Problem 2 - 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. $$ \begin{aligned} 3 x_{1}-2 x_{2} &=4 \\ x_{1}-x_{2} &=-2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \((8, 10)\).

Step 1: Write the Augmented Matrix

Start by writing the system of equations as an augmented matrix. For the system \( 3x_1 - 2x_2 = 4 \) and \( x_1 - x_2 = -2 \), the augmented matrix is: \[\begin{pmatrix} 3 & -2 & | & 4 \1 & -1 & | & -2 \end{pmatrix}\]

Step 2: Apply Row Operations to Obtain Upper Triangular Form

We perform Gaussian elimination to transform the matrix into an upper triangular form. First, we'll subtract 1/3 times the first row from the second row to create a 0 below the first element of the first column:\[R_2 = R_2 - \frac{1}{3}R_1 \]The resulting matrix is:\[\begin{pmatrix} 3 & -2 & | & 4 \0 & -\frac{1}{3} & | & -\frac{10}{3} \end{pmatrix}\]

Step 3: Solve for One Variable

Now that we have the matrix in upper triangular form, solve for the variable represented in the second row. From row 2, we get:\[-\frac{1}{3}x_2 = -\frac{10}{3} \]By multiplying both sides by \(-3\), we find:\[x_2 = 10\]

Step 4: Substitute Back to Find the Other Variable

Substitute \( x_2 = 10 \) back into one of the original equations to find \( x_1 \). Using the second equation, \( x_1 - x_2 = -2 \):\[x_1 - 10 = -2 \x_1 = 8\]

Step 5: Write the Solution

The solution to the system is \( x_1 = 8 \) and \( x_2 = 10 \). Therefore, the ordered pair solution is \((8, 10)\).

Key Concepts

Gauss-Jordan elimination

Gauss-Jordan elimination is an efficient method for solving systems of linear equations. It extends the process of Gaussian elimination further to simplify a matrix into what is known as row-reduced echelon form. This method makes it easier to directly interpret the solutions of the equations.

When using Gauss-Jordan elimination, we aim to achieve a matrix where the entries below and above each pivot are zero, leaving a diagonal of ones across the main diagonal. This way, the solution is trivial to read directly from the matrix without further back-substitution.

Key steps in Gauss-Jordan elimination include:

• Reducing each pivot to 1 by dividing the entire row by the pivot value.
• Making all elements above and below each pivot zero by adding or subtracting rows appropriately.

This method is particularly useful when dealing with complex systems or when solving multiple systems using similar matrices.

augmented matrix

An augmented matrix is a key concept when solving systems of linear equations. It simplifies the process by combining the coefficients and constants from each equation into a single matrix format.

For the system given in the original exercise, \[\begin{aligned}3x_1 - 2x_2 &= 4 \x_1 - x_2 &= -2\end{aligned}\]The augmented matrix representation looks like this:\[\begin{pmatrix} 3 & -2 & | & 4 \1 & -1 & | & -2 \end{pmatrix}\]Each row corresponds to one equation, with the vertical line separating the coefficients on the left from the constants on the right.

• The left side of the augmented matrix contains the coefficients of the variables.
• The right side contains the constants from each equation.

This format makes it easy to apply row operations to find the solution.

upper triangular form

The upper triangular form is an important step in the process of Gaussian elimination used to solve systems of equations. It refers to a specific type of matrix where all of the entries below the main diagonal are zero. This simplifies the process of solving the equations by back-substitution.

To convert an augmented matrix into upper triangular form:

• Use row operations to create zeros below each pivot element (non-zero entry in each row starting from the top left).
• This often involves scaling rows and subtracting multiples of one row from another.

For example, in step 2 of the original solution, after the second row operation, the matrix becomes:\[\begin{pmatrix} 3 & -2 & | & 4 \0 & -\frac{1}{3} & | & -\frac{10}{3} \end{pmatrix}\]With zeros below the main diagonal, it's easier to solve for the variables beginning from the bottom row upwards.

linear systems of equations

Linear systems of equations are collections of two or more linear equations that involve the same set of variables. They are fundamental in various areas of mathematics and applied sciences, allowing us to model and solve problems where multiple conditions or constraints interact.

The example provided from the original exercise is a simple linear system:\[\begin{aligned}3x_1 - 2x_2 &= 4 \x_1 - x_2 &= -2\end{aligned}\]These types of systems can be solved using various methods such as substitution, elimination, or more advanced matrix techniques like Gaussian and Gauss-Jordan elimination.

The goal when solving a linear system is to find the values of the variables that make all the equations true simultaneously.

• Simple systems are often solved by graphing or direct manipulation.
• More complex systems necessitate systematic methods like Gaussian elimination for simplicity and accuracy.

Linear systems are ubiquitous in both academic settings and real-life applications, from engineering to economics.