Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{ll|l} 1 & 0 & 1 \\ 1 & 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The system is inconsistent and has no solutions.

Step 1: Understand the Augmented Matrix

The given augmented matrix represents a system of linear equations. Each row corresponds to an equation, and by reading the matrix, we can write down the equations as: \( x = 1 \) and \( x = 2 \).

Step 2: Identify the Inconsistency

Both equations have the same variable \( x \) in the same position, but the constants on the right side are different (1 and 2). This implies that the same \( x \)-value cannot satisfy both equations simultaneously, indicating an inconsistency.

Step 3: Conclude on the Nature of the System

Since there is no possible value for \( x \) that satisfies both equations due to the contradictory constants, the system is inconsistent and therefore has no solutions.

Key Concepts

Augmented Matrix

An augmented matrix is a way of representing a system of linear equations in a compact form. It combines the coefficients of the variables and the constants from each equation into a single matrix. This matrix makes it easier to apply methods such as Gaussian elimination to solve the system.
Each row in the augmented matrix corresponds to an equation in the system, while each column represents either a coefficient of a variable or a constant from the equations. The vertical bar separates the coefficients from the constants. In our example:

• The first row represents the equation: \( 1x + 0y = 1 \)
• The second row represents the equation: \( 1x + 0y = 2 \)

The augmented matrix helps visualize how the equations interact and allows systematic manipulation to find solutions or to determine whether a solution exists. It is an essential tool in solving systems of equations using row operations.

Systems of Linear Equations

A system of linear equations is a set of two or more linear equations with common variables. The goal is to find a set of values for these variables that satisfy all the equations simultaneously. In simple terms, this means finding the point(s) of intersection if the equations are graphed in coordinate space.
The equations usually involve two or more unknowns, like \( x \) and \( y \), and can be represented as:

• \( ax + by = c \)
• \( dx + ey = f \)

Solving a system might involve methods like substitution, elimination, or matrix operations such as Gaussian elimination. Gaussian elimination is particularly beneficial because it systematically reduces the system to an upper-triangular form, making it easier to find solutions.
In our given exercise, we have the system reduced to equations, and using Gaussian elimination means simplifying it to check for possible solutions or inconsistencies.

Inconsistent Systems

An inconsistent system is one in which no set of variable values satisfies all the equations simultaneously. In other words, the system has no solution. This situation can be identified using various methods, including inspecting the reduced form of the augmented matrix during Gaussian elimination.
When working with matrices, an inconsistent system may be revealed when a row showcases contradictory statements, such as \( 0 = 1 \). In our exercise, the augmented matrix was:

• \( x = 1 \)
• \( x = 2 \)

Here, the inconsistency is clear because a single \( x \) cannot be equal to two different values simultaneously.
Detecting inconsistent systems early helps save time, as it indicates that no solution exists for the given set of equations. Recognizing these inconsistencies is crucial for thorough and efficient mathematical problem-solving.