Problem 58 - Elementary Algebra

Question

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} 5 x+3 y=4 \\ 2 x-3 y=5 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has exactly one unique solution and is classified as consistent and independent.

Step 1: Write the System of Equations

First, consider the given system of equations: \[ \begin{cases} 5x + 3y = 4 \ 2x - 3y = 5 \end{cases} \].

Step 2: Find the Determinant of the Coefficient Matrix

To determine if the system has a unique solution, no solution, or infinitely many solutions, calculate the determinant of the coefficient matrix. The coefficient matrix is: \[ \begin{bmatrix} 5 & 3 \ 2 & -3 \end{bmatrix} \] The determinant is given by \( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \).

Step 3: Calculate the Determinant

Substitute the coefficients into the determinant formula: \[ \text{det}(A) = (5 \times -3) - (3 \times 2) \] \[ \text{det}(A) = -15 - 6 = -21 \].

Step 4: Determine the Number of Solutions

Since the determinant of the coefficient matrix is non-zero (\( \text{det}(A) = -21 \)), the system has a unique solution. The system of equations is therefore consistent and independent.

Step 5: Classify the System

For this system: - The coefficient determinant is non-zero, indicating a unique solution. - The system is classified as consistent and independent.

Key Concepts

Determinant of Coefficient Matrix

Understanding the determinant of a coefficient matrix is critical in determining the nature of solutions for a linear system.
For the given system of equations: