Problem 41 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I omitted row 3 from \(\left[\begin{array}{rrr|r}1 & -1 & -2 & 2 \\ 0 & 1 & -10 & -1 \\ 0 & 0 & 0 & 5\end{array}\right]\) and expressed the system in the form \(\left[\begin{array}{rrr|r}1 & -1 & -2 & 2 \\ 0 & 1 & -10 & -1\end{array}\right]\).

Step-by-Step Solution

Answer

The given statement does not make sense because it neglects the contradiction inherent in the system of equations that dictates no solution exists.

Step 1: Analyze the given matrix

The given system of equations in matrix form is: \[ \left[\begin{array}{rrr|r}1 & -1 & -2 & 2 \ 0 & 1 & -10 & -1 \ 0 & 0 & 0 & 5\end{array}\right] \] This corresponds to the system of equations: \[ \begin{align*} x - y - 2z & = 2, \ y - 10z & = -1, \ 0 & = 5. \end{align*} \] The third equation is a contradiction and indicates that the system has no solution.

Step 2: Evaluate the statement

The statement suggests omitting the third row and expressing the system in the form: \[ \left[\begin{array}{rrr|r}1 & -1 & -2 & 2 \ 0 & 1 & -10 & -1\end{array}\right], \] which would correspond to the system of equations x - y - 2z = 2, y - 10z = -1.

Step 3: Determine validity of the statement

Omitting the third row disregards the crucial piece of information about the system - the inherent contradiction. Consequently, the omitted system would be inconclusive about the solutions to this system of equations and thus the provided statement does not make sense.

Key Concepts

Matrix

A matrix is a grid of numbers or symbols arranged in rows and columns. It's a useful tool for organizing data, especially for complex calculations like systems of equations. In terms of systems, a matrix can succinctly represent multiple linear equations.

In our specific exercise, each row of the matrix corresponds to an equation. The first part of each row represents the coefficients of the variables, while the numbers after the vertical line (|) represent the constants. This form is known as an augmented matrix. Using matrices like this one can simplify the process of solving systems by focusing on numerical relationships rather than verbal equations.

No Solution

Sometimes, a system of equations has no solution. This can occur when the equations describe parallel lines without intersection (in two dimensions), or more generally, when they describe a situation that's logically impossible.

In our exercise, the system initially includes a row \(0 = 5\). This represents a statement that is always false because zero cannot equal five. This false statement indicates that the original system has no solution. Such contradictions mean that there is no consistent set of variable values that satisfy all the equations simultaneously.

Row Operations

Row operations are simple manipulations you can perform on matrices to make solving systems of equations easier. There are three main types of row operations: swapping two rows, multiplying a row by a nonzero constant, and adding or subtracting rows.

These operations help transform the matrix into forms that make solutions more visible, especially when formally solving using methods like Gaussian elimination or row reduction. Remember, proper use of row operations preserves the fundamental relationships within the system, but manipulating them incorrectly can disrupt the logic, as noted by trying to omit important information like our contradictory \(0 = 5\) statement.

Contradiction in Systems

A contradiction in a system of equations happens when you derive an equation that is impossible, such as \(0 = 5\). This signals that the system is inconsistent.

When faced with a contradiction, it's a clear indication that the system does not have a solution. In matrix terms, if reducing your system of equations leads to a row of zeros on the left side and a non-zero constant on the right (such as \[0 \, 0 \, 0 \, | \, 5\]), it's a red flag of a contradiction. Recognizing contradictions is crucial because they highlight logical issues within the system, guiding you toward understanding the system's inconsistency.