Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{r} 2 x+3 y=2 \\ x-2 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -2\).

Step 1: Understand System of Equations

We are given a system of two linear equations: \[\begin{align*}2x + 3y &= 2 \x - 2y &= 8\end{align*}\] Our task is to solve this system using Cramer's rule, which is applicable when the coefficient matrix determinant is non-zero.

Step 2: Set Up Coefficient Matrix and Constants

Identify the coefficient matrix \(A\) and constant matrix \(B\):\[A = \begin{bmatrix} 2 & 3 \ 1 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \ 8 \end{bmatrix}\] We will need to find determinants in subsequent steps.

Step 3: Calculate Determinant of Coefficient Matrix

The determinant of matrix \(A\), denoted \(|A|\), is calculated as follows:\[|A| = \begin{vmatrix} 2 & 3 \ 1 & -2 \end{vmatrix} = (2)(-2) - (3)(1) = -4 - 3 = -7\]Since \(|A| eq 0\), Cramer's rule can be applied.

Step 4: Determine Matrix for x (\(D_x\)) and Find Det(D_x)

Replace the first column of \(A\) with the constants from \(B\) to get \(D_x\):\[D_x = \begin{bmatrix} 2 & 3 \ 8 & -2 \end{bmatrix}\]Calculate its determinant:\[|D_x| = (2)(-2) - (3)(8) = -4 - 24 = -28\]

Step 5: Determine Matrix for y (\(D_y\)) and Find Det(D_y)

Replace the second column of \(A\) with the constants from \(B\) to get \(D_y\):\[D_y = \begin{bmatrix} 2 & 2 \ 1 & 8 \end{bmatrix}\]Calculate its determinant:\[|D_y| = (2)(8) - (2)(1) = 16 - 2 = 14\]

Step 6: Apply Cramer's Rule to Find Solutions for x and y

According to Cramer's Rule:\[ x = \frac{|D_x|}{|A|}, \quad y = \frac{|D_y|}{|A|} \]Substitute the determined values:\[ x = \frac{-28}{-7} = 4, \quad y = \frac{14}{-7} = -2 \]

Step 7: Verify the Solution in the Original Equations

Substitute \(x = 4\) and \(y = -2\) back into the original equations:1. \(2(4) + 3(-2) = 8 - 6 = 2\), which is true.2. \(4 - 2(-2) = 4 + 4 = 8\), which is also true.Thus the solution \(x = 4, y = -2\) is verified.

Key Concepts

Understanding Systems of Linear Equations

A system of linear equations is simply a collection of two or more linear equations that share a common set of variables, often represented as \(x\) and \(y\) in basic problems. The goal is to find the values for these variables that satisfy all equations simultaneously.
In our exercise, we are given the system:

• \(2x + 3y = 2\)
• \(x - 2y = 8\)

This represents two lines on a two-dimensional plane, and solving the system means finding the intersection point of these two lines. Cramer's Rule is one method to achieve this, provided the system can be represented in a specific matrix form with a non-zero determinant.

The Determinant in Matrix Algebra

The determinant is a special number calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation described by the matrix.
In the context of Cramer's Rule, the determinant of the coefficient matrix tells us if the system of equations has a unique solution. If the determinant is zero, the matrix does not have an inverse, and the equations are either dependent or have no solution.
For our system, the coefficient matrix is \[A = \begin{bmatrix} 2 & 3 \ 1 & -2 \end{bmatrix}\]The determinant \(|A|\) is calculated as:\[|A| = (2)(-2) - (3)(1) = -7\]Since \(|A| eq 0\), Cramer's Rule can be used to find a unique solution.

Role of the Coefficient Matrix

The coefficient matrix in a system of linear equations is a matrix constructed from the coefficients of the variables from each equation in the system. This structure helps in managing the equations in a compact form.
In our particular problem, the coefficient matrix \(A\) is:\[A = \begin{bmatrix} 2 & 3 \ 1 & -2 \end{bmatrix}\]This matrix aids in the quick computation of the determinant and facilitates the formation of modified matrices required by Cramer's Rule. These modified matrices, known as \(D_x\) and \(D_y\), are formed by replacing specific columns of \(A\) with the constant matrix, derived from the right side of the equation system, enabling the computation of individual variable solutions.

Verification of the Solution

Solution verification is an essential step after computing results through Cramer's Rule or any other method. It involves substituting the determined values of \(x\) and \(y\) back into the original system of equations to ensure they satisfy both equations.
For the obtained solutions \(x = 4\) and \(y = -2\), let's verify:

• Substitute into the first equation: \(2(4) + 3(-2) = 8 - 6 = 2\), which is correct.
• Substitute into the second equation: \(4 - 2(-2) = 4 + 4 = 8\), which is also correct.

This verification confirms that the solutions are correct as they satisfy the original equations. Without verification, there could be mistakes in calculation, leading to incorrect conclusions about the system's solutions.