Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the row operation and write the equivalent system. What did the operation accomplish? Add Equation 1 to Equation 2. $$\left\\{\begin{aligned} x-2 y+3 z &=5 \\ -x+3 y-5 z &=4 \\ 2 x &-3 z=0 \end{aligned}\right.$$ Equation 1 Equation 2 Equation 3

Step-by-Step Solution

Answer

The equivalent system after adding Equation 1 to Equation 2 is \[\left\{\begin{aligned} x-2 y+3 z &=5 \ y - 2z &=9 \ 2 x -3 z&=0 \end{aligned}\right.\] This operation eliminated the x term from the second equation.

Step 1: Perform the Row Operation

Start by adding the terms of Equation 1 to the terms of Equation 2. This gives: \[(x - x) + (-2y + 3y) + (3z - 5z) = 5 + 4\]

Step 2: Simplify the Result

Simplify the resulting expression to get the new Equation 2 as: \[y - 2z = 9\]

Step 3: Write the Equivalent System

Now, replace the old Equation 2 with the new Equation 2 in the original system. The equivalent system of equations becomes: \[\left\{\begin{aligned} x-2 y+3 z &=5 \ y - 2z &=9 \ 2 x -3 z&=0 \end{aligned}\right.\]

Step 4: Analyze the Effect of the Operation

The row operation succeeded in eliminating the x term from the second equation, making the system easier to solve.

Key Concepts

Systems of Equations

In linear algebra, a system of equations is a collection of two or more equations with a common set of variables. These systems are a central topic because they can model a wide variety of real-world scenarios, such as predicting outcomes and optimizing resources.
Systems can be categorized mainly into three types based on the number of solutions they possess:

• Consistent: A system with at least one solution.
• Inconsistent: A system with no solutions.
• Dependent: A system with infinitely many solutions, meaning one equation can be derived from another.

When dealing with systems, our goal is to find all possible solutions that satisfy each equation simultaneously. Usually, the equations are linear, containing variables raised only to the first power, making them easier to solve linear algebraically.
Our current example involves three equations with three variables \(x\), \(y\), and \(z\), which we aim to simplify using various techniques.

Row Operations

Row operations are elementary transformations used in methods like Gaussian elimination to solve systems of equations. They are powerful tools that help convert a complex system into a more manageable form, typically to row-echelon form, where back substitution becomes straightforward. There are three main types of row operations:

• Row swapping: Interchanging two rows.
• Row scaling: Multiplying a row by a non-zero constant.
• Row addition: Adding or subtracting a multiple of one row to another.

In our exercise, we used the row addition operation. By adding Equation 1 to Equation 2, the \(x\) term in Equation 2 was eliminated. This step simplifies the system making it easier to solve. Simplification through row operations does not change the solution of the system, it just makes the calculations more approachable.

Equivalent Systems

Equivalent systems are systems of equations that have the same solutions. Even though they might look different, they essentially describe the same mathematical conditions to be satisfied by the variables.
Performing row operations on a system of equations yields an equivalent system. This means that while the appearance of the equations might change, the set of solutions remains constant. Thus, any simplification or transformation done through authorized operations merely eases our work without affecting the solution integrity.
In our task example, by modifying Equation 2 from its original form to the simplified one \(y - 2z = 9\), we create an equivalent system. The new system copies the original in terms of its solution set but is more direct to work with, thus aiding in faster problem-solving. This concept is crucial for validating any transformations applied to original equations.