Problem 25 - Algebra and Trigonometry

Question

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} x+y+z =2 \\ y-3 z &=1 \\ 2 x+y+5 z =0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent (no solutions).

Step 1: Write the system in matrix form

Represent the given system of equations as an augmented matrix. The system is: \[ \begin{aligned} x + y + z &= 2 \ y - 3z &= 1 \ 2x + y + 5z &= 0 \end{aligned} \] The augmented matrix is: \[\begin{bmatrix} 1 & 1 & 1 & | & 2 \ 0 & 1 & -3 & | & 1 \ 2 & 1 & 5 & | & 0 \ \end{bmatrix}\]

Step 2: Perform row operations to simplify matrix

Perform row operations to achieve upper triangular form by eliminating the variable \( x \) from the third row. Subtract 2 times the first row from the third row: \[ R_3 = R_3 - 2R_1 \] This gives: \[\begin{bmatrix} 1 & 1 & 1 & | & 2 \ 0 & 1 & -3 & | & 1 \ 0 & -1 & 3 & | & -4 \ \end{bmatrix}\]

Step 3: Further simplify the matrix

Eliminate the variable \( y \) from the third row by adding the second row to the third row: \[ R_3 = R_3 + R_2 \] This results in: \[\begin{bmatrix} 1 & 1 & 1 & | & 2 \ 0 & 1 & -3 & | & 1 \ 0 & 0 & 0 & | & -3 \ \end{bmatrix}\]

Step 4: Analyze for consistency and dependency

The final row of the matrix \( \begin{bmatrix} 0 & 0 & 0 & | & -3 \end{bmatrix} \) indicates a contradiction, as 0 cannot equal -3. This implies that the system does not have any solutions and is inconsistent.

Key Concepts

Augmented Matrix

When dealing with systems of linear equations, an augmented matrix is a powerful tool. It provides a simpler way to organize and manipulate these systems. The augmented matrix includes both the coefficients of the variables and the constants from each equation. This is achieved by placing the constants in an additional column on the right. For example, the system of equations:

• \( x + y + z = 2 \)
• \( y - 3z = 1 \)
• \( 2x + y + 5z = 0 \)

can be written in an augmented matrix form as:\[\begin{bmatrix} 1 & 1 & 1 & | & 2 \0 & 1 & -3 & | & 1 \2 & 1 & 5 & | & 0 \\end{bmatrix}\]The beauty of this method lies in its ability to visually separate the system's coefficients from its solutions. This helps immensely in performing subsequent calculations to find the solution of the system.

Row Operations

Row operations are crucial in simplifying augmented matrices in order to solve systems of equations. These operations are used to transform a matrix into a simpler equivalent form, typically reduced row-echelon form or upper triangular form. The main row operations include:

• Swapping two rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a multiple of one row from another row.

For example, in the original step-by-step solution, row operations were performed to simplify the augmented matrix. First, to eliminate the variable \( x \) from the third row, two times the first row was subtracted from the third row:\[R_3 = R_3 - 2R_1\]Then, to eliminate \( y \) from the third row, the second row was added to it:\[R_3 = R_3 + R_2\]These transformations help make the matrix easier to analyze for solutions or inconsistencies.

Inconsistent Systems

An inconsistent system of equations is one that has no solutions. This is identified when, after performing row operations, one of the rows in the augmented matrix translates into a false statement, such as \( 0 = -3 \). This indicates a contradiction, meaning there are no values of the variables that simultaneously satisfy all the equations in the system.
In the detailed step-by-step solution, the final row of the matrix was \( \begin{bmatrix} 0 & 0 & 0 & | & -3 \end{bmatrix} \). This row translates to \( 0x + 0y + 0z = -3 \), which is impossible. Therefore, the system is inconsistent, having no solutions. Detecting inconsistencies early can save time and help focus efforts on verifying initial calculations.

Dependent Systems

Dependent systems are systems of equations where the equations are linearly dependent. Essentially, this means that one equation can be derived from another, resulting in an infinite number of solutions. When represented as an augmented matrix, dependent systems will have at least one row that implies a true, yet redundant, statement like \( 0 = 0 \).
In such cases, the number of equations is fewer than the number of variables, or the system allows for a free variable, giving rise to multiple solutions.
Identifying dependent systems involves reducing the system to its simplest form and noting replacement of any of the rows with a zero row (other than the last column). Recognizing dependency can provide insight into the relationships between variables and their collective solutions. This is the opposite scenario compared to inconsistent systems and requires different strategies for finding the complete solutions.