Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}2 x+3 y= & 2 \\ x+3 y= & -2 \\ x-y= & 3\end{array}\)

Step-by-Step Solution

Answer

The solution to the given system of linear equations using the Gauss-Jordan elimination method is \(x = 3\) and \(y = 1\).

Step 1: Write the augmented matrix corresponding to the system

: Associate each equation with a row in the matrix and write it down as follows: \[ \begin{array}{ cc | c } 2 & 3 & 2 \\ 1 & 3 & -2 \\ 1 & -1 & 3 \end{array} \]

Step 2: Transform the matrix into its reduced row-echelon form (diagonal form)

: Perform the following row operations: 1. Swap row 1 and row 2 2. Multiply row 1 by 2, and subtract it from row 3 to eliminate x from row 3 3. Add row 2 to row 3 to eliminate y from row 3 4. Divide row 3 by 16 to make the element in the second position of row 3 equal to 1 5. Multiply row 3 by 3, and add it to row 2 to eliminate y from row 2 The transformed matrix will be: \[ \begin{array}{ cc | c } 1 & 3 & -2 \\ 0 & -1 & 4 \\ 0 & 0 & 16 \end{array} \]

Step 3: Get the solutions from the matrix in diagonal form

: Now, we can read the solutions directly from the matrix: 1. From the third row, we have \(16y = 16\), so \(y = 1\). 2. From the second row, we have \(-y = 4\), which means, using y = 1, \(x = 3\). Therefore, the solution of the system of linear equations is \(x = 3\) and \(y = 1\).

Key Concepts

Linear Equations

Linear equations are mathematical statements that express a relationship of equality between algebraic expressions. These equations involve variables with a degree of one, hence the term 'linear'. A simple example of a linear equation is:

• \( ax + by = c \)

where \( a, b, \) and \( c \) are constants, and \( x \) and \( y \) are variables.
Linear equations are fundamental in algebra because they allow us to describe real-world situations using mathematical language. They often appear in a system, allowing us to find values of the variables that satisfy all equations simultaneously. Solving these systems definitively identifies whether there is a unique solution, infinite solutions, or no solution, which provides insight into the problem at hand.

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra when dealing with systems of equations. This matrix combines the coefficients of the variables and the constant terms from the equations. Consider this handy format:

• For the linear system \[\begin{align*}ax + by &= c \dx + ey &= f\end{align*}\]

the augmented matrix becomes \[\begin{bmatrix}a & b & | & c \d & e & | & f\end{bmatrix}\]
The vertical bar separates the coefficients from the constants, highlighting the role of each term in the equations. Working with matrices allows for streamlined computational methods, like the Gauss-Jordan elimination method, which simplifies complex systems into manageable forms.

Row-Echelon Form

The row-echelon form of a matrix is crucial in solving systems of linear equations using methods like Gauss-Jordan elimination. A matrix is in row-echelon form when:

• All nonzero rows are above any rows of all zeros.
• The leading entry (first nonzero number from the left) of each nonzero row is to the right of the leading entry of the previous row.
• The leading entry in any row is 1.

Transforming an augmented matrix into row-echelon form simplifies the system, allowing easy identification of solutions. As each step reduces complexity, finding variable values becomes straightforward. This transformation involves systematic operations such as row swaps, scaling rows, and adding multiples of one row to another.

Row Operations

Row operations are the techniques used to manipulate matrices when solving systems of equations. They are pivotal in approaches like the Gauss-Jordan elimination process. There are three fundamental types of row operations:

• Swapping two rows: Changes the order without altering the solution.
• Multiplying a row by a nonzero scalar: Adjusts a row’s magnitude, maintaining its equation’s validity.
• Adding or subtracting a multiple of one row to another: Aims to eliminate variables from certain positions.

These operations enable the transformation of a matrix to its simplified forms, such as row-echelon or reduced row-echelon form, highlighting variable solutions clearly. This systematic method is crucial for reducing complex systems into solvable formats efficiently.