Problem 4 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 0.21 x_{1}+0.57 x_{2} &=0.369 \\ 0.1 x_{1}+0.2 x_{2} &=0.135 \end{aligned} $$

Step-by-Step Solution

Answer

\(x_1 = 0.21\), \(x_2 = 0.57\)

Step 1: Write the system in matrix form

First, write the system of equations in matrix form: \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the vector of variables, and \(B\) is the right-hand side vector. For our system: \[A = \begin{bmatrix} 0.21 & 0.57 \ 0.1 & 0.2 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad B = \begin{bmatrix} 0.369 \ 0.135 \end{bmatrix}\]

Step 2: Compute the determinant of matrix A

The determinant of matrix \(A\) (denoted as \(|A|\)) is calculated using the formula: \(|A| = a_{11}a_{22} - a_{12}a_{21}\). For the given matrix:\[|A| = (0.21)(0.2) - (0.57)(0.1) = 0.042 - 0.057 = -0.015\]

Step 3: Compute the determinant of matrix A1

Create matrix \(A_1\) by replacing the first column of \(A\) with vector \(B\): \[A_1 = \begin{bmatrix} 0.369 & 0.57 \ 0.135 & 0.2 \end{bmatrix}\]. The determinant \(|A_1|\) is: \[|A_1| = (0.369)(0.2) - (0.57)(0.135) = 0.0738 - 0.07695 = -0.00315\]

Step 4: Compute the determinant of matrix A2

Create matrix \(A_2\) by replacing the second column of \(A\) with vector \(B\): \[A_2 = \begin{bmatrix} 0.21 & 0.369 \ 0.1 & 0.135 \end{bmatrix}\]. The determinant \(|A_2|\) is: \[|A_2| = (0.21)(0.135) - (0.369)(0.1) = 0.02835 - 0.0369 = -0.00855\]

Step 5: Solve for the variables using Cramer's Rule

Cramer's Rule states that \(x_1 = \frac{|A_1|}{|A|}\) and \(x_2 = \frac{|A_2|}{|A|}\). Compute \(x_1\): \[x_1 = \frac{-0.00315}{-0.015} = 0.21\]Compute \(x_2\): \[x_2 = \frac{-0.00855}{-0.015} = 0.57\]

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. In the context of a 2x2 matrix, which is common in solving systems of two linear equations, the determinant helps us understand the properties of the matrix. It is a scalar value computed using the formula \[|A| = a_{11}a_{22} - a_{12}a_{21}\]where each \(a_{ij}\) represents the elements of the matrix.
For example, in the given exercise:\[A = \begin{bmatrix} 0.21 & 0.57 \ 0.1 & 0.2 \end{bmatrix}\]the determinant \[|A| = (0.21)(0.2) - (0.57)(0.1) = 0.042 - 0.057 = -0.015\]
Calculating the determinant is crucial when using Cramer's Rule since it indicates whether a unique solution exists for the system of equations. When the determinant is not zero, the system has a unique solution; however, if the determinant is zero, the system may not have a unique solution or might be inconsistent or dependent.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with operations involving matrices. It provides tools to solve system of equations efficiently. In this context, matrices are arrays of numbers, and they can represent coefficients of a linear system in a compact form.
When solving a system of linear equations, we often represent the equations in matrix form as \(AX = B\):

• \(A\) is the coefficient matrix, holding the coefficients of the variables.
• \(X\) is a column vector containing the variables of the system.
• \(B\) is the column vector of constants from the right side of the equations.

For example, in the exercise:\[A = \begin{bmatrix} 0.21 & 0.57 \ 0.1 & 0.2 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad B = \begin{bmatrix} 0.369 \ 0.135 \end{bmatrix}\]
This representation simplifies complex operations and allows us to use various matrix methods, including Cramer's Rule, to find solutions.

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear, and all equations involve the same set of variables. In simpler terms, it is a collection of equations that can be represented graphically as straight lines.
The objective is usually to find the values of the variables that satisfy all equations simultaneously. There are several methods for solving these systems, such as:

• Graphical method, which involves plotting on a coordinate plane
• Substitution method, replacing variables to simplify the system
• Elimination method, adding or subtracting equations to eliminate variables
• Cramer's Rule, a straightforward technique using determinants when the system is square and the determinant is non-zero.

For the exercise at hand, Cramer's Rule is applied to a 2x2 system. This rule is especially handy because it offers a way to calculate the values of variables directly by using determinants, simplifying the process considerably.