Problem 62 - College Algebra

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. Using row operations on an augmented matrix, I obtain a row in which 0 s appear to the left of the vertical bar, but 6 appears on the right, so the system I'm working with has no solution.

Step-by-Step Solution

Answer

The statement makes sense. When performing row operations on an augmented matrix, if a row results in zeroes on the left of the vertical bar and a non-zero number on the right, it means the system is inconsistent and does not have a solution. This is because it corresponds to an impossible equation, in this case, \(0 = 6\).

Step 1: Understand the Situation

In row operations of an augmented matrix, each row corresponds to an equation in the system. If a row is left with zeroes on the left of the vertical bar and a non-zero number on the right, the corresponding equation will be \(0 = \text{any non-zero number}\). For example, in this case, the row mentions \(0 = 6\).

Step 2: Interpretation

The equation \(0 = 6\) is not possible because zero cannot be equal to six. This inconsistency indicates that the original system of equations does not have a solution.

Key Concepts

Row Operations

Row operations are essential procedures used to simplify augmented matrices, which represent systems of equations. These operations include row addition, row multiplication by a non-zero number, and row switching. When applied effectively, row operations can help us determine whether a system has one solution, infinitely many solutions, or no solution at all.

Using row operations, our goal is to transform the matrix into a row-echelon form or even better, a reduced row-echelon form. This is where each row has more leading zeros than the row above it and the leading coefficient (the first non-zero number from the left in a row) is 1. Once in this form, the system of equations is easier to solve or analyze for consistency.

System of Equations

A system of equations is a set of two or more equations with a common set of variables. The solutions to the system are the values of these variables that satisfy all the equations simultaneously. Systems of equations can be represented and solved in various ways, including graphically, algebraically, and through the use of augmented matrices.

When using the augmented matrix method, each equation in the system is written as a row. The coefficients of the variables, along with the constants from the equality, make up the augmented matrix. Solving the system then involves using row operations to simplify the matrix and reveal the solution or lack thereof. By reaching a point where one row indicates an impossibility (like an equation that simplifies to \(0 = 6\)), it becomes clear the system has no solution.

Inconsistent Systems

An inconsistent system of equations is a system that has no solution. This inconsistency arises when the equations represent parallel lines in a two-variable case, or more generally, when at least one equation in the system contradicts the others. This means that no set of values exists that satisfies all the equations simultaneously.

In terms of augmented matrices, inconsistency is identified during the row reduction process. A row that simplifies to all zeros on the left of the vertical bar (representing the coefficients of the variables) and a non-zero number on the right (the constant) is the hallmark of an inconsistent system. This row essentially represents the false statement \(0 = \text{non-zero number}\), signaling that the system has no solution.