Problem 4 - Advanced Engineering Mathematics

Question

Solve the given system of equations by Cramer's rule. $$ \begin{array}{r} 0.21 x_{1}+0.57 x_{2}=0.369 \\ 0.1 x_{1}+0.2 x_{2}=0.135 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x_1 = 0.21 \) and \( x_2 = 0.57 \).

Step 1: Understand the System of Equations

We have a system of two linear equations: \( 0.21x_1 + 0.57x_2 = 0.369 \) and \( 0.1x_1 + 0.2x_2 = 0.135 \). Our goal is to solve for \( x_1 \) and \( x_2 \) using Cramer's Rule.

Step 2: Write Down the Coefficient Matrix

Identify the coefficients of the system and write them as a matrix: \[ A = \begin{bmatrix} 0.21 & 0.57 \ 0.1 & 0.2 \end{bmatrix} \]. This is known as the coefficient matrix.

Step 3: Find the Determinant of Matrix A

Compute the determinant of matrix \( A \) using the formula for a 2x2 matrix: \( \det(A) = a_{11}a_{22} - a_{12}a_{21} \). Substituting the values, we get \( \det(A) = (0.21)(0.2) - (0.57)(0.1) = 0.042 - 0.057 = -0.015 \).

Step 4: Construct the Matrix A1 and A2

Replace the first column of the matrix \( A \) with the constant terms \([0.369, 0.135]\) to form \( A_1 \) and replace the second column with \([0.369, 0.135]\) to form \( A_2 \). \[ A_1 = \begin{bmatrix} 0.369 & 0.57 \ 0.135 & 0.2 \end{bmatrix} \] \[ A_2 = \begin{bmatrix} 0.21 & 0.369 \ 0.1 & 0.135 \end{bmatrix} \].

Step 5: Calculate the Determinants of A1 and A2

Calculate \( \det(A_1) = (0.369)(0.2) - (0.135)(0.57) = 0.0738 - 0.07695 = -0.00315 \). Calculate \( \det(A_2) = (0.21)(0.135) - (0.1)(0.369) = 0.02835 - 0.0369 = -0.00855 \).

Step 6: Apply Cramer's Rule to Find x1 and x2

Using Cramer's rule, \( x_1 = \frac{\det(A_1)}{\det(A)} = \frac{-0.00315}{-0.015} = 0.21 \) and \( x_2 = \frac{\det(A_2)}{\det(A)} = \frac{-0.00855}{-0.015} = 0.57 \).

Key Concepts

Determinant Calculation

The determinant is a key component in matrix algebra, especially when using Cramer's Rule to solve linear equations. In simple terms, the determinant provides information about a matrix's properties, such as whether the matrix is invertible or describes the scale transformation in space.

For a 2x2 matrix, the calculation of the determinant is straightforward:

• Identify the elements of the matrix, often labeled as \( a_{11}, a_{12}, a_{21}, \) and \( a_{22} \).
• Apply the formula \( \det(A) = a_{11}a_{22} - a_{12}a_{21} \).

In our exercise, with matrix \( A = \begin{bmatrix} 0.21 & 0.57 \ 0.1 & 0.2 \end{bmatrix} \), the determinant is \( -0.015 \). This negative value suggests the transformation performed by matrix \( A \) reflects and scales down the coordinates of any vector in the space spanned by \( A \).

Linear Equations

Linear equations are mathematical expressions that represent lines when plotted on a graph. They take the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

These equations can be solved using various methods, including substitution, elimination, or matrix-based approaches like Cramer's Rule. For example, the system of linear equations in the exercise:

• \( 0.21x_1 + 0.57x_2 = 0.369 \)
• \( 0.1x_1 + 0.2x_2 = 0.135 \)

represents two lines. The solution to these equations will give the intersection point, which is essentially the values of \( x_1 \) and \( x_2 \) that satisfy both equations. Through Cramer's Rule, we ensure an efficient path to finding these solutions in the realm of matrix algebra.

Matrix Algebra

Matrix algebra offers a structured way to handle systems of equations. Each system of linear equations can be represented in matrix form, making them easier to manipulate mathematically.

For the given exercise, the coefficient matrix \( A \) is a representation of the constants from the equations. To manage such systems using matrices:

• Define the coefficient matrix \( A \) with elements from the linear equations.
• Create matrices \( A_1 \) and \( A_2 \) by substituting the constant terms into the appropriate columns of \( A \).
• Calculate the determinants of these matrices to solve the variables using Cramer's Rule.

Utilizing matrix algebra simplifies computation, especially for more complex systems with multiple variables. It allows us to leverage determinants and other mathematical properties, enabling us to find solutions efficiently.