Problem 17 - Precalculus with Limits

Question

Performing Row Operations In Exercises 17 and \(18 ,\) perform the row operation and write the equivalent system. Add Equation 1 to Equation 2 $$\left\\{ \begin{aligned} x - 2 y + 3 z = 5 & \text { Equation } 1 \\ - x + 3 y - 5 z = 4 & \text { Equation } 2 \\ 2 x - 3 z = 0 & \text { Equation } 3 \end{aligned} \right.$$ What did this operation accomplish?

Step-by-Step Solution

Answer

The row operation combined Equations 1 and 2, resulting in a new Equation 2 (\(y - 2z = 9\)). The equivalent system is:\[ \begin{aligned} x - 2y + 3z &= 5 \ y - 2z &= 9 \ 2x - 3z &= 0 \end{aligned} \]

Step 1: Understand the Equation

The first step is to understand the system of equations. The system given includes three equations: \(x - 2y + 3z = 5\), \(-x + 3y - 5z = 4\), and \(2x - 3z = 0\). We are asked to add the first equation to the second one.

Step 2: Perform the Row Operation

To add equations together, add the corresponding terms on both sides of the equals sign. This row operation will replace Equation 2 with the result. The operation will be: \((x - 2y + 3z) + (-x + 3y - 5z) = (5 + 4)\)Simplify this to:\(x - 2y + 3z - x + 3y - 5z = 9\)Combine like terms, leading to \(y - 2z = 9\). This now becomes the new Equation 2.

Step 3: Write the Equivalent System

After the row operation, the equivalent system of the equations become: \[ \begin{aligned} x - 2y + 3z &= 5 \ y - 2z &= 9 \ 2x - 3z &= 0 \end{aligned} \]

Key Concepts

Row Operations

Row operations are fundamental tools in solving systems of linear equations. They allow us to manipulate the equations to simplify and solve them effectively. A row operation involves modifying the rows of a matrix or, equivalently, equations in a system.
There are three main types of row operations:

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

In the given exercise, the operation performed is adding one row to another. The goal of row operations, such as the one performed, is to make the equations easier to solve by reducing them to simpler forms. This helps in isolating variables, thereby progressively leading the system towards a solution. As seen, equation manipulation can result in easier-to-handle equations like the transformation from y - 2z = 9.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It utilizes row operations to transform the matrix of coefficients into an upper triangular form. This form has zeros below the main diagonal, making the system straightforward to solve through back-substitution.
The process typically involves:

• Using row operations to create zeros under the pivot elements
• Ensuring that leading coefficients are 1 for simplified calculations
• Transforming the matrix until only left with diagonal and upper triangle values

In our exercise, by adding equations, we were performing a step akin to Gaussian elimination, where simplifying comparisons between equations can help eliminate variables systematically. Following these steps reduces complexity, which is pivotal in solving larger systems of equations.

Linear Algebra

Linear Algebra is a branch of mathematics concerned with vectors, vector spaces, and linear equations, among others. It forms the fundamentals for understanding and solving systems of equations efficiently.
In the context of linear equations, linear algebra provides the framework for analyzing and solving equations using rows, matrices, and operations like those demonstrated in Gaussian elimination. The concepts of vector spaces help to visualize solutions in graphs, where each equation might represent a vector, and their intersection points solutions to the system. Linear algebra underlines the theory behind methods like row operations and Gaussian elimination. By representing equations as matrices, we can unlock computational techniques that are exceptionally powerful and efficient for both two-dimensional and multi-dimensional problems. This background is essential for any mathematical, engineering, or computational discipline that involves solving complex systems of linear equations.