Problem 6 - Advanced Engineering Mathematics

Question

Solve the given system of equations by Cramer's rule. $$ \begin{gathered} 5 r+4 s=-1 \\ 10 r-6 s=5 \end{gathered} $$

Step-by-Step Solution

Answer

The solution is \(r = \frac{1}{5}\) and \(s = -\frac{1}{2}\).

Step 1: Identify the Coefficients

Start by identifying the coefficient matrix from the system of equations. The given equations are: 1. \(5r + 4s = -1\) 2. \(10r - 6s = 5\).So, the coefficient matrix \(A\) is: \[A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix}.\]

Step 2: Compute the Determinant of Coefficient Matrix

Calculate the determinant of matrix \(A\). The formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(ad - bc\).So for \(A\):\[\det(A) = (5)(-6) - (4)(10) = -30 - 40 = -70.\]

Step 3: Set Up and Solve for Determinant of Modified Matrices

Using Cramer's Rule, replace each column of \(A\) with the constants from the equations to create new matrices and find their determinants. Matrix for \(r\) (replace the first column with constants):\[A_r = \begin{bmatrix} -1 & 4 \ 5 & -6 \end{bmatrix}\] \[\det(A_r) = (-1)(-6) - (4)(5) = 6 - 20 = -14.\]Matrix for \(s\) (replace the second column with constants):\[A_s = \begin{bmatrix} 5 & -1 \ 10 & 5 \end{bmatrix}\] \[\det(A_s) = (5)(5) - (-1)(10) = 25 + 10 = 35.\]

Step 4: Calculate the Solution Values for Variables

Using Cramer's Rule formulas \(r = \frac{\det(A_r)}{\det(A)}\) and \(s = \frac{\det(A_s)}{\det(A)}\), substitute determinants to find \(r\) and \(s\).For \(r\):\[r = \frac{-14}{-70} = \frac{1}{5}.\]For \(s\):\[s = \frac{35}{-70} = -\frac{1}{2}.\]

Key Concepts

System of Equations

A system of equations involves finding the values of unknown variables that satisfy multiple equations simultaneously. These systems can have two or more equations involving the same set of unknowns. In linear algebra, such equations are particularly important, and different methods exist to solve them, including substitution, elimination, and Cramer's Rule.

Cramer's Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, using determinants. It's particularly useful when dealing with smaller systems, generally 2x2 or 3x3, due to the computational complexity of larger matrices.

The system of equations we are considering is in two variables, \(r\) and \(s\):

• \(5r + 4s = -1\)
• \(10r - 6s = 5\)

These equations are paired, meaning they need to be solved together to find a single set of values for \(r\) and \(s\) that works in both equations.

Determinant of a Matrix

The determinant is a special number calculated from a square matrix. It provides key insights into the properties of the matrix, such as whether it's invertible, and it is essential in Cramer's Rule.

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula \(ad - bc\). This calculation can indicate if a system of equations has a unique solution: if the determinant is zero, the system doesn't have a unique solution.

In our exercise, we have the coefficient matrix
\[ A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix} \]
The determinant is computed as \((5)(-6) - (4)(10) = -30 - 40 = -70\). Since the determinant \(eq 0\), the system has a unique solution, meaning that Cramer's Rule is applicable here.

Linear Algebra

Linear Algebra is the branch of mathematics dealing with vectors, vector spaces, and systems of linear equations. It involves the study of lines, planes, and subspaces, but can become complex when dealing with systems containing two or more variables.

This discipline provides the foundation for solving various real-world problems using matrices and vectors, crucially impacting fields ranging from physics to economics. Concepts like matrix operations, eigenvectors, and eigenvalues all stem from linear algebra and are used to solve equations like the ones in our exercise.

Within this framework, we use determinants and matrix representation to handle and solve the system of equations. Cramer's Rule, as applied here, is a classic example of linear algebra, leveraging matrix properties to find solutions efficiently.

Matrix Operations

Matrix operations are procedures used to manipulate matrices, crucial for solving systems of linear equations in linear algebra. These operations include addition, subtraction, multiplication, and finding the determinant or inverse of matrices.

Cramer's Rule involves several key matrix operations:

• Firstly, identifying the coefficient matrix from the system of equations.
• Then calculating the determinant of this matrix, a process which determines whether a unique solution exists.
• Next, creating modified matrices by replacing columns with the constants from the system.
• Finally, calculating the determinants of these modified matrices to solve for the variables.

In our case, after finding the determinant of our coefficient matrix \(A\), we modified the matrix to create \(A_r\) and \(A_s\), calculated their determinants, and then used these in Cramer's Rule to find the values of \(r\) and \(s\). These operations show how tools from linear algebra can simplify complex problems and enable precise solutions.