Problem 29 - 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. $$\left\\{\begin{aligned} x-y+3 z &=3 \\ 4 x-8 y+32 z &=24 \\ 2 x-3 y+11 z &=4 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent, as it has no solution.

Step 1: Write the system of equations

The system of equations given is: \( x - y + 3z = 3 \), \( 4x - 8y + 32z = 24 \), \( 2x - 3y + 11z = 4 \). We need to determine if this system is inconsistent or dependent, or if it has a unique solution.

Step 2: Write the system in matrix form

Express the system using matrices. Our coefficient matrix \( A \) will look like:\[A = \begin{bmatrix}1 & -1 & 3 \4 & -8 & 32 \2 & -3 & 11\end{bmatrix}\]And the augmented matrix \([A | B]\) is:\[[A | B] = \begin{bmatrix}1 & -1 & 3 & | & 3 \4 & -8 & 32 & | & 24 \2 & -3 & 11 & | & 4\end{bmatrix}\]

Step 3: Perform Row Operations for Row Echelon Form

Use row operations to simplify the augmented matrix. Subtract 4 times row 1 from row 2 to get a new row 2.\[R2 = R2 - 4R1 \\begin{bmatrix}1 & -1 & 3 & | & 3 \0 & -4 & 20 & | & 12 \2 & -3 & 11 & | & 4\end{bmatrix}\]Next, subtract 2 times row 1 from row 3:\[R3 = R3 - 2R1 \\begin{bmatrix}1 & -1 & 3 & | & 3 \0 & -4 & 20 & | & 12 \0 & -1 & 5 & | & -2\end{bmatrix}\]

Step 4: Further Simplify to Row Echelon Form

Now, multiply row 3 by 4 and subtract it from row 2:\[R2 = R2 - 4R3 \\begin{bmatrix}1 & -1 & 3 & | & 3 \0 & 0 & 0 & | & 20 \0 & -1 & 5 & | & -2\end{bmatrix}\]Observe that we have an invalid equation \(0 = 20\) in row 2, indicating that the system is inconsistent.

Step 5: Conclusion on the System's Consistency

The presence of the equation \(0 = 20\) in the row-reduced matrix signifies that the system is inconsistent, which means there is no solution that satisfies all three equations simultaneously.

Key Concepts

System of Equations

A system of linear equations consists of multiple equations that involve the same set of variables. In our example, we have three equations with three variables: \( x \), \( y \), and \( z \). Each equation provides a relationship among the three variables. To find a solution, we look for values of \( x \), \( y \), and \( z \) that satisfy all equations simultaneously.

• The goal is to determine whether these equations intersect at one point (a unique solution), coincide (infinitely many solutions), or don't intersect at all (no solution).
• In this exercise, the system is examined for consistency and dependence.

Understanding how these equations work together helps us predict whether we'll have one solution, no solution, or infinitely many solutions.

Matrix Row Operations

Matrix row operations are techniques used to simplify matrices, with the aim to solve systems of equations efficiently. By converting the system into a matrix form, we can perform operations like row addition, row subtraction, and scaling to reach a simpler, reduced form.

• Row Addition/Subtraction: This involves adding or subtracting multiples of one row to another to eliminate coefficients or simplify the matrix.
• Scaling: Multiplying a row by a non-zero scalar to simplify the equation.
• The target is to reach Row Echelon Form, where the matrix becomes more manageable and reveals important properties of the system.

In our solution, these operations were used to identify whether the system is inconsistent or dependent.

Inconsistent System

An inconsistent system of equations is one where there are no solutions that satisfy all equations simultaneously. This typically occurs when simplifying the matrix reveals a contradiction, like the appearance of an impossible statement (e.g., \( 0 = 20 \)).

• This contradiction indicates that the equations are in conflict, depicting parallel planes that never intersect.
• In our exercise, we found such a contradiction, indicating the system is inconsistent.

Recognizing an inconsistent system is crucial because it tells us that trying to find solutions is pointless due to fundamental conflicts in the equations themselves.

Dependent System

A dependent system indicates that the equations describe the same plane or line, essentially leading to infinitely many solutions. The equations are linear combinations of each other.

• In matrix form, a dependent system will often show a row of zeros in the row-reduced form without reaching a contradiction.
• This leads to infinitely many solutions because the equations overlap, describing the same relationships in different forms.

If a system were dependent, finding a complete solution involves expressing one variable in terms of the others, identifying a parameter that provides all potential solutions. In our exercise, the system was not dependent, as we found a contradiction highlighting its inconsistency.