Problem 8 - Algebra for College Students

Question

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{l}-3 x-4 y=14 \\ -2 x+3 y=-19\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = 2\), \(y = -5\).

Step 1: Write the System of Equations in Matrix Form

First, take the given system of equations:\[-3x - 4y = 14\]\[-2x + 3y = -19\]We can express this system in terms of matrix form \(AX = B\), where:\[A = \begin{pmatrix} -3 & -4 \ -2 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 14 \ -19 \end{pmatrix}\]

Step 2: Calculate the Determinant of Matrix A

To use Cramer's rule, first calculate the determinant \(\Delta\) of matrix \(A\):\[\Delta = \begin{vmatrix} -3 & -4 \ -2 & 3 \end{vmatrix} = (-3)(3) - (-4)(-2) = -9 - 8 = -17\]Since \(\Delta eq 0\), the system has a unique solution.

Step 3: Calculate Determinant \(\Delta_x\) by Replacing x-column

Replace the first column of \(A\) with the matrix \(B\) to find \(\Delta_x\):\[\Delta_x = \begin{vmatrix} 14 & -4 \ -19 & 3 \end{vmatrix} = (14)(3) - (-4)(-19) = 42 - 76 = -34\]

Step 4: Calculate Determinant \(\Delta_y\) by Replacing y-column

Replace the second column of \(A\) with the matrix \(B\) to find \(\Delta_y\):\[\Delta_y = \begin{vmatrix} -3 & 14 \ -2 & -19 \end{vmatrix} = (-3)(-19) - (14)(-2) = 57 + 28 = 85\]

Step 5: Find Values for x and y using Cramer's Rule

Using Cramer's Rule:\[x = \frac{\Delta_x}{\Delta} = \frac{-34}{-17} = 2\]\[y = \frac{\Delta_y}{\Delta} = \frac{85}{-17} = -5\]The solution to the system is \(x = 2\) and \(y = -5\).

Key Concepts

Determinant Calculation

One of the fundamental steps in applying Cramer's Rule to solve a system of linear equations involves calculating the determinant of a matrix. A determinant is a special number that can be calculated from a square matrix. It provides critical insights into the properties of the matrix, such as whether the matrix has an inverse or not.

For a 2x2 matrix, the determinant is calculated using a simple formula. Suppose you have a matrix:

• \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\)

The determinant \(\Delta\) is computed as:

• \(\Delta = ad - bc\)

In the provided exercise, the matrix \(A\) is:

• \(\begin{pmatrix} -3 & -4 \ -2 & 3 \end{pmatrix}\)

Thus, the determinant calculation in this specific case is:

• \((-3)(3) - (-4)(-2) = -9 - 8 = -17\)

A non-zero determinant indicates that the system of equations has a unique solution. This is a crucial step to apply Cramer's Rule effectively.

System of Equations

A system of equations consists of multiple equations that share common variables. Solving a system of equations involves finding the values of these variables that satisfy all equations simultaneously. There are several ways to solve them, including substitution, elimination, and using matrices like in Cramer's Rule.

The given problem involves a simple linear system of equations:

• \(-3x - 4y = 14\)
• \(-2x + 3y = -19\)

There are three possible outcomes when solving a system of linear equations:

• A unique solution, which occurs when the determinant of the matrix is non-zero as in this case.
• Infinitely many solutions, which happens when the equations are dependent, often indicated by the determinant being zero and leading to coinciding lines.
• No solution, which typically occurs when equations are inconsistent and parallel lines that never meet.

Cramer's Rule provides a method to find the unique solution when the determinant is non-zero, making it a powerful tool for solving linear systems efficiently.

Matrix Representation of Equations

Representing a system of equations in matrix form is an elegant and systematic approach. It simplifies the process of solving equations like in Cramer's Rule because it allows for the use of determinant calculations and matrix manipulations.

To convert the given system of equations into matrix form, we need to interpret the coefficients and constants as distinct matrices:

• Matrix \(A\) consists of the coefficients of the variables: \(A = \begin{pmatrix} -3 & -4 \ -2 & 3 \end{pmatrix}\)
• Matrix \(X\) consists of the variables themselves: \(X = \begin{pmatrix} x \ y \end{pmatrix}\)
• Matrix \(B\) consists of the constants from each equation: \(B = \begin{pmatrix} 14 \ -19 \end{pmatrix}\)

The matrix equation then takes the form of \(AX = B\), making it easier to apply matrix algebra methods such as Cramer's Rule for larger systems. This representation is crucial, not only for manual calculations but also for implementations in computational tools and software, where matrices are key data structures.