Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{aligned} 2 x-3 y &=5 \\ -6 x+9 y &=12 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent with no solution.

Step 1: Write the system in matrix form

First, we represent the given system of equations in matrix form. The system is:\[ \begin{aligned} 2x - 3y &= 5 \ -6x + 9y &= 12 \end{aligned} \] This can be written as \( AX = B \), where: \[ A = \begin{bmatrix} 2 & -3 \ -6 & 9 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \end{bmatrix}, \quad B = \begin{bmatrix} 5 \ 12 \end{bmatrix} \]

Step 2: Calculate the determinant of matrix A

To use Cramer's Rule, we first calculate the determinant of matrix \( A \). The determinant \( \text{det}(A) \) is given by: \[ \text{det}(A) = (2)(9) - (-3)(-6) = 18 - 18 = 0 \] Since \( \text{det}(A) = 0 \), the system does not have a unique solution with Cramer's Rule. When the determinant is zero, the system of equations is either dependent or inconsistent.

Step 3: Analyze the system for consistency or dependency

Given \( \text{det}(A) = 0 \), we must further investigate the system to decide if it is dependent or inconsistent. We do this by checking if one equation is a multiple of the other. Notice that multiplying the first equation by \(-3\) gives: \[ -6x + 9y = -15 \] This modified equation does not match the second equation \(-6x + 9y = 12\). Therefore, the equations are inconsistent, meaning there is no solution that satisfies both equations simultaneously.

Key Concepts

Determinant

A determinant is a special number that can be calculated from a square matrix. It provides significant information about the matrix. In the context of solving systems of equations using Cramer's Rule, the determinant plays a crucial role:

• If the determinant is non-zero, the system has a unique solution.
• If the determinant is zero, the system might be dependent or inconsistent.

To find the determinant of a 2x2 matrix, such as matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula is \( ad - bc \).
In our original exercise, after applying this formula to matrix \( A \), we found \( \text{det}(A) = 0 \). This zero determinant became the pivot point for determining the nature of the system's solutions.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables that you aim to solve simultaneously. Solving these equations can help in finding the variable values that satisfy all equations.
There are different methods to solve such systems, for example:

• Substitution.
• Elimination.
• Graphical methods.
• Cramer's Rule (when applicable).

Cramer's Rule is particularly effective for linear systems that result in square matrices with non-zero determinants, as it can provide a direct solution. However, when the determinant is zero, other methods or additional analysis is required to understand the nature of the solutions.

Inconsistent Systems

Inconsistent systems contain equations that cannot be satisfied by the same set of variable values. This means there is no solution that makes all the equations true simultaneously.
In the exercise, we found the determinant of matrix \( A \) was zero. To check consistency, we compared the transformed equations. Multiplying one equation by a constant might make it identical to the other, indicating a dependent system. However, if this transformed equation does not match the other exactly, as happened in our case, it indicates the system is inconsistent.
Such inconsistency means there's no intersection point when these equations are plotted graphically, showing they don't cross over or touch each other at any point.