Problem 35 - Algebra and Trigonometry

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} x+4 y-2 z &=-3 \\ 2 x-y+5 z &=12 \\ 8 x+5 y+11 z &=30 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is dependent, with solutions \(x = -t + 5\), \(y = t - 2\), \(z = t\).

Step 1: Write the Augmented Matrix

First, write the system of equations as an augmented matrix. This matrix consists of the coefficients of the variables and the constants on the right-hand side:\[\begin{bmatrix}1 & 4 & -2 & | & -3 \2 & -1 & 5 & | & 12 \8 & 5 & 11 & | & 30\end{bmatrix}\]

Step 2: Perform Row Operations to Achieve Row Echelon Form

Use row operations to convert the matrix into row echelon form. Start by eliminating the first column values under the first leading 1:1. \( R_2 = R_2 - 2R_1 \)2. \( R_3 = R_3 - 8R_1 \)The new matrix is:\[\begin{bmatrix}1 & 4 & -2 & | & -3 \0 & -9 & 9 & | & 18 \0 & -27 & 27 & | & 54\end{bmatrix}\]

Step 3: Simplify the Matrix Further

Continue performing row operations to simplify the matrix. We aim to make the matrix triangular:1. Notice that \( R_3 = 3R_2 \). Thus, simplify by dividing the second row by \(-9\):2. \( R_2 = \frac{1}{-9}R_2 \)Resulting matrix:\[\begin{bmatrix}1 & 4 & -2 & | & -3 \0 & 1 & -1 & | & -2 \0 & 0 & 0 & | & 0 \end{bmatrix}\]

Step 4: Look for Independence or Dependency

The matrix in row echelon form reveals a row of all zeros \( (0, 0, 0, |, 0) \) suggesting a dependent system. This means there are infinitely many solutions. The equations are not inconsistent as there is no row with all zeros equal to a non-zero value.

Step 5: Express the Solution in Terms of Free Variables

Since we have two pivot variables (leading 1s) and one free variable, express the solution in terms of the free variable. Assign \( z = t \):From the second row: \( y - z = -2 \) implies \( y = t - 2 \).From the first row: \( x + 4y - 2z = -3 \), substitute \( y = t - 2 \) and \( z = t \) to find:\( x + 4(t - 2) - 2t = -3 \) simplifies to:\( x = -t + 5 \).Thus, the solution set in parametric form is:\[\begin{align*}x &= -t + 5, \y &= t - 2, \z &= t.\end{align*}\]

Key Concepts

Inconsistent System

When working with systems of linear equations, an inconsistent system is one where there are no solutions that satisfy all the equations. This happens when the system contains at least one row in its row echelon form with all zero coefficients for the variables but a non-zero constant term. Put simply, the system tries to suggest that something impossible is true. For example, you'd end up with a row like this: \[0x + 0y + 0z = c\] where \(c\) is a non-zero number, which is never feasible. In our given system, we don't find such a contradictory row, indicating that the system is not inconsistent.

Dependent System

A dependent system of linear equations is one that has infinitely many solutions. This type occurs when there is redundancy among the equations, meaning one equation can be expressed as a combination of the others. For a system to be dependent, there must be at least one free variable in its row echelon form. In our example, the row echelon form demonstrated a zero row, i.e., \[ \begin{bmatrix} 1 & 4 & -2 & | & -3 \ 0 & 1 & -1 & | & -2 \ 0 & 0 & 0 & | & 0 \end{bmatrix} \] This indicates a dependent system since not all variables are bound by pivot positions, allowing for infinite solutions. In this case, we introduce a parameter \(t\) and express the other variables in terms of \(t\).

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations. It includes both the coefficients of the variables and the constants from the equations but omits the variables themselves. For the system of equations: \[\begin{align*} x+4y-2z &=-3 \2x-y+5z &=12 \8x+5y+11z &=30 \end{align*}\] The augmented matrix is: \[ \begin{bmatrix} 1 & 4 & -2 & | & -3 \ 2 & -1 & 5 & | & 12 \ 8 & 5 & 11 & | & 30 \end{bmatrix} \] This matrix setup provides a straightforward approach for applying matrix operations, crucial for solving the system effectively. By manipulating this matrix through row operations, we are able to disclose the nature of the system, whether it is consistent, inconsistent, dependent, or independent.

Row Echelon Form

Row echelon form (REF) is a type of matrix transformation used in linear algebra to simplify equations, making it easier to analyze and solve the system. When transforming a matrix into REF, the goal is to have leading 1s (pivots) and zeros below these pivots to form an upper triangular matrix. In our exercise, the system was adjusted into REF: \[ \begin{bmatrix} 1 & 4 & -2 & | & -3 \ 0 & 1 & -1 & | & -2 \ 0 & 0 & 0 & | & 0 \end{bmatrix} \] Key characteristics of REF:

• The leading entry of each row is 1 (known as a pivot).
• All entries below each pivot are zero.
• Each pivot is to the right of the pivot in the preceding row.

The REF of this system reveals one free variable, indicating dependent equations. This form is instrumental for expressing the system's solutions in terms of free variables.