Problem 37 - Precalculus: Functions and Graphs

Question

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

Cramer's Rule is not applicable; the system is inconsistent.

Step 1: Determine if Cramer's Rule is Applicable

First, convert the system of equations into matrix form, which is: \[ A = \begin{bmatrix} 2 & -3 \ -6 & 9 \end{bmatrix} \] and \[ B = \begin{bmatrix} 5 \ 12 \end{bmatrix} \]. The determinant of matrix \( A \) is calculated as follows: \[ \det(A) = (2)(9) - (-3)(-6) = 18 - 18 = 0 \]. Since the determinant is zero, Cramer's rule is not applicable.

Step 2: Conclude about the System

Since Cramer's rule is not applicable due to the determinant being zero, this indicates that the system of equations is either dependent (infinitely many solutions) or inconsistent (no solution). Further analysis reveals that multiplying the first equation by 3, we get \(-6x + 9y = 15\), which conflicts with the second equation \(-6x + 9y = 12\). This shows the system is inconsistent.

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided certain conditions are met. This method makes use of determinants of matrices to find the unique solution. Cramer's rule requires:

• That the system has the same number of equations as unknown variables.
• A non-zero determinant of the coefficient matrix.

When the determinant is zero, Cramer's rule cannot be applied, indicating the system may either have infinitely many solutions or no solution at all. To use Cramer's rule, the determinant of the matrix for each variable needs to be computed, offering a straightforward calculation for unique solutions when applicable.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It is particularly useful in the analysis of linear equations, providing insight into the system's solvability:

• A determinant of zero indicates potential dependency or inconsistency.
• A non-zero determinant suggests the system potentially has a unique solution.

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed as \( ad - bc \). This computation is pivotal in determining the applicability of Cramer’s rule and assessing the nature of solutions in linear systems.

Inconsistent System

An inconsistent system is a group of linear equations with no solution. This occurs when the equations contradict each other, as they describe parallel lines that never intersect in two-dimensional space. For example, in the given system:

• The equations \( 2x - 3y = 5 \) and \( -6x + 9y = 12 \) have no points in common despite their algebraic similarities.

This is confirmed by converting both equations into similar forms, like multiplying the first equation by 3 results in \(-6x + 9y = 15\), different from the second equation \(-6x + 9y = 12\). Thus, these parallel lines, indicated by different constants, confirm the system as inconsistent.

Matrix Form

Matrix form is a compact and efficient way to represent systems of linear equations using matrices. In this representation:

• Each equation in the system is translated into a row of numbers.
• These numbers correspond to the coefficients of each variable in the equations.

For the given problem, the system \( 2x - 3y = 5 \) and \( -6x + 9y = 12 \) becomes:\[ A = \begin{bmatrix} 2 & -3 \ -6 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 5 \ 12 \end{bmatrix} \]Using this format simplifies calculations and allows for techniques like Cramer's Rule or matrix operations such as row reduction. Matrix form is essential in linear algebra, aiding in understanding and solving equations more efficiently.