Problem 8 - Advanced Engineering Mathematics

Question

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}+2 x_{2}-4 x_{3}=9 \\ 5 x_{1}-x_{2}+2 x_{3}=1 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions parameterized by \(x_3 = t\): \(x_1 = 1, x_2 = 2t + 4, x_3 = t\).

Step 1: Write the System of Equations in Matrix Form

Start by converting the given system of equations into an augmented matrix. The system of equations is:\[\begin{align*}x_{1} + 2x_{2} - 4x_{3} &= 9 \5x_{1} - x_{2} + 2x_{3} &= 1\end{align*}\]The augmented matrix form is:\[\begin{bmatrix} 1 & 2 & -4 & | & 9 \5 & -1 & 2 & | & 1 \end{bmatrix}\]

Step 2: Perform Row 1 Operations

We will first work with Row 1. Currently, it does not need changes, so we will use it as a pivot row:\[R_1 = \begin{bmatrix} 1 & 2 & -4 & | & 9 \end{bmatrix}\]

Step 3: Eliminate Variable from Row 2

To eliminate the variable \(x_1\) from Row 2, use row operations. Multiply Row 1 by 5 and subtract it from Row 2:\[R_2 = R_2 - 5 \times R_1\]This gives:\[R_2 = (5, -1, 2, |, 1) - (5, 10, -20, |, 45) = (0, -11, 22, |, -44)\]

Step 4: Simplify Row 2

Simplify Row 2 by dividing through by -11 to make leading coefficient 1:\[R_2 = \begin{bmatrix} 0 & -11 & 22 & | & -44 \end{bmatrix} \rightarrow \begin{bmatrix} 0 & 1 & -2 & | & 4 \end{bmatrix}\]

Step 5: Back Substitute to Solution or Interpret

We find that now we have:\[\begin{align*}x_{1} + 2x_{2} - 4x_{3} &= 9 \x_{2} - 2x_{3} &= 4\end{align*}\]This system expresses two equations where one of the variables can be a free variable (since there are two equations and three variables). It implies an infinite set of solutions, parameterized by \(x_3\).

Step 6: Express Solution in Parametric Form

Let \(x_3 = t\), a parameter. Then from the second equation we have \(x_2 = 2t + 4\). Substitute \(x_2\) and \(x_3\) into the first equation:\[x_1 + 2(2t + 4) - 4t = 9\]Simplifying, we find \(x_1 = 9 - 8 = 1\), implying:\[x_1 = 1, \quad x_2 = 2t + 4, \quad x_3 = t\]

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vector spaces and the linear mappings between these spaces. It forms the foundation for various other fields of study and is essential for solving systems of linear equations. This discipline provides tools for dealing with vectors, matrices, and transformations that can be applied across different dimensions.

In linear algebra, we often work with systems of linear equations. These equations can be expressed in matrix form, allowing for efficient solution methods such as Gaussian elimination. Matrices serve as a compact representation and make complex computations easier to handle.

• **Vectors:** Fundamental elements stored in matrix rows or columns, representing points in space.
• **Matrices:** Rectangular arrays of numbers representing a linear transformation or system of equations.
• **Transformations:** Functions that map vectors to other vectors, preserving the structures of vector spaces.

Understanding these core components is crucial to mastering Gaussian elimination and solving larger systems efficiently.

Systems of Equations

Systems of equations are collections of two or more equations with a common set of variables. These equations can be linear or nonlinear, but in the context of linear algebra, we focus on systems of linear equations. Solving these systems involves finding the set of all possible values for the variables that satisfy all equations simultaneously. This process can be carried out using various methods, including substitution, elimination, and matrix operations like Gaussian elimination.

A system of equations can have different types of solutions:

• **Unique solution:** A single set of values satisfies all equations. This occurs when there is a unique intersection point of all equation lines.
• **No solution:** No set of values satisfies all equations, often resulting from parallel lines with different intercepts.
• **Infinite solutions:** Multiple values satisfy the equations, typically arising from coincident lines or having more variables than equations, leading to underdetermined systems.

Solving these systems is key in numerous applications, from engineering calculations to economics modeling.

Matrix Operations

Matrix operations are the set of rules that govern the manipulation of matrices. These operations allow us to perform arithmetic on matrices, solve systems of equations, and find solutions to various mathematical problems efficiently. In the context of Gaussian elimination, understanding matrix operations is essential as it provides a structured approach to simplifying systems of equations.

Key matrix operations include:

• **Addition and Subtraction:** Matrices can be added or subtracted element-wise if they have the same dimensions.
• **Multiplication:** Multiplying matrices involves row-by-column calculations and is necessary for many linear transformations.
• **Row Operations:** Specific manipulations like swapping, multiplying, or adding rows are used during Gaussian elimination to streamline a matrix into row-echelon form.

These operations form the basis for algorithms that efficiently handle large datasets and enable solutions to complex linear systems.