Problem 62 - Intermediate Algebra

Question

If the system represented by the augmented matrix at the right has no solution, what do you know about \(k ?\) Explain your answer. $$ \left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & k \end{array}\right] $$

Step-by-Step Solution

Answer

The system has no solution if \(k \neq 0\).

Step 1: Understand the System of Equations

The augmented matrix provided represents the system of linear equations. The goal is to determine when this system has no solutions, which would occur if the matrix represents an inconsistent system.

Step 2: Identify the Row of Interest

Observe the third row of the matrix: \[ 0 \,\, 0 \,\, 0 \,\, k \] This row equates to the equation \(0x + 0y + 0z = k\). For the system to have no solution, this equation must be inconsistent.

Step 3: Determine the Inconsistency Condition

An equation of the form \(0 = k\) is inconsistent when \(k\) is a non-zero value. This is because a true statement, such as \(0 = 0\), is transformed into a false statement when \(k\) is not zero.

Step 4: Conclude the Condition on \(k\)

The system of equations has no solution if and only if the equation represented by the third row, \(0 = k\), is false. Therefore, \(k\) must not equal zero for the system to be inconsistent and have no solution.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. For example, consider a system with three variables:

• Equation 1: \(x + y = 1\)
• Equation 2: \(z = 2\)

The goal is to find values for the variables that satisfy all the equations simultaneously. Here, the equations correspond to the rows of the augmented matrix from the exercise:

• The first row, \([1 \, 1 \, 0 \, | \, 1]\), represents \(x + y = 1\).
• The second row, \([0 \, 0 \, 1 \, | \, 2]\), represents \(z = 2\).
• The third row, \([0 \, 0 \, 0 \, | \, k]\), encodes the equation \(0x + 0y + 0z = k\).

When graphed, each equation in a two-variable system can represent a line in a plane, and their intersection represents the solution of the system. With three dimensions, the equations represent planes in space. For the system to be consistent, these planes must intersect at a point, line, or overlap entirely.

Inconsistent System

An inconsistent system is one in which no solution exists. In terms of an augmented matrix, inconsistency occurs when there is a row where all the coefficients of the variables are zero, but the constant term is non-zero. Consider the third row of the matrix from the exercise, which forms the equation:\[ 0x + 0y + 0z = k \]This simplifies to \(0 = k\). If \(k\) is non-zero, this equation becomes impossible because 0 cannot equal any non-zero number. This impossibility means the system has no points in common that satisfy all equations, leading to an inconsistent system.

• An inconsistent system indicates a contradiction, like having parallel lines that never meet.
• In higher dimensions, it means the represented planes do not intersect at any common point.

Solution of Equations

The solution of a system of linear equations is a set of values for the variables that satisfy every equation in the system. For a consistent system, possible solutions include:

• A unique solution: When lines or planes intersect at a single point.
• Infinite solutions: In case the lines or planes coincide perfectly.

However, when facing an inconsistent system like the one described with our augmented matrix, there is no solution. The contradiction \(0 = k\) (where \(k eq 0\)) suggests such, reinforcing the understanding that you cannot solve an impossible condition. It's crucial to identify these characteristics early in the problem-solving to avoid unnecessary calculations:

• Always inspect rows of zeros in matrices carefully—non-zero constants in these rows signal inconsistency.
• If the system is set up correctly, finding no solution indicates either a miscalculation or that no solution is possible under given conditions.