Problem 63 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 3 x-16=5 y \\ -3 x+5 y-33=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent or inconsistent (determinant is zero).

Step 1: Write the system in standard form

Rewrite the given system of equations so that all terms involving variables and constants are on one side of the equation. This gives us:\[\begin{align*}3x - 5y &= 16 \-3x + 5y &= 33\end{align*}\]

Step 2: Write down the coefficient matrix

Identify and write out the coefficient matrix for the system. The coefficients from the equations can be arranged in a matrix as follows:\[A = \begin{bmatrix} 3 & -5 \ -3 & 5 \end{bmatrix}\]

Step 3: Calculate the determinant of the coefficient matrix

Find the determinant of matrix \(A\). The determinant is calculated as follows:\[\text{det}(A) = (3)(5) - (-3)(-5) = 15 - 15 = 0\]The determinant equals zero, indicating either a dependent or inconsistent system.

Step 4: Conclude the nature of the system

Since the determinant of the coefficient matrix is zero, this system of linear equations is either dependent or inconsistent. This means it does not have a unique solution and indicates that the equations could potentially describe the same line (dependent) or parallel lines (inconsistent).

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. In a linear equation, each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in two variables can be expressed as \(Ax + By = C\). Here, \(x\) and \(y\) are variables, and \(A\), \(B\), and \(C\) are constants.
Linear equations can have various types of solutions depending on their arrangement and relationships with each other:

• A unique solution results when two lines intersect at a single point.
• No solution occurs when equations represent parallel lines that never meet.
• Infinite solutions arise when equations represent the same line, implying they overlap completely.

In the case of the given equations, we want to employ Cramer's Rule to determine the system's solution nature. However, before we use any method, understanding this underlying relation helps us predict the outcome.

Determinant and Its Role

The determinant is a special number that can be calculated from a square matrix, which provides insight into the system of equations. For a 2x2 matrix, defined as \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\], the determinant is calculated as \(det(A) = ad - bc\).
The determinant is crucial because:

• If the determinant is non-zero, the matrix is invertible, indicating potentially a unique solution for the system of linear equations.
• If the determinant is zero, it reveals that the matrix is not invertible, signaling that the system could either be dependent or inconsistent—this means no distinct solution exists.

In our given exercise, the determinant is zero, helping us conclude that the system might either be inconsistent or dependent, hence lacking a single solution point.

The Coefficient Matrix Overview

The coefficient matrix is a simplified way to represent the system of linear equations, focusing only on the coefficients of the variables. For the system \(3x - 5y = 16\) and \(-3x + 5y = 33\), the coefficient matrix would be:
\[\begin{bmatrix} 3 & -5 \ -3 & 5 \end{bmatrix}\]
The coefficient matrix helps us efficiently organize and solve the system, particularly when applying techniques like Cramer's Rule or matrix inversion. By isolating the coefficients, it becomes feasible to apply mathematical operations without the distraction of variables and constants from the equations. Importantly, these matrices serve as the foundation to compute determinants, which then lead us to determine the nature of the solutions.

Dependent and Inconsistent Systems Explained

In linear algebra, when analyzing systems of equations, understanding the difference between dependent and inconsistent systems is essential:

• A **dependent system** consists of equations that do not provide unique information because they are multiples of each other. Graphically, they represent the same line, meaning they have infinitely many solutions along that line.
• An **inconsistent system** contains equations that depict parallel lines, leading to no intersection. Since parallel lines never meet, there are no solutions to satisfy both equations simultaneously.

In our exercise, the determinant being zero suggests that the system is either dependent or inconsistent. This result tells us there is no single solution point, demanding further analysis or a different approach to comprehend the outcome fully.