Problem 86 - Precalculus with Limits

Question

In Exercises 85-90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. \( \left\\{ \begin{array}{l} 2x + 10y + 2z = 6 \\ x + 5y + 2z = 6 \\ x + 5y + z = 3 \\ -3x - 15y - 3z = -9 \\ \end{array} \right. \)

Step-by-Step Solution

Answer

The system of equations is inconsistent and hence, has no solution.

Step 1: Write the Equations in Augmented Matrix Form

Take each equation of the system, and write it as a matrix, where each row represents an equation and columns represent coefficients of the variables in the exact same order in all the equations. The augmented matrix form for this system is:\n\n\[\begin{bmatrix}2 & 10 & 2 & 6\ 1 & 5 & 2 & 6\ 1 & 5 & 1 & 3\ -3 & -15 & -3 & -9\end{bmatrix}\]

Step 2: Reduce the Matrices

To solve the system, we must transform this matrix into a reduced row echelon form (RREF), using a graphing utility or manually by row operations. Operations include swapping two rows, multiply a row by a constant, or adding/subtracting one row from another. The reduced matrix obtained will be:\n\n\[\begin{bmatrix}1 & 5 & 0 & 0\ 0 & 0 & 1 & 1\ 0 & 0 & 0 & 1\ 0 & 0 & 0 & 0\end{bmatrix}\]

Step 3: Translate Back Into System of Equations

We convert this RREF matrix back into system of equations. The first row implies \(x+5y = 0\), the second implies \(z = 1\) and the third row suggests that the system of equations is inconsistent as it would mean 0 = 1, which is not possible.

Step 4: Confirm the Inconsistency of the System

This means that there is a contradiction in the given system of equations and thus, no solution exists.

Key Concepts

Reduced Row Echelon Form

The concept of the reduced row echelon form (RREF) is an essential tool in solving systems of linear equations using matrices. When a matrix is in RREF, it is structured in a way that simplifies the process of finding the solutions to the equations represented by the matrix. To achieve the RREF of a matrix, you perform row operations to transform the original matrix. The process involves a few systematic steps:

• First, ensure each leading entry (the first non-zero number in each row) is 1.
• Next, make sure each leading 1 is the only non-zero entry in its column.
• Finally, the leading 1s should appear in a staircase pattern from left to right, top to bottom.

Achieving RREF can be done manually or using matrix capabilities in a graphing utility, which saves time and reduces the likelihood of algebraic errors. In the solved exercise, the matrix in RREF revealed an inconsistent system due to the row representing the impossibility of 0 equaling 1.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. For example, in the provided exercise, we have a system with four equations and three variables: x, y, and z. Translating this into a matrix form lays the groundwork for matrix operations to be performed. Systems can be categorized as:

• Consistent: Having at least one solution.
• Inconsistent: Having no solutions, often due to a contradictory result.
• Dependent: Multiple solutions exist due to overlapping equations, leading to free variables.

In our exercise, the system turned out to be inconsistent, as identified by the RREF form. Inconsistencies indicate that the original equations do not all intersect at a single shared solution point.

Matrix Operations

Matrix operations are the steps taken to manipulate a matrix to make solving for variables straightforward. These operations involve row manipulation, aiming to obtain a particular matrix form, such as the reduced row echelon form (RREF). Common operations include:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting one row from another using a multiplier.

These operations are based on certain rules of equivalence, meaning they do not change the solutions of the equations represented by the matrix. This methodology allows for consistent and efficient problem-solving when dealing with complex sets of equations. In the given exercise, applying these matrix operations was crucial to transforming the augmented matrix into RREF, subsequently helping to reveal the inconsistency of the system of equations. Such operations enable handling complex problems with multiple variables and equations with ease.