Problem 2 - Advanced Engineering Mathematics

Question

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

Solution {x_1 = -\frac{16}{3}, x_2 = -10}.

Step 1: Write Augmented Matrix

First, write the system of equations as an augmented matrix. The system is \( \begin{aligned} 3x_1 - 2x_2 &= 4 \ x_1 - x_2 &= -2 \end{aligned} \). The augmented matrix is: \[ \begin{bmatrix} 3 & -2 & | & 4 \ 1 & -1 & | & -2 \end{bmatrix} \]

Step 2: Perform Row Operations

We want to eliminate the first element of the second row. Use the first row to make the first element of the second row zero. Subtract \( \frac{1}{3} \times \text{Row 1} \) from Row 2 to get:\[ \begin{bmatrix} 3 & -2 & | & 4 \ 0 & \frac{1}{3} & | & -\frac{10}{3} \end{bmatrix} \]

Step 3: Make Leading Coefficient 1

Now, make the leading coefficient in the second row equal to 1 by multiplying the entire row by 3:\[ \begin{bmatrix} 3 & -2 & | & 4 \ 0 & 1 & | & -10 \end{bmatrix} \]

Step 4: Eliminate Second Variable from First Row

Use the second row to eliminate the second variable from the first row. Add \(2 \times \text{Row 2}\) to Row 1:\[ \begin{bmatrix} 3 & 0 & | & -16 \ 0 & 1 & | & -10 \end{bmatrix} \]

Step 5: Solve for Variables

Now that the matrix is in row-echelon form, read off the solutions:\( x_2 = -10 \) from the second row.Plug \( x_2 = -10 \) into Row 1 to solve for \( x_1 \):\( 3x_1 = -16 \rightarrow x_1 = -\frac{16}{3} \).

Step 6: Check Solution

Substitute \( x_1 = -\frac{16}{3} \) and \( x_2 = -10 \) back into the original equations to verify the solution.Equation 1: \( 3(-\frac{16}{3}) - 2(-10) = -16 + 20 = 4 \), correct.Equation 2: \( -\frac{16}{3} - (-10) = \frac{14}{3} = -2 \), correct.

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan Elimination is a powerful method used to solve systems of linear equations. It is an extension of Gaussian elimination and aims to transform a matrix into a form where the solution can be easily identified. Here's how it works:

• Begin with an augmented matrix, which includes the coefficients of the variables and the constants from the right-hand side of the equations.
• Use row operations to simplify the matrix into its Reduced Row Echelon Form (RREF), where the leading coefficient (the first non-zero number from the left, in each row) is 1, and all other entries in that column are zeros.
• In the end, the matrix is transformed in a way that each equation has one variable solved, making it straightforward to read off the solutions directly from the matrix.

By systematically using row operations, Gauss-Jordan elimination not only finds solutions but also checks their uniqueness or existence.

Augmented Matrix

The Augmented Matrix is an essential tool in solving linear equations using methods like Gaussian or Gauss-Jordan elimination. Here's what you need to know:

• An augmented matrix includes both the coefficients of the variables and the constants from the equations on the right-hand side.
• It is formed by writing the coefficients of each variable in a row for each equation, followed by a vertical `|`, and then the constant term(s).

For example, given a system of equations:
\(\begin{aligned}3x_1 - 2x_2 = 4 \x_1 - x_2 = -2\end{aligned}\)
The augmented matrix representation would be:
\[\begin{bmatrix}3 & -2 & | & 4 \1 & -1 & | & -2\end{bmatrix}\]
This representation allows us to apply row operations efficiently to solve the system.

Row Operations

Row Operations are crucial in transforming matrices when solving linear systems. There are three types you can perform to simplify the matrix:

• Swapping Rows: You can exchange the positions of any two rows.
• Multiplying a Row by a Non-zero Constant: Any row can be multiplied by a non-zero number to make calculations easier.
• Adding or Subtracting Rows: You can add or subtract the multiples of one row to another to eliminate variables and simplify the matrix.

These operations do not change the solutions of the system; they only simplify the equations to reach a form where the solution can be easily identified.

Linear Equations

Linear Equations are equations that represent straight lines when graphed on a coordinate plane. They have general forms such as:

• \(ax + by = c\)
• \(x = d\)

Where \(a\), \(b\), and \(c\) are constants. In linear systems, you solve multiple linear equations simultaneously. Each equation represents a line or plane, and the solution corresponds to the point(s) where these intersect.

• Solutions can be unique (a single point), infinite (the same line), or nonexistent (parallel lines that never intersect).
• Methods such as Gauss-Jordan elimination help determine these solutions efficiently by reducing the equations to simpler forms.

Understanding linear equations and how they intersect is fundamental in fields including economics, engineering, and sciences.