Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a System of Linear Equations (a) write the system of equations as a matrix equation \(A X=B\) and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix X. Use a graphing utility to check your solution. $$\left\\{\begin{array}{rr} x_{1}-x_{2}+4 x_{3}= & 17 \\ x_{1}+3 x_{2} & =-11 \\ -6 x_{2}+5 x_{3} & =40 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution matrix X to the system of equations is \[ X = \begin{pmatrix} -2 \ 3 \ 8 \end{pmatrix} \].

Step 1: Write the System of Equations as a Matrix Equation Ax=B

Firstly, the system of equations can be represented as a matrix. In matrix form, the set of equations becomes \[ \begin{pmatrix} 1 & -1 & 4 \ 1 & 3 & 0 \ 0 & -6 & 5 \end{pmatrix} \begin{pmatrix} x_{1} \ x_{2} \ x_{3} \end{pmatrix} = \begin{pmatrix} 17 \ -11 \ 40 \end{pmatrix} \]. Here, matrix A is a 3 x 3 matrix with the coefficients of the variables, matrix X is the variable matrix, and matrix B is the constants matrix.

Step 2: Form the Augmented Matrix [A: B]

Combine matrices A and B to form the augmented matrix as \[ \left[ \begin{array}{ccc|c} 1 & -1 & 4 & 17 \ 1 & 3 & 0 & -11 \ 0 & -6 & 5 & 40 \end{array} \right] \].

Step 3: Gauss-Jordan Elimination on the Augmented Matrix

Apply Gauss-Jordan elimination to this augmented matrix (add/subtract multiples of rows as necessary to get the form in which the leading element of each row is 1 (pivot 1) and all other elements in the column beneath and above the pivot 1 are zero). Let's subtract row 1 from row 2, multiply row 2 by -1/2, add row 2 and row 3 times 3, and add row 3 times 4 to row 1. - the final resulting matrix is \[ \left[ \begin{array}{ccc|c} 1 & 0 & 0 & -2 \ 0 & 1 & 0 & 3 \ 0 & 0 & 1 & 8 \end{array} \right] \]. This translates back into the system of equations as \(x_{1} = -2, x_{2} = 3, x_{3} = 8\).

Step 4: Checking the Solution using Graphing Utility (Optional)

You can verify the solution by substituting \(x_{1} = -2, x_{2} = 3, x_{3} = 8\) into the original system of equations and confirming it satisfies all three equations or by using a graphing utility although the use of a graphing utility is not described in this format.

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan elimination is a powerful method for solving systems of linear equations. It involves using elementary row operations on an "augmented matrix" to reduce it to a form where the solution can be easily identified. An augmented matrix combines the coefficients and constants of the system of equations into one matrix.
As you manipulate the rows using operations such as adding or subtracting multiples of other rows, you aim to reach "reduced row echelon form." In this form, each leading coefficient (also called a pivot) in every row is 1, and all other elements in the pivot's column are zeros. By achieving this, the system can be translated back into a simple form where each variable equals a constant.
To illustrate:

• Start with the matrix formed from your system of equations.
• Perform operations like row swapping, scaling rows (multiplying all elements by a non-zero scalar), and adding/subtracting multiples of rows.
• Continue until you reach a matrix where solutions are clear.

This method simplifies the solving process, making it straightforward to solve complex systems.

Matrix Representation

Matrix representation is a crucial concept in understanding linear equations, especially when dealing with multiple variables. It allows you to write systems of equations in a streamlined way, which not only makes them easier to handle but also to solve using computational methods.
When you write a system of equations like:

• \[x_{1}-x_{2}+4 x_{3} = 17\]
• \[x_{1}+3 x_{2} = -11\]
• \[-6 x_{2}+5 x_{3} = 40\]

into a matrix form, it becomes easier to visualize relationships and operations:

• The "coefficient matrix" contains the coefficients of the variables.
• The "variable matrix" stores the unknowns or variable names.
• The "constant matrix" represents the results or constants on the right side of the equations.

This method is efficient and serves as a foundation for various solving techniques, such as the Gauss-Jordan elimination.

System of Equations

A system of equations consists of two or more equations that need to be solved simultaneously. It's a common part of algebra that deals with finding values for multiple variables that satisfy all equations in the system.
In linear systems, each equation is a straight line when represented graphically, and the solutions are the intersection points of those lines.
There are a few potential outcomes when solving a system of linear equations:

• A unique solution: The system has one point of intersection where all lines meet.
• No solution: The system consists of parallel lines that never intersect.
• Infinite solutions: All equations represent the same line, thus sharing all points.

Understanding systems of equations is essential because they model many real-world situations. Using algebraic methods like substitution, elimination, and matrix operations, they can quickly be reduced to a form where solutions are apparent.