Problem 82 - Precalculus with Limits

Question

In Exercises 63-84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. \( \left\\{ \begin{array}{l} x +2y = 0 \\ x + y = 6 \\ 3x - 2y = 8 \end{array} \right. \)

Step-by-Step Solution

Answer

The system of equations is inconsistent and has no solution.

Step 1: Set Up the Augmented Matrix

First, the system of equations should be written in the form of an augmented matrix. The coefficients of the variables form the left side of the matrix, and the constants on the right side of the equal signs form the right column of the matrix. For the given system of equations, the augmented matrix is \[ \begin{bmatrix} 1 & 2 & 0 \ 1 & 1 & 6 \ 3 & -2 & 8 \end{bmatrix} \].

Step 2: Perform Row Operations Using Gauss-Jordan Elimination

Next, perform row operations to get the matrix in reduced row echelon form (RREF). Begin by swapping Row 1 and Row 2. Then replace Row 3 with \(3 \times \text{Row 1} - \text{Row3}\). The updated matrix after this operation is \[ \begin{bmatrix} 1 & 1 & 6 \ 1 & 2 & 0 \ 0 & 5 & 10 \end{bmatrix} \]. Replace Row 2 with \( \text{Row 2} - \text{Row 1}\) and Row 3 with \( \frac{1}{5} \times \text{Row 3}\). The updated matrix after these operations is \[ \begin{bmatrix} 1 & 1 & 6 \ 0 & 1 & -6 \ 0 & 1 & 2 \end{bmatrix} \] . Next replace Row 3 with \( \text{Row 3} - \text{Row 2}\) and Row 1 with \( \text{Row 1} - \text{Row2}\). The updated matrix after these operations is \[ \begin{bmatrix} 1 & 0 & 12 \ 0 & 1 & -6 \ 0 & 0 & 8 \end{bmatrix} \] .

Step 3: Read off the Solutions

Now that the matrix is in RREF, the solutions can be read off directly. The solutions are x = 12 and y = -6. The last row indicates that 0 = 8 which is a contradiction, meaning the system of equations is inconsistent and has no solution.

Key Concepts

Systems of Equations

A system of equations is essentially a set of two or more equations that have common variables. Solving such systems is important for situations where you need to find values for variables that satisfy all equations in the system simultaneously. This is frequently encountered in mathematics, engineering, and real-world problem-solving tasks.
A typical example might include equations to figure out the intersection point of two lines or to adjust various constraints in linear programming.
The system can have:

• a unique solution (one intersecting point),
• infinitely many solutions (lines coincide),
• or no solution at all (parallel lines).

Understanding how to determine which category a system of equations falls into is essential when deciding how to approach solving it.

Augmented Matrix

An augmented matrix is a crucial element in the process of solving a system of equations using either Gaussian elimination or its more refined cousin, Gauss-Jordan elimination.
To create an augmented matrix from a system of equations, you place the coefficients of each variable into rows, aligning them based on their variable counterpart. The constants from each equation are aligned in the next column over, separated by a distinction line.
For instance, given the system:

• \( x + 2y = 0 \)
• \( x + y = 6 \)
• \( 3x - 2y = 8 \)

The augmented matrix will be constructed like this: \[ \begin{bmatrix} 1 & 2 & 0 \ 1 & 1 & 6 \ 3 & -2 & 8 \end{bmatrix} \]This matrix will then undergo row operations that manipulate its structure without changing the solutions to the system, eventually simplifying this matrix into a form from which solutions can easily be read.

Inconsistent System

An inconsistent system is one where no solution exists to simultaneously satisfy all the given equations.
This typically happens when at least two of the equations in the system describe parallel lines or conflicting conditions. When you transform the system into its augmented matrix and reduce it using row operations, spotting an inconsistency comes down to discovering a row that represents a false statement.
For instance, if during matrix reduction you uncover a row of zeros except for a constant on the right side, such as \[ \begin{bmatrix} 0 & 0 & 8 \end{bmatrix} \], it equates to \( 0 = 8 \), a blatant contradiction, signaling that the system does not intersect at any point in the solution space.
Understanding when and why a system is inconsistent is key in interpreting solution sets and in adjusting models for more accurate results.