Problem 2 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{array}{r} x_{1}+x_{2}=4 \\ 2 x_{1}-x_{2}=2 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x_1 = 2\) and \(x_2 = 2\).

Step 1: Write the System in Matrix Form

The given equations are: \(x_1 + x_2 = 4\) and \(2x_1 - x_2 = 2\). To apply Cramer's Rule, we first write these in matrix form: \[ \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix} \] Let \(A\) be the matrix \(\begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix}\), \(\mathbf{x}\) be \(\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\), and \(\mathbf{b}\) be \(\begin{bmatrix} 4 \ 2 \end{bmatrix}\).

Step 2: Calculate the Determinant of the Coefficient Matrix

Next, we calculate the determinant of the matrix \(A\), denoted as \(\mathrm{det}(A)\):\[ \mathrm{det}(A) = \begin{vmatrix} 1 & 1 \ 2 & -1 \end{vmatrix} = (1)(-1) - (1)(2) = -1 - 2 = -3 \]

Step 3: Calculate the Determinant of the Matrix Substituting Each Column

Create matrices \(A_1\) and \(A_2\) by replacing columns of \(A\) with \(\mathbf{b}\):For \(x_1\):\[ A_1 = \begin{bmatrix} 4 & 1 \ 2 & -1 \end{bmatrix} \]\[ \mathrm{det}(A_1) = \begin{vmatrix} 4 & 1 \ 2 & -1 \end{vmatrix} = (4)(-1) - (1)(2) = -4 - 2 = -6 \]For \(x_2\):\[ A_2 = \begin{bmatrix} 1 & 4 \ 2 & 2 \end{bmatrix} \]\[ \mathrm{det}(A_2) = \begin{vmatrix} 1 & 4 \ 2 & 2 \end{vmatrix} = (1)(2) - (4)(2) = 2 - 8 = -6 \]

Step 4: Apply Cramer’s Rule to Find the Solution

Cramer's Rule states that \( x_i = \frac{\mathrm{det}(A_i)}{\mathrm{det}(A)}\). Calculate each variable:\(x_1\):\[ x_1 = \frac{\mathrm{det}(A_1)}{\mathrm{det}(A)} = \frac{-6}{-3} = 2 \]\(x_2\):\[ x_2 = \frac{\mathrm{det}(A_2)}{\mathrm{det}(A)} = \frac{-6}{-3} = 2 \]

Step 5: Conclusion

The solution for the system of equations using Cramer's Rule is \(x_1 = 2\) and \(x_2 = 2\).

Key Concepts

System of Equations

A system of equations consists of multiple equations that share the same set of unknowns. In our exercise, we have two equations with the variables \(x_1\) and \(x_2\). These are:

• \(x_1 + x_2 = 4\)
• \(2x_1 - x_2 = 2\)

The goal is to find values for \(x_1\) and \(x_2\) that satisfy both equations simultaneously. Such problems often appear in fields like physics and engineering, where multiple conditions or phenomena are interconnected.
Cramer's Rule provides an effective method for solving these linear systems, assuming the system meets certain criteria—mainly that the determinant of the coefficient matrix is not zero. By using matrices to represent the system, we simplify complex calculations and leverage matrix algebra for straightforward solutions.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It provides important properties regarding the matrix and the associated linear equations. For our 2x2 matrix:\[ A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \]The determinant is found using the formula:\[ \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \]Applying this, \( \text{det}(A) = (1)(-1) - (1)(2) = -1 - 2 = -3 \).
This determinant tells us about the system of equations:

• If \( \text{det}(A) eq 0 \), the system has a unique solution, which is the case here.
• If \( \text{det}(A) = 0 \), the system might have infinitely many solutions or none at all.

In using Cramer's Rule, the determinant plays a key role in calculating the solutions to the variables \(x_1\) and \(x_2\). It essentially acts as a divisor in the rule's formula, confirming whether a unique solution exists.

Matrix Algebra

Matrix algebra refers to operations on matrices, which are arrays of numbers that model systems of equations. Understanding matrices is crucial for solving complex systems efficiently. In this exercise, we used matrix representation and associated operations to find solutions to the system of equations.Our matrix representation is:

• Coefficient matrix: \[ \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \]
• Matrix of variables: \[ \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \]
• Constant matrix: \[ \begin{bmatrix} 4 \ 2 \end{bmatrix} \]

By transforming the system into matrix form, we utilized operations such as calculating determinants and performing matrix multiplications to solve for \(x_1\) and \(x_2\) through Cramer's Rule. This approach not only simplifies the solution process but also highlights the interconnected nature of algebraic systems, making it easier to handle larger systems or those with more complex structures.