Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{rr}2 x+6 y-4 z= & 1 \\ x+3 y-2 z= & 4 \\ 2 x+y-3 z= & -7\end{array}\right.\)

Step-by-Step Solution

Answer

The system is inconsistent; there are no solutions.

Step 1: Write the System in Matrix Form

Let's express the system of equations as a matrix equation. Define matrix A, vector x, and vector b as follows:Matrix A:\[A = \begin{pmatrix} 2 & 6 & -4 \ 1 & 3 & -2 \ 2 & 1 & -3 \end{pmatrix}\]Vector x:\[\mathbf{x} = \begin{pmatrix} x \ y \ z \end{pmatrix}\]Vector b:\[\mathbf{b} = \begin{pmatrix} 1 \ 4 \ -7 \end{pmatrix}\]Thus, the system can be written as \(A\mathbf{x} = \mathbf{b}\).

Step 2: Set Up and Simplify the Augmented Matrix

Combine matrix A and vector b to form an augmented matrix [A|b]:\[\left(\begin{array}{ccc|c} 2 & 6 & -4 & 1 \ 1 & 3 & -2 & 4 \ 2 & 1 & -3 & -7 \end{array}\right)\]

Step 3: Perform Row Operations to Achieve Row Echelon Form

We will perform row operations to simplify this matrix to row-echelon form:1. Start with R1 unchanged: \((2 \, 6 \, -4 \, | \, 1)\).2. Replace R2 with R2 - 0.5\times R1, resulting in \((0 \, 0 \, 0 \, | \, 3.5)\).3. Replace R3 with R3 - R1, resulting in \((0 \, -5 \, 1 \, | \, -8)\). This leads to the matrix:\[\left(\begin{array}{ccc|c} 2 & 6 & -4 & 1 \ 0 & 0 & 0 & 3.5 \ 0 & -5 & 1 & -8 \end{array}\right)\].

Step 4: Analyze the Matrix for Solutions

Observe the second row. It is inconsistent (0x + 0y + 0z = 3.5), implying no solutions. The system of equations does not have a solution because the augmented matrix implies a contradiction, making the system inconsistent.

Key Concepts

Matrix Form

When dealing with systems of equations, converting them into matrix form makes manipulation and computation much simpler. Essentially, this involves expressing the system as a single matrix equation of the form \(A\mathbf{x} = \mathbf{b}\), where:

• \(A\) is the coefficient matrix, which is formed from the coefficients of the variables in the equations.
• \(\mathbf{x}\) is the vector of unknowns, like \(x\), \(y\), and \(z\).
• \(\mathbf{b}\) is the constant vector, consisting of the constants from each equation.

By using matrix notation, complicated systems become more structured and are easier to handle using matrix operations, paving the way for solving the system efficiently.

Augmented Matrix

Once in matrix form, constructing an augmented matrix is an intuitive next step. An augmented matrix combines the coefficient matrix \(A\) and the constant vector \(\mathbf{b}\) into a single matrix. This is done by adding \(\mathbf{b}\) as an additional column to \(A\):

• The augmented matrix for our system is written as: \[\left(\begin{array}{ccc|c} 2 & 6 & -4 & 1 \ 1 & 3 & -2 & 4 \ 2 & 1 & -3 & -7 \\end{array}\right)\]
• Each row represents one equation from the system.

Using an augmented matrix is advantageous because it aligns all data in one place, aiding in performing systematic row operations.

Row-Echelon Form

The goal when working with matrices often involves transforming them into row-echelon form, a specific format that eases the process of solving systems of equations. To reach row-echelon form, use row operations like swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiples of rows from each other. The characteristics of the row-echelon form include:

• All zero rows, if any, are at the bottom of the matrix.
• The leading entry of each row is to the right of the leading entry of the row above it.
• The leading entry in each nonzero row is 1, known as the pivot position.

By carefully choosing and applying these operations, the matrix becomes easier to work with for further analysis or finding solutions.

Inconsistent System

An inconsistent system occurs when a system of equations has no solution. In the context of an augmented matrix, this typically reveals itself when a row results in a blatant contradiction, such as \(0 = 3.5\). Understanding an inconsistent system can be challenging, but some clues include:

• The presence of a row that equates to something false, meaning the left side of an equation has zero values for all variables while it's set equal to a non-zero constant.
• It means the original equations in the system contradict each other at some point, and no set of variables will satisfy all equations simultaneously.

Identifying an inconsistent system early can save time and allow students to focus on checking initial assumptions or constraints of the problem setup.