Problem 39 - Intermediate Algebra

Question

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rl} x+\frac{2}{3} y & =-6 \\ -\frac{1}{4} x+3 y & =-8 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = -4 \) and \( y = -3 \).

Step 1: Write the System in Matrix Form

The given system of equations can be written in the form \( AX = B \) where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. The system equations are: \[x + \frac{2}{3}y = -6\] \[-\frac{1}{4}x + 3y = -8\] The matrices are: \[A = \begin{pmatrix} 1 & \frac{2}{3} \ -\frac{1}{4} & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} -6 \ -8 \end{pmatrix} \]

Step 2: Calculate the Determinant of Matrix A

To use Cramer's Rule, calculate the determinant of matrix \( A \). The determinant \( \text{det}(A) \) is given by: \[ \text{det}(A) = \left(1\right)\left(3\right) - \left(-\frac{1}{4}\right)\left(\frac{2}{3}\right) \] Calculate each term: \[ 1 \cdot 3 = 3 \] \[ -\frac{1}{4} \cdot \frac{2}{3} = -\frac{1}{6} \] Now compute the determinant: \[ \text{det}(A) = 3 + \frac{1}{6} = \frac{19}{6} \] Since \( \text{det}(A) eq 0 \), the system has a unique solution.

Step 3: Calculate Determinants for Cramer's Rule

Now, calculate determinants required by Cramer's Rule: - Replace the first column of \( A \) with \( B \) to form \( A_x \): \[ A_x = \begin{pmatrix} -6 & \frac{2}{3} \ -8 & 3 \end{pmatrix} \] Calculate \( \text{det}(A_x) \): \[ \text{det}(A_x) = (-6)(3) - \left(-8\right)\left(\frac{2}{3}\right) = -18 + \frac{16}{3} = -\frac{38}{3} \] - Replace the second column of \( A \) with \( B \) to form \( A_y \): \[ A_y = \begin{pmatrix} 1 & -6 \ -\frac{1}{4} & -8 \end{pmatrix} \] Calculate \( \text{det}(A_y) \): \[ \text{det}(A_y) = (1)(-8) - (-\frac{1}{4})(-6) = -8 - \frac{3}{2} = -\frac{19}{2} \]

Step 4: Apply Cramer's Rule to Find Solutions

Cramer's Rule states that the variables are given by the following formulas: \[ x = \frac{\text{det}(A_x)}{\text{det}(A)} \] \[ y = \frac{\text{det}(A_y)}{\text{det}(A)} \] Substitute the computed determinants: \[ x = \frac{-\frac{38}{3}}{\frac{19}{6}} = \frac{-38}{3} \cdot \frac{6}{19} = -4 \] \[ y = \frac{-\frac{19}{2}}{\frac{19}{6}} = -\frac{19}{2} \cdot \frac{6}{19} = -3 \] Thus, the solutions are \( x = -4 \) and \( y = -3 \).

Key Concepts

Determinant Calculation

When dealing with matrix equations, the determinant plays a crucial role, especially in solving systems of linear equations using methods like Cramer's Rule. The determinant is a special number that can be calculated from a square matrix. For a matrix \( A \), its determinant (denoted as \( \text{det}(A) \)) provides information about the matrix, such as if it is invertible or not.
For a 2x2 matrix, the determinant \( \text{det} \) is calculated using the formula:\[ \text{det}(A) = a \cdot d - b \cdot c \]where \( a, b, c, \) and \( d \) are the elements of the matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).

• If \( \text{det}(A) = 0 \), the matrix does not have an inverse, and the system might have no solutions or infinitely many solutions.
• If \( \text{det}(A) eq 0 \), the matrix is invertible, and the system has a unique solution.

This calculation helps us determine whether Cramer's Rule can be applied to find unique solutions for the equations.

Matrix Equations

In solving systems of linear equations, expressing them in terms of matrices can simplify the process significantly. A system of equations can be written in matrix form as: \( AX = B \), where:

• \( A \) is the coefficient matrix, containing the coefficients of the variables.
• \( X \) is the column matrix of variables.
• \( B \) is the constant matrix.

Consider the example equations:
\[ x + \frac{2}{3}y = -6 \]
\[ -\frac{1}{4}x + 3y = -8 \]
We can express them in a matrix form:
\[ A = \begin{pmatrix} 1 & \frac{2}{3} \ -\frac{1}{4} & 3 \end{pmatrix}, \, X = \begin{pmatrix} x \ y \end{pmatrix}, \, B = \begin{pmatrix} -6 \ -8 \end{pmatrix} \]
This matrix equation helps to systematically solve the variables by applying algebraic rules such as Cramer's Rule.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations with the same set of variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. Solving such a system can be done using several methods, including:

• Graphical Method: Visual representation of lines on a graph to find the intersection point.
• Substitution Method: Solve one equation for one variable and substitute into the other.
• Elimination Method: Eliminate one variable by adding or subtracting equations.
• Cramer's Rule: A method using determinants that is applicable for systems with the same number of equations as variables.

Cramer's Rule specifically exploits the properties of determinants to find the unique solution of a system. It is suited for small systems where the calculation of determinants is straightforward, particularly 2x2 or 3x3 systems. For the system to have a solution, the determinant of the coefficient matrix must not be zero, ensuring that it has a unique solution.