Problem 112 - Algebra and Trigonometry Real Mathematics, Real People

Question

When using Gaussian elimination to solve a system of linear equations, explain how you can recognize that the system has no solution. Give an example that illustrates your answer.

Step-by-Step Solution

Answer

In Gaussian Elimination, a system of linear equations has no solution when an inconsistency or contradiction occurs. This situation arises when there is a row of all zeros on the left side of the augmented matrix, while the right side (augmented column) contains a non-zero value, e.g. \(0 = c\), \(c ≠ 0\). Example: the system of equations, \(2x + 3y = 30\) and \(4x + 6y = 60\), when represented in augmented matrix and reduced by Gaussian Elimination, results in such a row, illustrating the system has no solution.

Step 1: Define a System of Linear Equations

Consider the system of linear equations below: \ \(1) 2x + 3y = 30 \ \(2) 4x + 6y = 60\

Step 2: Form an Augmented Matrix

We form an augmented matrix from the system of equations. Our augmented matrix then becomes \[ [ 2, 3 | 30 ], \ [ 4, 6 | 60 ] \]

Step 3: Application of Gaussian Elimination

We apply the Gaussian elimination method. Start by swapping rows (if necessary) to ensure that an upper triangular matrix will be created. However, in this case, no swap is needed. Next, make the elements below the pivot (top left corner element of the matrix) to be zero by subtracting multiples of the first row from the others. Subtracting twice of row 1 from row 2 results into a row of all zeros on the left of the augmented matrix while the augmented column on its right does not contain zero \[ [ 2, 3 | 30],\ [0, 0 | 0 ]\].

Step 4: Recognition of No Solution

The form of the matrix after Gaussian elimination indicates that there's a contradiction or inconsistency in the system of equations, this is due to the presence of a row with non-zero element on the right side of the augmented column, and zeros on the left. Hence, the system of equations has no solution.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with multiple variables that we want to solve simultaneously. Each equation describes a straight line when graphed on a coordinate plane. For instance, with the given example:

• Equation (1) is: \(2x + 3y = 30\)
• Equation (2) is: \(4x + 6y = 60\)

Here, we see two equations with two unknowns, \(x\) and \(y\). The goal is to find values for \(x\) and \(y\) that satisfy both equations at the same time, meaning the intersection of these two lines. However, systems can sometimes be more complex or not have a solution at all. The solution, if it exists, means finding a point (or points) where all the lines intersect.

Augmented Matrix

An augmented matrix is a convenient tool to solve a system of linear equations using methods like Gaussian elimination. We write only the coefficients and constants of the equations in a matrix format to simplify calculations. Let's examine the system of equations above. The augmented matrix is written as:\[\begin{bmatrix} 2 & 3 & | & 30 \4 & 6 & | & 60 \end{bmatrix}\]Here:

• The first two columns are from the coefficients of \(x\) and \(y\).
• The third column, separated by a vertical bar, represents the constants (the numbers on the right of the equations).

This transformation helps to apply matrix operations, making solving the system more structured and organized, especially when dealing with multiple equations.

Inconsistent System

An inconsistent system is one that has no solutions. This means the equations describe lines that do not intersect on a graph. Inconsistent systems occur due to a mathematical contradiction that makes it impossible to find a common point that satisfies all equations in the system.
In the context of Gaussian elimination, we can recognize an inconsistent system when, during the row operations, a row reduces to have all zeros except for a non-zero entry in the constant column. In the provided example, the second row becomes \([0, 0 | 0]\). This implies:

• There is no combination of \(x\) and \(y\) that can satisfy both equations simultaneously.
• The equations essentially represent parallel lines, indicating they never intersect.

This explains why the system is considered inconsistent.

No Solution in Algebra

The phrase 'no solution' in algebra signifies that there are no values for the variables that satisfy all equations of a system simultaneously.
In the given example, after performing Gaussian elimination, we noticed that the row operations resulted in a row indicating a contradiction \([0, 0 | 0 ]\). The condition here can be summarized as:

• Zero equals a non-zero number: mathematically impossible.
• This contradiction signals that the original system of equations has no solution.

When an algebraic system has no solution, it often means that the geometric interpretation of such equations describes lines that do not share any common points, such as parallel lines. This understanding is crucial, as it quickly reveals whether further calculations are useless because no solution exists.