Problem 40 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The system is inconsistent.

Step 1: Set Up the Augmented Matrix

Write down the augmented matrix for the system of equations. The given system is:\[\begin{align*} x + 3z & = 3 \ 2x + y - 2z & = 5 \ -y + 8z & = 8 \ \end{align*}\] So the augmented matrix is: \[\begin{bmatrix} 1 & 0 & 3 & | & 3 \ 2 & 1 & -2 & | & 5 \ 0 & -1 & 8 & | & 8 \end{bmatrix}\]

Step 2: Perform Row Operations to Achieve Row-Echelon Form

The goal is to convert the matrix to row-echelon form. Start by using row operations to create leading 1s down the diagonal. First, subtract 2 times the first row from the second row: \[R_2 = R_2 - 2R_1 \Rightarrow \begin{bmatrix} 1 & 0 & 3 & | & 3 \ 0 & 1 & -8 & | & -1 \ 0 & -1 & 8 & | & 8 \end{bmatrix}\] Next, add the second row to the third row to eliminate the \(y\) term: \[R_3 = R_3 + R_2 \Rightarrow \begin{bmatrix} 1 & 0 & 3 & | & 3 \ 0 & 1 & -8 & | & -1 \ 0 & 0 & 0 & | & 7 \end{bmatrix}\]

Step 3: Analyze the Matrix for Consistency

The third row translates to the equation \(0x + 0y + 0z = 7\) which simplifies to \(0 = 7\). This is a contradiction, indicating that the system is inconsistent.

Key Concepts

Augmented Matrix

An augmented matrix is an essential tool for solving systems of linear equations. It represents both the coefficients of the variables and the constants from the equations in a compact form. To write an augmented matrix, you first extract the coefficients and constants of the system of equations. For instance, consider the following equations:

• \(x + 3z = 3\)
• \(2x + y - 2z = 5\)
• \(-y + 8z = 8\)

In augmented matrix form, these equations become:\[\begin{bmatrix} 1 & 0 & 3 & | & 3 \ 2 & 1 & -2 & | & 5 \ 0 & -1 & 8 & | & 8\end{bmatrix}\]The vertical line divides the coefficients from the constants, clearly delineating the two parts of each equation.

Row-Echelon Form

Achieving row-echelon form is a crucial step when using matrices to solve systems of equations. In this form, the matrix looks more like a 'staircase' where each leading entry (or pivot) is to the right of the leading entry in the row above, and any rows consisting entirely of zeros are at the bottom of the matrix. To convert our augmented matrix to row-echelon form, we apply elementary row operations such as swapping rows, multiplying rows by non-zero scalars, and adding or subtracting rows. For the given matrix:

• Convert the second row by subtracting 2 times the first row from it.
• Add the modified second row to the third row.

This manipulation results in:\[\begin{bmatrix} 1 & 0 & 3 & | & 3 \ 0 & 1 & -8 & | & -1 \ 0 & 0 & 0 & | & 7\end{bmatrix}\]The zeros in the last row's leading entries show that we are nearing a decision on the system's consistency.

Inconsistent System

A system of linear equations is termed inconsistent if there is no possible set of values that satisfies all equations simultaneously. When we analyze the transformed row-echelon matrix, any row which leads to a contradiction (for example, an equation like \(0 = 7\)) indicates an inconsistent system. This means that the set of equations, when translated back into an augmented matrix and row-reduced, lacks a solution entirely. No values for \(x\), \(y\), or \(z\) will satisfy all given equations without violating logical or mathematical principles.

Dependent System

In contrast to inconsistency, a dependent system involves equations that are not truly independent of each other. This often means that one equation can be derived from others, making the system have infinite solutions, lying on a common overlapping plane in space.For example, if during the row reduction process we arrive at a row of all zeros on the left but a zero on the right side as well,

• such as \(0x + 0y + 0z = 0\),

this indicates that we have a dependent system. This zero row doesn't affect the solution set, unlike in inconsistent systems where such a row produces a contradiction.In our original system, however, the contradiction revealed an inconsistency rather than dependence.