Problem 1 - Precalculus Mathematics for Calculus

Question

These exercises refer to the following system. $$ \left\\{\begin{aligned} x-y+z=& 2 \\ -x+2 y+z=&-3 \\ 3 x+y-2 z=& 2 \end{aligned}\right. $$ If we add 2 times the first equation to the second equation, the second equation becomes _________ = ________

Step-by-Step Solution

Answer

The second equation becomes \( x + 3z = 1 \).

Step 1: Write the Original Equations

We begin by identifying the original set of equations from the problem statement. The system of equations is given by:1. \( x - y + z = 2 \)2. \( -x + 2y + z = -3 \)3. \( 3x + y - 2z = 2 \)

Step 2: Multiply the First Equation by 2

To add two times the first equation to the second equation, we first multiply the entire first equation by 2. This yields:\[ 2(x - y + z) = 2 imes 2 \]Simplifying both sides, we get:\[ 2x - 2y + 2z = 4 \]

Step 3: Add the Result to the Second Equation

Next, we will add the transformed first equation to the original second equation. The second equation is:\[ -x + 2y + z = -3 \]Now, add the two equations:- First rearrange: \[-x + 2y + z + (2x - 2y + 2z)\]- Combine like terms: - \(-x + 2x = x \) - \(2y - 2y = 0 \) - \(z + 2z = 3z \)This gives us:\[ x + 3z \]The right side:\[ -3 + 4 = 1 \]So the new equation becomes:\[ x + 3z = 1 \]

Key Concepts

Linear Equation Manipulation

A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. Understanding how to manipulate linear equations is essential in solving a system of equations. Such manipulation includes adding, subtracting, or multiplying equations by numbers to simplify or combine them. This exercise demonstrates linear equation manipulation by multiplying and adding equations.

• **Multiplication of Equations:** Here, the first equation was multiplied by 2, transforming it into a new form that's easier to work with: \[ 2x - 2y + 2z = 4 \] This step was critical to align coefficients for the purpose of further simplification or elimination.
• **Addition of Equations:** Adding the modified first equation to the second allows for the elimination of certain terms, making the system simpler. The process is: \[-x + 2y + z + (2x - 2y + 2z) = -3 + 4 \] This results in a new equation: \[ x + 3z = 1 \]

Linear equation manipulation synergies involve familiar algebraic properties, helping unravel complex systems into simpler, more interpretable forms.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that share a common solution. The goal is to find the values for each variable that satisfy all equations at the same time. In the given system, we work with the equations:

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

By manipulating these equations, we aim to find values of \( x \), \( y \), and \( z \) that work for all at once. The more equations and variables involved, the more intricate the process. The method used often involves simplifying equations to solve for variables one at a time. Here, by transforming the second equation, we simplify it to solve for \( x \) and \( z \). Then, we can systematically approach solving for \( y \) using the remaining equations. Understanding and solving simultaneous equations enables tackling problems involving multiple unknowns systematically.

Elementary Row Operations

Elementary row operations are fundamental in solving systems of linear equations, especially when using matrices. They include row addition, row multiplication, and row swapping. Each operation is valid in altering an equation's form without changing its solution set. In our problem, we use what can be seen as row addition. We multiply the first equation by 2 (a form of row scaling) and add it to the second. This combination is akin to a row operation in matrix algebra where you're adding the adjusted row to another to zero out unwanted elements. These operations can:

• Simplify or eliminate coefficients to identify variable values directly.
• Transform systems into simpler forms, known as row-echelon or reduced row-echelon format.
• Create equivalent systems that are easier to solve or interpret.

Employing these operations strategically allows efficient solving of complex systems, converting them into more straightforward ones, one step at a time. Understanding and practicing elementary row operations is invaluable for those working with systems of equations, both simple and advanced.