Problem 6 - Algebra and Trigonometry

Question

\(3-6\) . State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y &=2 \\ y+2 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is linear because all equations are linear.

Step 1: Identify Each Equation Type

Start by identifying the format of each equation in the system presented. A linear equation is one in which each variable is raised to the power of one, and there are no products of variables or nonlinear terms.

Step 2: Examine the First Equation

The first equation is \( x - 2y + 3z = 10 \). This equation is linear because each term involves a variable raised to the first power, and there are no products or nonlinear terms.

Step 3: Examine the Second Equation

The second equation is \( 2x + 5y = 2 \). This equation is also linear for the same reasons as the first equation: all terms have variables raised to the first power.

Step 4: Examine the Third Equation

The third equation is \( y + 2z = 4 \). Like the others, this equation is linear because it only involves variables raised to the first power.

Step 5: Conclusion About the System

Since each of the three equations in the system is linear, the entire system of equations is linear.

Key Concepts

Understanding a System of Equations

In mathematics, a system of equations is a set of two or more equations that share common variables. These equations are solved together to find a set of values for the variables that satisfy all the given equations simultaneously.

In the exercise provided, you are given a system of three equations with three variables:

• The first equation is \( x - 2y + 3z = 10 \)
• The second equation is \( 2x + 5y = 2 \)
• The third equation is \( y + 2z = 4 \)

Each of these equations needs to be satisfied at the same time, which means finding values of \( x \), \( y \), and \( z \) that make all three equations true simultaneously. This is what makes solving systems of equations a powerful tool in mathematical problem-solving, helping to find solutions where multiple conditions must be considered at once.

The system of equations given in the exercise is linear because every equation is linear, which means they are easier to solve using standard methods like substitution or elimination.

Exploring Linear Algebra in Systems of Equations

Linear algebra is a significant area of mathematics that focuses on vector spaces and linear mappings between these spaces. It plays a crucial role in understanding systems of equations, particularly those that are linear.

The system given in the exercise includes linear equations, each representing a plane in three-dimensional space. Here's where linear algebra comes in handy:

• It offers methods such as matrices and determinants to deal with systems of linear equations efficiently.
• Utilizing matrices, one can transform a system of linear equations into a matrix form, making it easier to apply various algorithms to find solutions.
• It introduces concepts such as eigenvalues and eigenvectors, which can provide deeper insights into the properties of the system.

These methods allow us to analyze and solve complex systems by simplifying them into a form that can be more easily manipulated using algebraic operations.

The Role of Mathematical Problem Solving

Mathematical problem solving is the process of applying mathematics to find solutions to problems or answer questions. It involves not just computation but also a deep understanding of the underlying principles and the ability to apply them.

In the context of the system of equations in the exercise:

• You begin with understanding the structure of the equations and identifying them as linear.
• Next, you apply techniques such as substitution or elimination to simplify and solve the equations.
• Mathematical reasoning and critical thinking are crucial as you identify relationships between variables and their constraints.

Solving such systems sharpens your problem-solving skills, enhancing your ability to tackle more complex mathematical challenges. By practicing this process, you strengthen your logical reasoning and understanding, vital skills both academically and in real-world applications.