Problem 13 - Algebra and Trigonometry

Question

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} w-3 x+y-4 z &=4 \\ -2 w+x+2 y &=-2 \\ 3 w-2 x+y-6 z &=2 \\ -w+3 x+2 y-z &=-6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The complete solution to the given system of equations is \((w, x, y, z) = (1, -1, 3, -1)\)

Step 1: Write the augmented matrix

The systems of equations can be represented as a augmented matrix as follows: \[\begin{bmatrix} 1 & -3 & 1 & -4 & 4 \ -2 & 1 & 2 & 0 & -2 \ 3 & -2 & 1 & -6 & 2 \ -1 & 3 & 2 & -1 & -6 \end{bmatrix}\]

Step 2: Apply Gaussian Elimination

Use a sequence of row operations to transform the original matrix to row-echelon form. These operations include swapping the order of rows, multiplying a row by a nonzero number, and adding (or subtracting) a multiple of one row to another row.

Step 3: Make the leading coefficient '1'

For the row-echelon form, each leading coefficient should be 1 (to the right of any sequence of zeros). Use row operations to make these adjustments.

Step 4: Set the system of equations

After completing the Gaussian elimination, rewrite each row of the matrix as an equation and find the solution set.

Key Concepts

Systems of Equations

When we talk about systems of equations, we refer to a collection of two or more equations with the same set of variables. Our goal is usually to find the values of these variables that satisfy all equations simultaneously. Systems can have one solution, many solutions, or no solution at all. For example, the system given in the exercise

\[ \left\{\begin{aligned}w-3 x+y-4 z &=4 \-2 w+x+2 y &=-2 \3 w-2 x+y-6 z &=2 \-w+3 x+2 y-z &=-6\end{aligned}\right. \]

is a set of four equations in four variables: w, x, y, and z. Solving such a system manually could be daunting. This is where methods like Gaussian elimination come in handy, simplifying the process of finding solutions.

Row-Echelon Form

The row-echelon form is a particular arrangement of a matrix that arises from a system of linear equations. It has three essential features:

• All nonzero rows are above any row of all zeros.
• The leading entry of a nonzero row, called the pivot, is always strictly to the right of the leading entry of the row above it.
• All entries in a column below a pivot are zeros.

The row-echelon form does not have to have 1s as leading entries, but it sets the stage for simpler calculations. In the context of Gaussian elimination, we often modify the row-echelon form further into a reduced row-echelon form, where each pivot is indeed 1 and is the only non-zero entry in its column.

Augmented Matrix

An augmented matrix represents a system of equations in a compact form, combining the coefficients of the variables along with the equalities on the right-hand side into one matrix. In the exercise, the augmented matrix is given as

\[\begin{bmatrix} 1 & -3 & 1 & -4 & 4 \ -2 & 1 & 2 & 0 & -2 \ 3 & -2 & 1 & -6 & 2 \ -1 & 3 & 2 & -1 & -6 \end{bmatrix}\]

This matrix includes the coefficients of w, x, y, and z from each equation and the constant terms on the right side of the equations (after the equal sign). The vertical representation separates the variable coefficients from the solutions, which is critical in identifying the relationships between the operands.

Row Operations

Row operations are the tools we use in Gaussian elimination to transform the original matrix into row-echelon form, and eventually, into reduced row-echelon form. There are three types of row operations:

• Row swapping: exchanging two rows.
• Row multiplication: multiplying all the entries of a row by a non-zero constant.
• Row addition: adding to one row a multiple of another.

These operations are used to simplify the matrix step by step while ensuring that the equations the rows represent remain equivalent to the original ones. Through thoughtful application of these operations, we eventually simplify the matrix to reach a point where the variables' values can be easily deduced or where it becomes apparent that no solution exists.