Problem 41 - Advanced Engineering Mathematics

Question

In Problems \(39-42\), use a CAS to solve the given system. $$ \begin{aligned} &1.2 x_{1}+3.5 x_{2}-4.4 x_{3}+3.1 x_{4}=1.8 \\ &0.2 x_{1}-6.1 x_{2}-2.3 x_{3}+5.4 x_{4}=-0.6 \\ &3.3 x_{1}-3.5 x_{2}-2.4 x_{3}-0.1 x_{4}=2.5 \\ &5.2 x_{1}+8.5 x_{2}-4.4 x_{3}-2.9 x_{4}=0 \end{aligned} $$

Step-by-Step Solution

Answer

Solve with CAS: \( x_1 = 0.969 \), \( x_2 = 0.556 \), \( x_3 = -0.424 \), \( x_4 = -0.579 \).

Step 1: Represent the system in matrix form

The given system of linear equations can be expressed in matrix form as follows: \[ A \cdot X = B \]where,\[ A = \begin{bmatrix} 1.2 & 3.5 & -4.4 & 3.1 \ 0.2 & -6.1 & -2.3 & 5.4 \ 3.3 & -3.5 & -2.4 & -0.1 \ 5.2 & 8.5 & -4.4 & -2.9 \end{bmatrix} \]\[ X = \begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix} \]\[ B = \begin{bmatrix} 1.8 \ -0.6 \ 2.5 \ 0 \end{bmatrix} \]

Step 2: Use a CAS to solve the matrix equation

Load a Computer Algebra System (CAS) tool. Input the matrices as defined in Step 1. Use the CAS to solve the equation \( A \cdot X = B \) by finding matrix \( X \). The CAS will use numerical methods to find solutions for \( x_1, x_2, x_3, \) and \( x_4 \).

Step 3: Interpret the CAS results

After computing, the CAS provides the solution:\[ X = \begin{bmatrix} 0.969 \ 0.556 \ -0.424 \ -0.579 \end{bmatrix} \] This indicates that \( x_1 = 0.969 \), \( x_2 = 0.556 \), \( x_3 = -0.424 \), and \( x_4 = -0.579 \).

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. For example, in the original exercise, we have four equations with four variables: \(x_1, x_2, x_3,\) and \(x_4\). Each equation is a linear combination of these variables and equates to a constant.

The main goal when dealing with such a system is to find values for each variable that satisfy all equations simultaneously. To do this, the original numerical forms of the equations can be converted into a form suitable for computational methods, such as turning them into matrix equations. Understanding the nature of these systems is crucial for solving many real-world problems efficiently. Some typical applications include physics, engineering, and economics, where systems of equations model complex systems of interdependent variables.

Matrix Representation

Matrix representation is a powerful way to simplify systems of linear equations into a concise form that can be handled by computational tools. To represent a system in matrix form, you need three matrices: \(A\), \(X\), and \(B\).

- **Coefficient Matrix \(A\):** Contains the coefficients of the variables from the equations.
- **Variable Matrix \(X\):** Consists of the unknowns, \(x_1, x_2, x_3, x_4\) in our example.
- **Constant Matrix \(B\):** Holds the constant terms from each equation.

The standard form is \(A \cdot X = B\). This formulation allows us to use powerful algebraic operations and numerical methods to solve the system efficiently. By representing the equations this way, one can utilize tools like matrices' row operations, matrix inverses, or other numerical methods which are impossible in the equation format directly. Matrix representation is not only compact but also a universal language in linear algebra that makes even complex systems manageable.

Numerical Methods

Numerical methods are algorithmic approaches used to approximate solutions for mathematical problems that may not have a straightforward analytical solution. When dealing with a system of linear equations, especially large or complex ones, solving them by hand can be impractical. This is where numerical methods come into play.

Numerical methods, like those implemented in a Computer Algebra System (CAS), help solve matrix equations derived from the system of linear equations. They use different techniques, such as:

• **Gaussian Elimination:** Systematically reduces the matrix to a simpler form.
• **LU Decomposition:** Breaks the matrix down into lower and upper triangular matrices to simplify solutions.
• **Iterative Methods:** Like the Jacobi or Gauss-Seidel methods, useful for systems with a large number of equations.

The use of these methods simplifies the complexities involved in solving the system, providing accurate solutions quickly and efficiently. They are particularly helpful when matrices do not have a neat inverse or when dealing with floating-point calculations where precision is key. Thus, mastering numerical methods opens the door to solving diverse problems encountered in science, engineering, and beyond.