) \) is 0, Cramer's rule cannot be applied as it requires a non-zero determinant. This means the system may have no unique solution, implying that it could have either no solutions or infinitely many.

Key Concepts

Determinant

In linear algebra, the determinant is a special scalar value that is computed from a square matrix. It is a critical component when using Cramer's Rule for solving systems of equations. The determinant provides insight into the properties of the matrix, such as whether a system of linear equations has a unique solution. When the determinant is zero, the matrix is singular, and the system may not have a unique solution. Cramer's Rule requires the determinant to be non-zero to guarantee a single unique solution. For a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( \ ext{det}(A) = ad - bc \). This simple calculation helps gauge the behavior of the linear system associated with the matrix.

Coefficient Matrix

A coefficient matrix is derived from the coefficients of the variables in a system of linear equations. It compiles all the numerical coefficients into one single matrix. For example, in the system of equations: \( 3x - \frac{1}{2} y = 6 \-2x + \frac{1}{3} y = -4 \) The coefficient matrix \( A \) is:\[ A = \begin{pmatrix} 3 & -\frac{1}{2} \ -2 & \frac{1}{3} \end{pmatrix} \]This matrix includes the coefficients 3, -\(\frac{1}{2}\), -2, and \(\frac{1}{3}\) from the variables \(x\) and \(y\). The correct formation of this matrix is crucial, as it plays a key role in applying Cramer's Rule. It is important to note that only the coefficients are used, not the constants from the right-hand side of the equations.

Systems of Equations

Systems of equations appear when multiple equations need to be solved together for common variables. These systems can be linear or non-linear, and in this case, we are dealing with a linear system since the equations are linear in nature:- **Linear System**: Each equation represents a straight line in a two-dimensional space.- **Objective**: Find the values of the variables that satisfy all equations at the same time.The system provided earlier is a classic example of a linear system:\[ \begin{cases} 3x - \frac{1}{2}y = 6 \ -2x + \frac{1}{3}y = -4 \end{cases} \]To solve this, we aimed to use Cramer's Rule. However, because the determinant of the coefficient matrix was found to be zero, this approach was not applicable. Thus, we understand that a matrix with zero determinant indicates potential multiple or no solutions, as they do not intersect at a single unique point in their plotted graph.