Problem 42 - Advanced Engineering Mathematics

Question

Use a CAS to solve the given system. \(\begin{aligned} x_{1}-x_{2}-x_{3}+2 x_{4}-x_{5} &=5 \\ 6 x_{1}+9 x_{2}-6 x_{3}+17 x_{4}-x_{5} &=40 \\ 2 x_{1}+x_{2}-2 x_{3}+5 x_{4}-x_{5} &=12 \\\ x_{1}+2 x_{2}-x_{3}+3 x_{4} &=7 \\ x_{1}+2 x_{2}+x_{3}+3 x_{4} &=1 \end{aligned}\)

Step-by-Step Solution

Answer

Solve the system using a CAS to find the values of \(x_1, x_2, x_3, x_4,\) and \(x_5\).

Step 1: Understand the System of Equations

We have five linear equations involving five unknowns: \(x_1, x_2, x_3, x_4,\) and \(x_5\). Our goal is to find the values of these variables that satisfy all the equations simultaneously.

Step 2: Set Up the Augmented Matrix

Write the system of equations as an augmented matrix. Each row in the matrix represents one of the equations, and each column corresponds to one of the variables \[\left[\begin{array}{ccccc|c}1 & -1 & -1 & 2 & -1 & 5 \6 & 9 & -6 & 17 & -1 & 40 \2 & 1 & -2 & 5 & -1 & 12 \1 & 2 & -1 & 3 & 0 & 7 \1 & 2 & 1 & 3 & 0 & 1 \\end{array}\right]\]

Step 3: Use a Computer Algebra System (CAS)

Input the augmented matrix into a CAS such as MATLAB, Mathematica, or a graphing calculator. Use the tool's function for solving linear systems (such as `Solve` or `rref` for reduced row-echelon form) to find the solutions.

Step 4: Interpret the Results

The CAS will provide you with the solution set for \(x_1, x_2, x_3, x_4,\) and \(x_5\). Interpret these results as the values for each variable that satisfy the given system of equations. The output from the CAS is a step that directly provides the solutions.

Key Concepts

System of Linear Equations

A system of linear equations comprises multiple linear equations that share the same set of variables. The challenge here is to find the values of these variables that satisfy all the equations at once. With problems involving many variables and equations, the systems can become complex, but there are systematic ways to find solutions. For these types of problems, we often use methods like substitution or elimination. However, with larger systems, it becomes efficient to use matrices or computational tools to organize and solve them effectively. In the given exercise, we have five equations involving the variables:

• \(x_1\)
• \(x_2\)
• \(x_3\)
• \(x_4\)
• \(x_5\)

Each equation represents a linear relationship among these unknowns. Our objective is to determine the set of values for the variables \(x_1, x_2, x_3, x_4,\) and \(x_5\) that simultaneously satisfy all five equations. This can be tackled effectively using an augmented matrix and a CAS.

Augmented Matrix

The augmented matrix is a powerful tool when dealing with systems of linear equations. It essentially combines the coefficients of each variable from the system into a matrix form, with an additional column for the constants. Each row in the augmented matrix represents one of the equations, and by lining them up, we can handle all equations collectively. The matrix transforms our written system into a format that can be easily manipulated and solved using matrix operations. In the exercise provided, the augmented matrix looks like this:\[\begin{bmatrix}1 & -1 & -1 & 2 & -1 & \vert & 5 \6 & 9 & -6 & 17 & -1 & \vert & 40 \2 & 1 & -2 & 5 & -1 & \vert & 12 \1 & 2 & -1 & 3 & 0 & \vert & 7 \1 & 2 & 1 & 3 & 0 & \vert & 1\end{bmatrix}\]This format is particularly advantageous for computational purposes as it sets the stage for using tools like a Computer Algebra System (CAS) to find solutions efficiently.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool specifically designed to handle complex algebraic computations. CAS can execute operations that involve solving equations, simplifying expressions, and performing integrations or differentiations. When it comes to solving a system of linear equations, CAS is invaluable. Let’s see why:

• It automates the row operations needed to bring the augmented matrix to a form suitable for extracting solutions.
• CAS works efficiently with the reduced row-echelon form (RREF) method, simplifying the entire process.
• Systems of any size can be addressed swiftly, ensuring both accuracy and speed.

To solve the given exercise system, you input the augmented matrix into a CAS, such as MATLAB or Mathematica. Commands like `Solve` or `rref` help directly yield the solutions: the values for \(x_1, x_2, x_3, x_4,\) and \(x_5\) that satisfy all the equations. This process minimizes errors and ensures a smooth, accurate solving procedure. CAS tools not only simplify computations but also enhance our understanding of the relationships between variables in large systems.