Problem 1 - Algebra and Trigonometry

Question

\(1-2\) . 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 new second equation is \(x + 3z = 1\).

Step 1: Identify Given Equations

We have three equations: 1. \( x - y + z = 2 \)2. \( -x + 2y + z = -3 \)3. \( 3x + y - 2z = 2 \)

Step 2: Multiply the First Equation by 2

To apply the given operation, first multiply the entire first equation by 2:\( 2(x - y + z) = 2 \times 2 \), which simplifies to \( 2x - 2y + 2z = 4 \).

Step 3: Add the Results to the Second Equation

Now, add the newly obtained equation to the second equation:\( (-x + 2y + z) + (2x - 2y + 2z) = -3 + 4 \).Combine like terms: \( (-x + 2x) + (2y - 2y) + (z + 2z) = -3 + 4 \).This simplifies to \( x + 3z = 1 \).

Step 4: Record the New Second Equation

The new second equation formed by the addition is \( x + 3z = 1 \).

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It forms the backbone of many mathematical algorithms and applications. Understanding linear algebra helps simplify complex problems, which can then be solved using straightforward methods.
Linear algebra involves several core concepts, such as:

• Vectors: Objects with both magnitude and direction.
• Matrices: Rectangular arrays of numbers that represent linear transformations.
• Systems of Equations: Collections of equations that can be solved using matrix operations.

In the context of the given exercise, a system of linear equations is presented. Each equation is a linear combination of variables whose solutions define a vector space. Solving these systems requires a methodical approach using algebraic techniques and operations like matrix transformations and vector manipulations.

Equation Solving

Equation solving in the context of linear systems involves finding values for the variables that satisfy all equations simultaneously. The solution can be found using various techniques such as substitution, elimination, and matrix methods.
In the provided problem, we use the elimination method. The main goal here is to manipulate the system to simplify it, making it easier to find solutions.
Steps involved in solving a system of equations typically include:

• Expressing equations in standard form.
• Selecting an appropriate method to simplify or eliminate variables.
• Performing operations such as addition or multiplication to simplify the system.
• Rewriting the system in a form easy to solve, such as obtaining an identity.

The solution involves combining equations strategically to eliminate one or more variables, leading to a simpler equation that can be solved directly.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving linear equations. It involves performing legal mathematical operations to transform equations into a more convenient form. This includes distributing coefficients, combining like terms, and adding or subtracting equations.
To solve the given set of equations, algebraic manipulation is used extensively:

• First, the first equation is multiplied by 2, which requires distribution of the multiplier across all terms.
• Next, we combine this result with the second equation by adding both equations together, aiming to eliminate or simplify terms.
• Simplification involves reducing terms by combining like terms, such as adding or cancelling equal coefficients.

Through algebraic manipulation, we arrive at a simplified equation: \( x + 3z = 1 \). This step is crucial as it reduces the complexity, making it easier to identify the solution for the variables using other equations or methods.