Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}x-2 y= & 2 \\ 7 x-14 y= & 14 \\ 3 x-6 y= & 6\end{array}\)

Step-by-Step Solution

Answer

The solution of the given system of linear equations using the Gauss-Jordan elimination method is: \[ x = 2 + 2y, \quad y \in \mathbb{R} \] There are infinitely many solutions, with x depending on the value of y.

Step 1: Write the system in augmented matrix form

To start, we'll re-write the given system of equations in its augmented matrix form, meaning that the coefficients of the variables will be placed in a matrix and the right-hand side of the equation will be separated by a vertical bar. \[ \left[\begin{array}{cc|c} 1 & -2 & 2 \\ 7 & -14 & 14 \\ 3 & -6 & 6 \end{array}\right] \]

Step 2: Row reduce the matrix

Now, we will begin the row reduction process to obtain the matrix in RREF. We start by eliminating the '7' below the pivotal '1' in the first row. To do this, we replace the second row with (-7 * Row 1 + Row 2). \[ \begin{array}{c} R_2 = -7R_1 + R_2 \end{array} \] Matrix after performing the operation: \[ \left[\begin{array}{cc|c} 1 & -2 & 2 \\ 0 & 0 & 0 \\ 3 & -6 & 6 \end{array}\right] \] Next, we will eliminate the '3' below the pivotal '1' in the first row. Replace the third row with (-3 * Row 1 + Row 3). \[ \begin{array}{c} R_3 = -3R_1 + R_3 \\ \end{array} \] Matrix after performing the operation: \[ \left[\begin{array}{cc|c} 1 & -2 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \] The matrix is now in its RREF.

Step 3: Interpret the reduced row-echelon form

Interpreting the RREF matrix, the first equation corresponds to the original system as: \[ x - 2y = 2 \] As the last two rows of the matrix have 0's in all columns and the right-hand side, the system of equations is consistent, and there are infinitely many solutions since there's no unique value for \(y\). To express the solution of the system, we can isolate x in terms of y: \[ x = 2 + 2y \]

Step 4: Write the final solution

The solution of the given system of linear equations using the Gauss-Jordan elimination method is: \[ x = 2 + 2y, \quad y \in \mathbb{R} \] This means that there are infinitely many solutions, with x depending on the value of y.

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan Elimination is a systematic method used to solve systems of linear equations. It involves transforming a given matrix, derived from the system of equations, into a diagonal form called the Reduced Row Echelon Form (RREF). This form simplifies the process of finding solutions by making the relationship between the variables clear. The steps to accomplish this include

• creating an augmented matrix,
• performing row operations to simplify it, and
• identifying the solutions from the final RREF matrix.

An important part of this method is the strategic use of row operations - swapping, scaling, and row addition or subtraction - to achieve zeros below and, if necessary, above every leading 1 in the matrix's columns. This simplicity is key; it reveals the nature of the solutions, whether unique, infinite, or non-existent.

Augmented Matrix

An augmented matrix is used to represent a system of linear equations. It consolidates the coefficients of the variables and the constants from the equations into a single matrix form. For example, given the system:

• \( x - 2y = 2 \)
• \( 7x - 14y = 14 \)
• \( 3x - 6y = 6 \)

This system can be represented as:\[\begin{bmatrix} 1 & -2 & | & 2 \ 7 & -14 & | & 14 \ 3 & -6 & | & 6 \end{bmatrix}\]This matrix separates the coefficients (on the left) from the constants (on the right), making it easier to apply matrix operations. Augmented matrices play a critical role in methods like Gauss-Jordan Elimination because they allow for a straightforward manipulation of multiple equations at once.

Reduced Row Echelon Form (RREF)

Reduced Row Echelon Form (RREF) is the simplified form of a matrix, where each leading entry in a row is 1 and is positioned to be the only non-zero number in its column. RREF clarifies the relationship between variables, allowing for the straightforward extraction of solutions to the system of equations.To reach the RREF, one must transform the matrix using elementary row operations:

• swap rows to position leading ones as desired,
• multiply a row by a non-zero scalar to create leading ones, and
• add or subtract multiples of rows to form zeros where needed above and below leading ones.

For the given system, the elimination of the coefficients below the leading 1 in the first row results in:\[\begin{bmatrix} 1 & -2 & | & 2 \ 0 & 0 & | & 0 \ 0 & 0 & | & 0 \end{bmatrix}\]In RREF, it is clear whether there are no solutions, a unique solution, or infinitely many solutions.

Infinitely Many Solutions

A system of linear equations can have infinitely many solutions when there is a relationship between the variables that allows multiple solutions to satisfy all equations simultaneously. This scenario often occurs when at least one row of the matrix in RREF consists entirely of zeros, indicating dependent equations.The solution must be expressed in terms of the variables that can take any value, often called free variables. For example, in the system:

• Equation from RREF: \( x - 2y = 2 \)

The variable \( y \) can take any value in the real numbers set, \( \mathbb{R} \). Consequently, \( x \) is derived as a function of \( y \):\[ x = 2 + 2y \]This means that for every real number \( y \), there is a corresponding \( x \). Such solutions are graphical lines on a two-dimensional plane, representing a complete set of solutions for the system.