Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the system of linear equations represented by the augmented matrix. (Use the variables \(x, y, z,\) and \(w,\) if applicable.) $$\left[\begin{array}{rrrrrr} 6 & 2 & -1 & -5 & 5 & -25 \\ -1 & 0 & 7 & 3 & \vdots & 7 \\ 4 & -1 & -10 & 6 & \vdots & 23 \\ 0 & 8 & 1 & -11 & \vdots & -21 \end{array}\right]$$

Step-by-Step Solution

Answer

The system of linear equations derived from the augmented matrix is: \(6x + 2y - z - 5w = -25, -x + 7z + 3w = 7, 4x - y - 10z + 6w = 23, 8y + z - 11w = -21\).

Step 1: Conversion of First row to a Linear Equation

The first row of the augmented matrix is \(6, 2, -1, -5, 5, -25\). This translates to the equation \(6x + 2y - z - 5w = -25\).

Step 2: Conversion of Second row to a Linear Equation

The second row of the augmented matrix is \(-1, 0, 7, 3, 7\). This translates to the equation \(-1x + 0y + 7z + 3w = 7\).

Step 3: Conversion of Third row to a Linear Equation

The third row of the augmented matrix is \(4, -1, -10, 6, 23\). This translates to the equation \(4x - y - 10z + 6w = 23\).

Step 4: Conversion of Fourth row to a Linear Equation

The fourth row of the augmented matrix is \(0, 8, 1, -11, -21\). This translates to the equation \(0x + 8y + z - 11w = -21\).

Key Concepts

Linear Algebra

Linear algebra is a fundamental area of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is the branch of algebra that allows us to represent and solve a system of linear equations in a compact and efficient way. Understanding linear algebra is crucial for disciplines such as engineering, physics, computer science, and economics, where relationships between variables need to be studied and modeled.

The use of linear algebra simplifies complex problems and serves as a gateway to more advanced mathematical concepts. It helps in breaking down problems into manageable parts, where linear equations represent relationships with a straight-line graph. This framework is particularly powerful because many real-world scenarios can be approximated as linear, making linear algebra an indispensable tool in problem-solving and logical reasoning.

System of Equations

A system of equations is a set of two or more equations that share a common solution. In essence, a system reflects multiple constraints that must be satisfied simultaneously. Finding the solution of such a system involves identifying the values of the variables that make all equations true at the same time.

Solving a system of linear equations can be approached in various ways, such as graphing, substitution, elimination, or matrix methods like Row Reduced Echelon Form (RREF). The versatility of solving systems is important as it offers distinct advantages in computational efficiency and practical applications. It's this system of interconnected equations where each contributes to the final solution that makes systems of equations impactful in depicting complex situations through algebraic expressions.

Matrix Representation

Matrix representation is a powerful tool in linear algebra that provides a convenient way to organize and manipulate systems of linear equations. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. When dealing with linear equations, matrices can be particularly useful for several reasons:

• They allow for the concise representation of a system of linear equations, needing less space than writing out the equations individually.
• Operations on matrices can correspond to meaningful algebraic manipulation of the systems they represent.
• Matrices can be used for various methods of solving systems, like Gaussian elimination or finding inverses for certain types of matrices.

Understanding the concept of an augmented matrix is key as it incorporates both the coefficients of the variables and the constants from each equation into a single matrix framework, thus enabling a more streamlined approach to solution finding.

Algebraic Manipulation

Algebraic manipulation involves rearranging, simplifying, or rewriting algebraic expressions using algebraic rules and properties. It's an essential skill in linear algebra used to solve equations, simplify expressions, and understand the relationships between variables. Algebraic manipulation can be simple, like combining like terms, or more complex, involving factoring, expanding expressions, or working with polynomial division.

In the context of solving systems of linear equations, algebraic manipulation includes techniques such as elimination, substitution, and using inverse operations. These methods allow for the systematic simplification of equations to reduce them to a form where the solution is evident or can easily be computed. It's the refinement of these techniques that enable students to pivot from learning basic algebra to engaging with more sophisticated mathematical challenges present in linear algebra.