Problem 6 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 5 r+4 s &=-1 \\ 10 r-6 s &=5 \end{aligned} $$

Step-by-Step Solution

Answer

\( r = \frac{1}{5} \), \( s = -\frac{1}{2} \) using Cramer's Rule.

Step 1: Write the System in Matrix Form

First, we represent the given system of equations in matrix form. For the equations \(5r + 4s = -1\) and \(10r - 6s = 5\), the coefficient matrix is \(A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix}\), the variable matrix is \(X = \begin{bmatrix} r \ s \end{bmatrix}\), and the constant matrix is \(B = \begin{bmatrix} -1 \ 5 \end{bmatrix}\). Thus, the system can be written as \(AX = B\).

Step 2: Calculate the Determinant of Coefficient Matrix

Calculate \( \det(A) \), the determinant of the coefficient matrix \(A\). For \(A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix}\), the determinant is \( \det(A) = (5)(-6) - (4)(10) = -30 - 40 = -70 \).

Step 3: Formulate the Matrices for Cramer's Rule

According to Cramer's Rule, you need \( \det(A_r) \) and \( \det(A_s) \). Replace the first column of \(A\) with \(B\) to form \(A_r\) and the second column to form \(A_s\). For \(A_r = \begin{bmatrix} -1 & 4 \ 5 & -6 \end{bmatrix}\) and \(A_s = \begin{bmatrix} 5 & -1 \ 10 & 5 \end{bmatrix}\).

Step 4: Calculate Determinants for Adjusted Matrices

Calculate \( \det(A_r) \) and \( \det(A_s) \):- \(\det(A_r) = (-1)(-6) - (4)(5) = 6 - 20 = -14 \)- \(\det(A_s) = (5)(5) - (-1)(10) = 25 + 10 = 35 \)

Step 5: Solve for Variables Using Cramer's Rule

Using Cramer's Rule, \( r = \frac{\det(A_r)}{\det(A)} = \frac{-14}{-70} = \frac{1}{5} \) and \( s = \frac{\det(A_s)}{\det(A)} = \frac{35}{-70} = -\frac{1}{2} \).

Step 6: Final Answer Summary

The solution to the system using Cramer's Rule is \( r = \frac{1}{5} \) and \( s = -\frac{1}{2} \).

Key Concepts

Determinant Calculation

A determinant is a special number that can be calculated from a square matrix. Determinant calculation plays a crucial role in matrix algebra, especially when solving systems of linear equations using methods like Cramer's Rule.

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) is computed as:

• \( \det(A) = ad - bc \)

This formula is derived from the cross-multiplication of the matrix's elements.
For the given exercise, the coefficient matrix is \( A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix} \).

• Calculate \( \det(A) = (5)(-6) - (4)(10) = -30 - 40 = -70 \)

Understanding how to compute the determinant allows you to evaluate if a matrix is invertible, which is essential in solving equations using Cramer's Rule.

Matrix Algebra

Matrix algebra is the study of matrices and their operations, including addition, subtraction, multiplication, and finding inverses. It provides powerful tools for solving systems of linear equations. In matrix algebra, a system of equations can be expressed conveniently in matrix form as \( AX = B \).Consider a system of equations:

• \( 5r + 4s = -1 \)
• \( 10r - 6s = 5 \)

It can be rewritten in matrix form where:

• The coefficient matrix \( A = \begin{bmatrix} 5 & 4 \ 10 & -6 \end{bmatrix} \)
• The variable matrix \( X = \begin{bmatrix} r \ s \end{bmatrix} \)
• The constant matrix \( B = \begin{bmatrix} -1 \ 5 \end{bmatrix} \)

This matrix representation simplifies calculations and allows us to use algebraic methods like Cramer's Rule or matrix inversion to find solutions efficiently. The process of expressing systems in matrix form is a critical step in exploring more advanced mathematical subjects.

Systems of Linear Equations

A system of linear equations comprises two or more linear equations that share the same set of variables. The main goal is often to find values for these variables that satisfy all the equations simultaneously.Using Cramer's Rule, we have a structured approach for systems where the number of equations equals the number of unknowns. To apply Cramer's Rule, we need:

• A non-zero determinant of the coefficient matrix, \( \det(A) ot= 0 \).
• Adjusted matrices for each variable, replacing one column at a time with the constant matrix.

In the provided exercise, Cramer's Rule is used to find values for \( r \) and \( s \) by:

• \( r = \frac{\det(A_r)}{\det(A)} = \frac{-14}{-70} = \frac{1}{5} \)
• \( s = \frac{\det(A_s)}{\det(A)} = \frac{35}{-70} = -\frac{1}{2} \)

This rule is applicable only when \( \det(A) eq 0 \), ensuring a unique solution exists for the system.